# Interface control of electronic transport across the magnetic phase
transition in SrRuO3/SrTiO3 heterointerface

S. Roy1, C. Autieri2, B. Sanyal2, T. Banerjee1<EMAIL_ADDRESS>1 Physics of
Nanodevices, Zernike Institute for Advanced Materials, University of
Groningen, Groningen, The Netherlands
2 Department of Physics and Astronomy, Uppsala University, Box-516, 75120
Uppsala, Sweden

###### Abstract

The emerging material class of complex-oxides, where manipulation of physical
properties lead to new functionalities at their heterointerfaces, is expected
to open new frontiers in Spintronics. For example, SrRuO3 is a promising
material where external stimuli like strain, temperature and structural
distortions control the stability of electronic and magnetic states, across
its magnetic phase transition, useful for Spintronics. Despite this, not much
has been studied to understand such correlations in SrRuO3. Here we explore
the influence of electron-lattice correlation to electron-transport, at
interfaces between SrRuO3 and Nb:SrTiO3 across its ferromagnetic transition,
using a nanoscale transport probe and first-principles calculations. We find
that the geometrical reconstructions at the interface and hence modifications
in electronic structures dominate the transmission across its ferromagnetic
transition, eventually flipping the charge-transport length-scale in SrRuO3.
This approach can be easily extended to other devices where competing ground
states can lead to different functional properties across their
heterointerfaces.

## Introduction

Complex oxides have propelled a vast research field, in establishing itself as
next frontiers in electronic materials by offering unique prospects to control
and manipulate new functionalities across their heterointerfaces Dagotto ;
Huijben ; Zhuravlev ; Gruverman ; Garcia . These materials, typically of the
form ABO3, exhibit strong correlations between the charge, spin, orbital and
lattice degrees of freedom, leading to modulation of their device properties
and are emerging as strong contenders for Beyond Moore technology. Interfaces
coupling complex oxides exhibit novel functionalities, not present in their
parent compounds Hwang . These primarily stem from broken symmetries at the
interface, which when carefully tailored, can lead to unique device
functionalities. In this regard, SrRuO3 is a canonical metallic system that
has been studied extensively for its growth and physical properties and is
widely used as an electrode in complex oxide devices Sanchez ; Herranz ; Bern
; Aso ; Koster . Interestingly, SrRuO3 undergoes a ferromagnetic transition at
140 K (TC) and presents a distorted orthorhombic structure Pbnm below 820 K.
The microscopic origin of ferromagnetism is supported by the distortions of
the oxygen octahedra that are coupled to the degree of hybridization between O
$\textit{2}p$ and the Ru $\textit{4}d$ orbitals Vailionis1 .This strongly
influences its magnetic and electronic properties on either side of the TC and
provides an interesting opportunity to study the correlation between magnetism
and transport in SrRuO3 across its TC when integrated in an electronic device.

In this work, we employ a nanoscale transport probe to study such complex
correlation across an interface between SrRuO3 on semiconducting Nb:SrTiO3
(Nb:STO). Principally, this technique allows us to study any material system,
whose electronic properties are susceptible to a temperature driven phase
transition, while simultaneously allowing us to investigate its homogeneity
across such buried interfaces with a semiconductor, at the nanoscale. This is
an important approach for material systems such as complex oxides where small
changes in the geometrical properties with thickness or temperature can
produce large changes in the electric and magnetic properties. We perform a
systematic experimental study to understand the effect of thickness and
temperature on electronic transport across thin films of SrRuO3 (t = 6, 8, 9
and 10 unit cells) with Nb:STO and observe an order of magnitude increase in
the transmission in the same device, across its ferromagnetic phase
transition. Surprisingly though, we find that this increase originates from an
enhanced transmission at the interface between SrRuO3 and Nb:STO, rather than
in the bulk of SrRuO3, and manifests as two different transport lengths at
room temperature and below TC. We combine these findings with density
functional theory (DFT) calculations to quantitatively understand the role of
interface reconstruction, octahedral distortions and exchange splitting in the
observed features in electronic transport.

## Results

### Film growth and device scheme

We have grown high quality films of SrRuO3 of varying thicknesses, 6, 8, 9 and
10 unit cells (u.c.) on 0.01 wt.% Nb:STO substrates using Pulsed Laser
Deposition (PLD). The substrates are chemically treated to ensure a uniform
TiO2 termination and characterized for their surface quality with atomic force
microscopy (AFM) Koster1 . This process assumes critical importance because a
clean, defect free starting surface is highly desirable for the growth of
crystalline epitaxial films of SrRuO3 and determines a uniform electronic
transport across such a metal-semiconductor (M-S) interface. However, thermal
annealing often causes local SrO terminations commonly occurring at the
terrace edges of the annealed substrates. With unequal growth rate of SrRuO3
on SrO and TiO2 terminations, clear trenches along local SrO terminating sites
are observed where SrRuO3 grown is thinner than on TiO2 terminations Koster .
Figure 1a,b shows the AFM image of the deposited films of different
thicknesses (6 and 10 u.c.). The root-mean-square surface roughness of the 6
u.c. and 10 u.c. films, outside the trenches, are found to be 0.21 nm and 0.15
nm respectively. The grown SrRuO3 films were characterized for their
electronic transport and magnetic properties using standard van der Pauw
method and superconducting quantum interference device (SQUID) magnetometry,
respectively. The electrical resistivity studies on the films show a typical
metallic behavior for all thicknesses. Resistivities of 6 and 10 u.c. SrRuO3
films, measured from 300 K down to 15 K, are shown in Figure 1c. Their
electrical properties are strongly affected by the film thickness, evidenced
by a higher resistivity observed for the 6 u.c. film that decreases for the 10
u.c. film. A study of the temperature variation of resistivity shows a kink
indicating the paramagnetic to ferromagnetic transition, with the
corresponding TC (marked by black arrows) derived from the temperature
dependence of the derivative of resistivity plots (not shown). The Curie
temperature (TC) determined for the 6 u.c. film is 135 K, and for 10 u.c. is
142 K. From SQUID magnetometry studies, the magnetic properties of the films
were studied in an applied field of 0.1 T on field cooling for a temperature
range between 5 K to 300 K (Figure 1d). The TC for the 6 u.c. and 10 u.c.
films determined from these plots (derivative of magnetic moment with
temperature, not shown) are found to be 135 K and 142 K respectively, and
demonstrates the coupled magnetic and electric properties in SrRuO3. The high
value of the saturation magnetic moment obtained from experiments arises due
to the strained films on (001) SrTiO3, which decreases the orbital overlap via
increased bond angle in Ru-O bonds, in such ultrathin films of SrRuO3. This
has been reported earlier in SrRuO3 films Grutter ; Grutter1 . Thus, through a
combination of surface characterization technique and physical property
measurements, we establish the high quality of the films used in our work.

In order to probe electronic transport of the deposited SrRuO3 films, they are
patterned with standard UV lithography and further ion beam etched to be
defined as devices as shown in Figure 1e. Our experimental nanoscale transport
probe is known as ballistic electron emission microscopy (BEEM) (Figure 1c)
which uses the scanning tunneling microscope (STM) tip to inject a
distribution of hot electrons with energies higher than the Fermi energy (EF)
Kaiser ; Bell . In this study, we use a modified Ultra High Vacuum (UHV) STM
system from RHK Technology and the measurements are performed at 300 K and 120
K using PtIr metal tips. An electrical contact between the STM tip and SrRuO3
measures the tunnel current. The device design necessitates an additional
contact at the rear-end of Nb:STO, for the collection of the transmitted
electrons. These injected hot electrons, driven by the applied bias VT with
respect to the Fermi level of the metal, travel across the metallic base to
reach the M-S interface; if their energy is sufficient to overcome the
Schottky barrier, they can be collected into the semiconductor conduction
bands (Figure 1f). To obtain the BEEM spectra, the STM tip is held at a fixed
location while VT is varied within a voltage range and the collector current
is recorded in the constant current mode. As these hot electrons propagate
through the SrRuO3 films, they undergo scattering, which influences the
collected current. The collected electrons in Nb:STO are primarily those whose
momentum parallel to the interface is conserved, the rest are back-scattered
into the SrRuO3 base layer. These electrons serve as an important parameter to
study strong correlations in such metallic oxide thin films and provide
information on the homogeneity of the interface electronic structure with high
spatial resolution Prietsch ; Kanel ; Parui1 ; Haq . The determination of the
relative strengths of such scattering processes (elastic/inelastic) enables a
comprehensive understanding of electron transport in complex oxides and allows
for designing of new oxide based electronic devices.

### Thickness and temperature dependent BEEM transmission

The hot electron distribution encounters elastic and inelastic scattering
while transmitting through the SrRuO3 metallic base. In this process, the
momentum distribution at the M-S interface is broadened, as compared to the
injection interface, adding to the reduction of the collected BEEM current
(IB) Subir .These effects get more pronounced as the thickness
(d${}_{SrRuO_{3}}$) of the base layer is increased. The normalized BEEM
transmission follows an exponential behavior as described by Kaiser :

$\frac{I_{B}}{I_{T}}=C\times
exp\left[-\frac{d_{SrRuO_{3}}}{\lambda_{SrRuO_{3}}(E)}\right]$ (1)

where $\lambda_{SrRuO_{3}}(E)$ is the energy-dependent hot electron
attenuation length, IB is the collected BEEM current, IT is the injected
tunnel current and C is a constant that is the kinematic transmission factor
representative of electron transport across the M-S interface. Figure 2 shows
the BEEM transmission data for various thickness of SrRuO3 thin films, in
their paramagnetic (300 K) and ferromagnetic (120 K) phases. First we discuss
the observation for the paramagnetic phase (Figure 2a). The transmission for
all film thicknesses is low up to a certain bias voltage beyond which, the
transmission increases rapidly. This onset in bias voltage corresponds to the
local Schottky barrier height (SBH) at the M-S interface. In accordance to the
Bell-Kaiser (B-K) model, a plot of the square root of BEEM current versus the
bias voltage yields the value of SBH, which in this case is 1.13$\pm$0.03 eV
(Inset, Figure 2a) Rana . It matches well with our previously extracted values
for such an interface Roy . As the thickness of SrRuO3 films is increased, a
decrease in transmission is observed. However, the local SBH extracted from
measurements on films of different thicknesses are found to be the same, thus
providing yet another proof of a high quality growth of atomically sharp M-S
interfaces. The representative BEEM spectra shown for each thickness are
averages of several spectra taken at different devices and locations.

Electronic transport across the ferromagnetic phase transition in SrRuO3 thin
films was studied across its interface with Nb:STO, in the same set of
devices, by performing similar BEEM measurements at 120 K. The characteristic
BEEM spectra are shown in Figure 2b. Similar to our earlier observations, the
BEEM transmission progressively decreases with increasing thickness of the
films. We find that the hot electron transmission increases by an order of
magnitude as compared to its corresponding values at 300 K. The extracted SBH
at 120 K, at these interfaces, is found to be 1.14$\pm$0.03 eV (Inset, Figure
2b), which is the same as that obtained in its paramagnetic phase. One can
thus safely discard the origin of the enhanced BEEM transmission in SrRuO3 at
120 K, due to differences in the SBH across its magnetic phase transition.

### Influence of interface and film thickness on electronic transport

An exponential fit of the BEEM transmission for varying thicknesses of SrRuO3
films allows us to extract the (energy dependent) transport length scale known
as attenuation length (Equation 1). Above EF, the transmission of hot
electrons in SrRuO3 films is mostly governed by inelastic scattering,
originating from the availability of increased phase space for hot electrons
to decay into Banerjee ; Prietsch . The extracted attenuation length of the
carriers reflects the combined effects of inelastic, elastic as well as quasi-
elastic scatterings. We plot the transmissions for the paramagnetic and
ferromagnetic phase of SrRuO3 at a certain bias voltage of -2 V, shown in
Figure 3a. An important observation is that the BEEM transmission starts to
deviate appreciably at 120 K from that at 300 K, with decreasing thickness.
This has not been observed earlier in other metallic systems using BEEM.
Extrapolation of the data for zero SrRuO3 thickness shows two orders higher
transmission, at the interface for ferromagnetic SrRuO3. An exponential fit
using Equation 1, yields an attenuation length of 1.6$\pm$0.2 u.c. in the
paramagnetic phase and 0.88$\pm$0.4 u.c. in the ferromagnetic phase at -2 V.
In Figure 3b, we plot the energy dependence of the attenuation length by
fitting the data obtained in Figure 2 with Equation 1 for energies above the
SBH. We find that the attenuation length decreases with increasing energy,
consistent with the enhancement of the density of states (DOS) at higher
energies Guedes . What is surprising is that in-spite of the enhanced
transmission in the ferromagnetic phase, the attenuation length in SrRuO3 is
shorter, for all energies at 120 K, than at 300 K. This apparent conundrum is
explained by our ab initio calculations, taking into account the
characteristics of geometries and electronic structures at the interface
between SrRuO3 and SrTiO3 along with the quasiparticle renormalization of
attenuation length.

###  Ab initio study of SrRuO3/SrTiO3(001) interfaces

We have performed ab initio density functional calculations with a specific
focus on correlating the structural and electronic properties at the interface
to the BEEM transmission and to understand the origin of the flipping of the
attenuation lengths at 120 K and 300 K. In order to simulate SrRuO3 thin films
grown on SrTiO3 (001) substrates, we fix the lattice parameter $a$ to the
experimental value of the SrTiO3 substrate. 3, 6 and 9 unit cells of SrRuO3
have been considered in the low temperature ferromagnetic and room temperature
(RT) paramagnetic phases. As a paramagnetic phase with disordered local
moments is difficult to simulate within our computational technique, we study
the nonmagnetic phase as the RT phase and ferromagnetic phase calculated at 0
K will represent the low temperature (LT) phase. We show in Figure 4a the
supercell with 6 layers of SrRuO3 on 3 layers of SrTiO3. In the following
discussions, we will follow the notation shown in Figure 4a, where we denote
STOL (or SROL) with L=1,2,…, the L-th layer from the SrTiO3/SrRuO3 interface.

To determine the energy dependent attenuation length $\lambda^{LDA}(E)$, we
calculate the mean value over the bands and k-points, at a given energy $E$,
of the product of the modulus of the group velocity along the $z$ axis
$v^{z}_{n\textbf{k}}$ and the lifetime $\tau_{n\textbf{k}}$, which is
expressed as:

$\lambda^{LDA}(E)=\langle
v^{z}_{n\textbf{k}}\tau_{n\textbf{k}}\rangle\propto\frac{\langle\frac{1}{\hbar}\frac{\partial\varepsilon_{n}(\textbf{k})}{\partial{k_{z}}}\rangle}{\int_{0}^{E}N(\varepsilon)d\varepsilon}$
(2)

where the lifetime is proportional to the inverse of the cumulative density of
states from the Fermi level to the energy $E$Zhukov . We have only considered
the z-component, as the BEEM transmissions across the M-S interface are
measured along this direction. As seen in Figure 4b, the attenuation length
$\lambda^{LDA}(E)$ is almost constant in the range between 1.6 and 2.2 eV
(above EF) and is greater at 120 K than at RT. In recent works Renormalization
; Renormalization2 , the importance of dynamical correlations in SrRuO3 have
been discussed with the aid of dynamical mean field theory. Whereas the
dynamical correlations and hence, the temperature dependent quasiparticle
weight Z are beyond the scope of LDA, we note that this weight is 1 in LDA and
is a multiplicative factor of the spectrum and the velocity. Assuming the same
quasiparticle weight for the $e_{g}$ manifold and neglecting its influence on
$\tau$, we get a reduction of the attenuation length, which is more pronounced
at lower temperatures. In Figure 4b we plot the attenuation length multiplying
$\lambda^{LDA}$ by $Z$ and observe that under these assumptions, the
attenuation length at RT is greater than at 120 K, in agreement with the
experimental findings.

### Variation of the electronic and structural properties of the interface at
LT and RT

To understand the experimentally observed enhancement in the interface
transmissions at 120 K and 300 K, we calculate the electronic and geometric
structures of SrRuO3/SrTiO3 interfaces. Considering the variation in the
geometric and magnetic properties, we can divide the SrRuO3 thin films in
three regions along the c-axis: surface, inner layer and interface region. Our
calculated densities of states of SrRuO3 along with that of SrTiO3 are shown
in Figure 5a for 9 u.c. of SrRuO3 in the LT ferromagnetic phase and the RT
nonmagnetic phases. It is observed that between 1 eV and 2.2 eV (above EF),
the DOS in SrRuO3 is dominated by $e_{g}$ electrons whereas in SrTiO3, the
$t_{2g}$ electrons dominate with a larger DOS between 1.2 and 1.4 eV (above
EF) in the nonmagnetic phase of SrRuO3. The exchange split $e_{g}$ state
produces a smaller DOS at LT between 1.2 and 1.4 eV (above EF). While we
observe the differences between LT and RT phases in all SRO layers, SRO1 layer
at the interface needs an extra attention. A distinct feature of
$d_{x^{2}-y^{2}}$ character (marked by the shaded area) is observed for RT
phase in the SRO1 layer along with the renormalization of $e_{g}$ DOS between
1.2 and 1.3 eV (above EF). This is not an energy shift but is due to an
enlargement of the bandwidth. Our calculated hopping parameters, for example,
for the 6 u.c. at RT are 580 meV between the $x^{2}$-$y^{2}$ orbitals
(corresponding Wannier function shown in Figure 4a) in the SRO1 layer and
540-543 meV for other layers. Also, the hopping mismatch between interface
(575 meV) and inner layers (554-561 meV) is reduced at LT. Thus, we can
clearly see that the mismatch between the interface and inner layers is larger
at RT. We further note from our DOS calculations that the mismatch of the
electronic states at the interface between SrRuO3 and SrTiO3 is larger at RT
than at LT. Such elastic scattering effects will lead to a reduction in the
BEEM transmission at RT. All these differences in the electronic structures of
the two phases strongly suggest the importance of the prefactor C in Equation
1, which influences the BEEM transmission at LT and RT phases.

The difference in the characteristics at the interface for LT and RT phases is
quite evident also in the structural properties shown in Figure 5b. The
interface reconstruction makes this aspect more evident in the in-plane bond
angles compared to the out-of-plane bond angles. For a slab of reasonable
thickness (9 u.c. in our case), one can quantitatively identify the surface,
inner layers and interface regions by the evolution of the Ru-O-Ru bond angle
in the ab plane along the c-axis as shown in Figure 5b. For the 9 u.c. case, a
large region of inner layers (properties similar to bulk) exists while this
region is not distinctly observed in the 3 u.c. case. Due to the interface
reconstruction, the interface layer SRO1 is characterized by a Ru-O-Ru in-
plane bond angle having a value between the SrRuO3 and SrTiO3 bulk values. The
properties of the inner layers are closer to the bulk. At the surface, we
always obtain a bond angle larger than the bulk. Apart from the general
differences between bond angles for different regions of the slab, an
important observation can be made regarding the bond angles for the LT and RT
phases.

It is clearly seen that the difference in bond angle between SRO1 and other
SRO layers is larger for the RT phase compared to the LT one. For example, in
the 6 u.c. case, the difference in the bond angle between SRO1 and SRO5 is
1.5∘ for the LT phase, while it is 3.0∘ for the RT phase (Figure 5b). This
influences the hopping of electrons (as evident from the calculated hopping
parameters mentioned above) and hence the transport properties at the
interfaces Autieri14 . Therefore, we can conclude that the differences in the
geometries and electronic structures between SRO1 and other SRO layers are
less dramatic in the LT phase compared to the RT one, which may effectively
lead to an enhanced transmission of hot electrons across the interface at LT.
The larger mismatch at RT between interface and inner layers will yield a
smaller transmission factor (C factor in Equation 1) in the RT phase. Such a
substantial influence of geometric and structural changes at the interface
leading to strong temperature dependent transport characteristics across the
magnetic phase transition in SrRuO3 is a remarkable finding, not observed
before in any oxide heterostructure.

## Discussion

In conclusion, we have studied the BEEM transmission through thin films of
SrRuO3 of various thicknesses, interfaced with SrTiO3 across its magnetic
transition. The increase in BEEM transmission at 120 K compared to RT is
accompanied by a surprising flipping of the attenuation length. Our first
principles calculations indicate strong geometrical reconstructions at the
interfaces characterized by Ru-O-Ru bond angles along with distinct features
in electronic structures of the Ru-d orbitals, both of which are strongly
dependent on the magnetic phases. As a consequence, the importance of the
transmission factor in the expression for BEEM current is realized to explain
the experimentally observed BEEM transmission data. Moreover, the inclusion of
the temperature dependent quasiparticle weight for the calculation of the
attenuation length correctly describes the experimental observation at 120 K
and RT.

The evolution of such unique features in electronic transport in complex oxide
heterostructures, across a magnetic phase transition, opens new possibilities
to manipulate the electronic and spin degrees of freedom by a selective choice
and tailoring of their interfaces. This approach of tuning heterointerfaces by
coupling structural, electronic and magnetic properties can be extended to
other material systems with promising prospects for future oxide electronic
and spintronic devices.

## Methods

Experimental details We grow SrRuO3 films of different thickness by pulsed
laser deposition. All the devices were grown on [001] Nb doped SrTiO3
substrates with 0.01 wt.$\%$ of Nb. The Nb:STO substrates used for deposition
were treated chemically to obtain uniform TiO2 terminations, and were annealed
at 960∘ C which rendered a clean surface with uniform straight terraces. The
cleanliness and uniformity of the interface becomes critical in electronic
transport across the heterointerface. The films were grown using PLD with a
KrF excimer laser with a repetition rate of 1 Hz at an oxygen pressure of 0.13
mbar at 600∘ C. Their corresponding thicknesses were determined in-situ by
reflection high energy electron diffraction. Electrical resistivities of the
films were obtained by standard van der Pauw method within a temperature range
of 15 K to 300 K. Magnetic measurements were performed using a Quantum Design
MPMS XL-7 SQUID. In order to fabricate devices, patterns of 100x150 $\mu$ m2
areas were defined using standard ultraviolet (UV) lithography technique where
100 nm of Au was deposited for making the top contact, providing the pathway
for the STM current. Thereafter, device areas of 200x300 $\mu$ m2 were defined
with the Au pads on top of it, and the samples were Ion Beam Etched (IBE) with
a pre-calibrated etch-stop. This removed the SrRuO3 from the unpatterned
regions isolating the adjacent devices on the sample. An Ohmic back contact
was created by depositing Ti (70nm)/Au (70nm) on the rear side of the Nb:STO
substrate for collecting the BEEM current. The photoresist was lifted-off
carefully and the topography of SrRuO3 was checked using AFM.

Computational details We perform spin polarized first-principles density
functional calculations within the LSDA (Local Spin Density Approximation)
Kohn64  by using the plane wave VASP VASP  DFT package and the Perdew-Zunger
Perdew  parametrization of the Ceperly-Alder data Ceperley  for the exchange-
correlation functional. The choice of LSDA exchange functional is suggested in
a recent paper Etz12  where the Generalized Gradient Approximation was shown
to perform worse than LSDA for SrRuO3. The interaction between the core and
the valence electrons was treated with the projector augmented wave (PAW)
method BlochlPAW  and a cutoff of 430 eV was used for the plane wave basis.The
computational unit cells are constructed as supercells with three SrTiO3
octahedra having a 12 Å of vacuum. Convergence was checked with respect to the
thickness of the vacuum layer. Depending on the thickness of SrRuO3 films, 3,
6 and 9 octahedra of SrRuO3 are placed on top of SrTiO3. For Brillouin zone
integrations, a 8$\times$8$\times$1 k-point grid is used for geometry
relaxation and a 10$\times$10$\times$2 k-point grid for the determination of
the density of states (DOS). In all cases, the tetrahedron method with Blöchl
corrections BlochlCORR  is used for the Brillouin zone integrations. We
optimize the internal degrees of freedom by minimizing the total energy to be
less than 10-5 eV and the Hellmann-Feynman forces to be less than 7 meV/Å. The
Hubbard $U$ effects at the Ru and Ti sites were included in the LSDA+$U$
Anisimov91 ; Anisimov93  approach using the rotational invariant scheme
proposed by Liechtenstein Anisimov95 . We have used $U$=5 eV and $J_{H}$=0.64
eV for the Ti 3$d$ states Pavarini04  while $U$=0.30 eV and $J_{H}$=0.05 eV
are considered for the Ru 4$d$ states. In the literature, a value of the
Coulomb repulsion of 0.50 eV for ruthenates Etz12  was suggested to reproduce
the magnetization of the bulk, but we use a slightly smaller value because a
smaller magnetization is found for the thin filmsTian07 . To extract the
character of the electronic bands at low energies, we used the Slater-Koster
interpolation scheme based on Wannier functions.Marzari97 ; Souza01  Such an
approach is applied to determine the real space Hamiltonian matrix elements in
the eg-like Wannier function basis using a 8$\times$8$\times$2 k-point grid.
After obtaining the Bloch bands in density functional theory, the Wannier
functions are constructed using the WANNIER90 code.Mostofi08  Starting from an
initial projection of atomic $d$ basis functions belonging to the eg manifold
and centered on Ru sites, we get the two eg-like Wannier functions. The group
velocity is obtained using a 49$\times$49$\times$49 k-point grid. We got
qualitatively similar results for the Fig. 4b using the quasiparticle weight Z
calculated in different referencesRenormalization ; Renormalization2 .

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## Acknowledgements

We thank B. Noheda and T. T. M. Palstra for use of the Pulsed Laser Deposition
system. Technical support from J. Baas and J. G. Holstein is thankfully
acknowledged. This work is supported by the Netherlands Organization for
Scientific Research NWO-FOM (nano) and the Rosalind Franklin Fellowship
program. C. A. and B. S. acknowledge financial support from Carl Tryggers
Stiftelse (grant no. CTS 12:419 and 13:413) and supercomputing allocation by
Swedish National Infrastructure for Computing.

## Author contributions statement

S. R. carried out the device fabrication, all experimental measurements and
contributed to the analysis of the data. T. B. assisted in experimental
planning, contributed to the data analysis and supervised the overall work. C.
A. did the theoretical calculations and took part in the analysis of
theoretical data. B. S. supervised the theoretical part of the project and
took part in the analysis of theoretical data. All coauthors extensively
discussed the results and wrote the paper.

## Additional information

Competing financial interests: The authors declare no competing financial
interests.

Figure 1:  Atomic force microscopy (AFM) images of grown films and device
design. (a) 6 u.c. of SrRuO3 grown on Nb:STO. Clearly, the presence of local
SrO terminations of the substrate is observed as dark patches where SrRuO3
growth rate is lower than on TiO2 terminations of the substrate. (b)
Topography image after 10 u.c. of SrRuO3 growth. As the film grows thicker, it
starts covering the local trenches with lower SrRuO3 growth ultimately
replicating the substrate morphology. We observe a dependence of thickness on
the uniform coverage of SrRuO3 on the substrate. (c) Temperature dependence of
resistivity of 6 and 10 u.c. SrRuO3 films. The TC obtained from the kink in
the curves indicates that it decreases with decreasing film thickness. Black
arrows indicate the TC of the films, which is 135 K and 142 K respectively for
the 6 u.c and 10 u.c of SrRuO3. Also, 10 u.c. SrRuO3 is more metallic than 6
u.c. SrRuO3. (d) Magnetic moment dependence on temperature is recorded on
field cooling in a magnetic field of 0.1 T. Extracted TC for 6 u.c. and 10
u.c. SrRuO3 films are 135 K and 142 K, respectively. These values are
consistent with the findings from temperature dependent resistivity
measurements. (e) Schematic design of SrRuO3/Nb:STO device and measurement
scheme. Electrons are injected by the STM tip and BEEM current is collected
from Nb:STO. (f) Energy band profile of the measurement scheme. The Schottky
barrier at the SrRuO3/Nb:STO interface acts as an energy filter for the
transmitted hot electrons. Figure 2:  Thickness dependent BEEM transmission in
SrRuO3/Nb:STO devices at 300 K and in the ferromagnetic phase at 120 K. (a) A
plot of BEEM current vs. bias voltage is shown for different thicknesses of
SrRuO3 at 300 K. Clearly, the transmission decreases progressively with
increasing film thickness. The curve for 6 u.c. SrRuO3 has been multiplied by
0.5 factor. The inset shows representative local Schottky barrier height (SBH)
extracted by the Bell-Kaiser model, for the interface of metallic SrRuO3 and
semiconducting Nb:SrTiO3. For all the thicknesses, it is found to be 1.13$\pm$
0.02 eV. The error bars reflect the standard deviation of the measured onset
of the BEEM current at a location. (b) The plot of BEEM current vs. bias
voltage shows an thickness dependence, similar to what we observe at 300 K. An
increased BEEM transmission for all thicknesses is observed compared to 300 K.
The curves for 6 u.c. and 8 u.c. of SrRuO3 have been multiplied by 0.5 and the
9 u.c. SrRuO3 curve by a factor 2. The extracted local SBHs for the devices
are 1.14$\pm$ 0.03 eV, similar to what was observed at 300 K. Figure 3:  Plot
of attenuation length with applied bias and energy dependence. (a) The BEEM
transmission at -2 V is plotted against film thicknesses for both at 300 K
(RT) and at 120 K. An exponential fit (solid lines) show the dependence of
BEEM current with film thickness. An extrapolation of the transmission
indicates that the interface transmission has increased by two orders of
magnitude at 120 K. (b) Energy dependence of attenuation length at 120 K and
300 K is plotted. Contrary to the higher BEEM transmission across
SrRuO3/Nb:STO devices at 120 K, the corresponding attenuation length is
consistently shorter than the attenuation length measured at 300 K. The error
bar in attenuation length comes as a fitting error from the deviation of the
best fit. Figure 4: Side view of the slab and calculated attenuation length.
(a) Side view of the slab with 3 layers of SrTiO3 and 6 layers of SrRuO3. Sr,
O, Ru and Ti atoms are shown as green, fuchsia, grey and blue balls
respectively. In the SRO1 layer, we show the Wannier function corresponding to
the $d_{x^{2}-y^{2}}$ orbital where red and blue contours are for isosurfaces
of identical absolute values but opposite signs. (b) Attenuation length
calculated within LDA without and with the quasiparticle weight Z. The top and
bottom panels show $\lambda^{LDA}$ and Z$\lambda^{LDA}$ respectively in the
energy range between 1.6 and 2.2 eV (above EF) for the case with 6 layers of
SrRuO3. This energy range is the relevant experimental energy range where the
attenuation length was measured. The LT (RT) phase is shown in blue (red).
Figure 5:  Layer-resolved electronic and structural properties. (a) Layer
projected DOS (spin-up + spin-down) closer to the interface from STO2 to SRO3
for the 9 u.c. case. The LT (RT) phase is shown in blue (red). The shaded DOS
below 1.2 eV in the RT SRO1 layer has $x^{2}$-$y^{2}$ character. The energy
range is from 1.0 eV to 2.2 eV (above EF). (b) Ru-O-Ru bond angle in the ab
plane as a function of the layer index for LT (solid lines) and RT (dashed
lines) phases. The red, green and blue lines represent respectively the 3
u.c., 6 u.c. and 9 u.c. cases. The black dashed line represents the calculated
LT bulk value in the same volume setup. The values for the SRO1 layer are
indicated by an arrow.


# Microlocal Projector for Complete Flow

Sheng-Fu Chiu<EMAIL_ADDRESS>Institute of Mathematics, Academia
Sinica, Taiwan

###### Abstract.

In this note we generalize the construction of microlocal projector to the
sublevel set of autonomous function with complete Hamiltonian flow under some
mild conditions. Furthermore, we mention that the condition of being complete
is optimal, by providing a non-example that violates the projector property.

###### Key words and phrases:

Microlocal projector, Hamiltonian dynamics, Derived Categories

###### 2020 Mathematics Subject Classification:

53E50, 57R17, 18G80

###### Contents

  1. 1 Introduction
  2. 2 Theory of microlocal sheaves
    1. 2.1 Sheaves and their microlocal supports
    2. 2.2 The Tamarkin categories $\mathcal{D}$
    3. 2.3 Sheaf quantization
  3. 3 Microlocal Projector
    1. 3.1 Projectors associated to sublevel sets
    2. 3.2 Proof of Theorem 3.2
  4. 4 A Non-example

##  1\. Introduction

In symplectic geometry and topology one often considers Hamiltonian
diffeomorphisms with compact supports on a symplectic manifold. There are
natural reasons regarding such assumption. First, the flow of such Hamiltonian
is always complete. Second, they are bounded deformations of the identity and
thus distinguish symplectic diffeomorphisms from conformal symplectic
diffeomorphisms. Third, the natural compact-open topology on the group of
Hamiltonian diffeomorphisms is metrizable and has a countable basis.
Nevertheless, relaxations of the condition of compact support are available
under certain circumstances.

Let $Q$ be a connected smooth manifold and denote the coordinates of its
cotangent bundle $T^{*}Q$ by $(q,p)$. In [2] the author constructed a
microlocal projector and used it to prove a nonsqueezing result of contact
balls. The existence of the microlocal ball projector relies on the geometric
fact that on $T^{*}\mathds{R}^{n}$ the Hamiltonian function
$H_{1}(q,p)=q^{2}+p^{2}$ satisfies the following condition:

For each $b\in\mathds{R}$, the set $H_{1}^{-1}((-\infty,b])$ is compact.

We also refer the reader to [4, Definition 1.4] where Kawasaki and Orita made
the same assumption to the moment map on $T^{*}Q$ and studied its rigid
fibers.

On the other hand, even the simplest linear motion in Hamiltonian dynamics has
its Hamiltonian $H_{2}(q,p)=\frac{1}{2}p^{2}$, which does not satisfy the
above condition. Instead it satisfies the following coercive condition:

For each compact set $K\subset Q$ and each $b\in\mathds{R}$,

the set $\\{(q,p)|q\in K,H_{2}(q,p)\leq b\\}$ is compact.

Likewisely in Lagrangian dynamics there are convex and superlinear conditions
for the sake of the well-definition of the Legendre transformation.

One could also consider other Hamiltonians such as the superposition of two
harmonic oscillators
$H_{3}(q,p)=({q_{1}}^{2}+{p_{1}}^{2})-2({q_{2}}^{2}+{p_{2}}^{2})$. The
function $H_{3}$ does not obey the above coercive condition, nor does it
possess any compact level surface. That being said, it is easy to see that
every single level surface of $H_{3}$ is foliated by compact invariant tori,
and on each such leaf the Hamiltonian flow lives and lasts forever.

A rather natural condition one could ask for is the following condition
($\star$):

######  Definition 1.1.

A time-independent Hamiltonian function $H:T^{*}Q\rightarrow\mathds{R}$
satisfies the condition ($\star$) if:

  1. (1)

(Long-term Completness) The solution $h_{s}\in Ham(T^{*}Q)$ of the Hamiltonian
equations $(\dot{q}(s),\dot{p}(s))=X_{H}(q(s),p(s))$ are defined for all
$s\in\mathds{R}$.

  2. (2)

(Short-term Separating) There is a connected open interval $J$, with $J=-J$,
such that for any given $0<s\in J$ and $q_{1},q_{2}\in Q$ there exists at most
one Hamiltonian trajectory $\gamma$ from $T^{*}_{q_{1}}Q$ to $T^{*}_{q_{2}}Q$
with time $s$.

One sees that the above condition ($\star$) applies to many circumstances in
symplectic geometry and dynamical systems. In this note we refine the argument
in [2] to construct microlocal projectors for those $H$ satisfying the
condition ($\star$). Our result is:

######  Theorem 1.2 (See Theorem 3.2 for its definition of mircolocal
projector).

Suppose $H:T^{*}Q\rightarrow\mathds{R}$ is a function satisfying the condition
($\star$). Let $B_{C}=H^{-1}((-\infty,C))$ be an open sublevel set of level
$C$. Then there exists a microlocal projector $\mathscr{P}_{C}$ for $B_{C}$.

A motivation for complete Hamiltonian flow comes from quantum mechanics. By
Stone’s theorem the quantum Hamiltonian observable $\hat{H}$ is always given
self-adjoint, and its evolution on the quantum system is always governed by
the one-parameter unitary group $e^{is\hat{H}}$, which is automatically
defined for all times $s\in\mathds{R}$. The microlocal projector in a derived
category, in a loose sense, is a classical geometric analogue of the quantum
projection operator that plays an essential role in quantum measurements.

We would like to refer the reader to [6] that partial quasi-morphisms and
quasi-states, obtained from spectral invariants, can be defined for
Hamiltonians with complete flow. The authors of [6] also suggested a possible
generalization of Mather’s alpha function for such Hamiltonians. We hope that
in the future our construction of microlocal projector could shed some light
on this matter.

##  2\. Theory of microlocal sheaves

###  2.1. Sheaves and their microlocal supports

Let $\mathds{k}$ be a ring. Let $D(Q)$ be the derived category of sheaves of
$\mathds{k}$-modules on $Q$. For any object $\mathscr{F}\in\mathcal{D}(Q)$,
Kashiwara and Schapira [5] define its _microsupport_ (denoted by
$SS(\mathscr{F}))$ to be the closure of those $(q_{0},p_{0})\in T^{*}Q$ such
that there exists a $C^{1}$ function $\phi:Q\rightarrow\mathbb{R}$ with
$d\phi(q_{0})=p_{0}$ and satisfying the following ill propagation condition:

(1) $(R\Gamma_{\\{q\in Q|\phi(q)\geq\phi(q_{0})\\}}\mathscr{F})_{q_{0}}\ncong
0.$

Functorial relations between microsupports are extensively used in the
determination of subcategories of certain sheaves. Their explanations and
proofs can be found in [3, 5, 7]. For the convenience of the reader, we list
several of them which take roles in the present paper.

Let $Y\xrightarrow{f}X$ be a morphism of smooth manifolds. It induces the
natural morphisms

$T^{*}X\xleftarrow{f_{\pi}}T^{*}X\times_{X}Y\xrightarrow{f^{t}}T^{*}Y.$

The microsupports of the images under Grothendieck’s six
functors($Rf_{*},Cf_{!},f^{-1}$, $f^{!},\otimes,\underline{Rhom}$) are bounded
by the microsupports of their preimages, via $f^{t}$ and $f_{\pi}$, in the
sense of the following propositions:

######  Proposition 2.1.

[5, Proposition 5.4.4] Let $\mathscr{G}\in D(Y)$ and assume $f$ is proper on
$supp(\mathscr{G})$, then

$SS(Rf_{*}\mathscr{G})=SS(Rf_{!}\mathscr{G})\subset
f_{\pi}((f^{t})^{-1}(SS(\mathscr{G}))).$

######  Proposition 2.2.

[5, Proposition 5.4.13] Let $\mathscr{F}\in D(X)$ and assume $f$ is non-
characteristic for $SS(\mathscr{F})$, i.e., $f_{\pi}^{-1}(SS(\mathscr{F}))\cap
T_{Y}^{*}X\subset Y\times_{X}T_{X}^{*}X$. Then

  1. (1)

$SS(f^{-1}\mathscr{F})\subset(f^{t})(f_{\pi}^{-1}(SS(\mathscr{F})))$.

  2. (2)

$f^{-1}\mathscr{F}\otimes\omega_{Y/X}\cong f^{!}\mathscr{F}$.

######  Proposition 2.3.

[5, Proposition 5.4.14] Let $\mathscr{F},\mathscr{G}\in D(X)$. Then

  1. (1)

$SS(\mathscr{F})\cap SS(\mathscr{G})^{a}\subset T_{X}^{*}X\Rightarrow
SS(\mathscr{F}\otimes\mathscr{G})\subset SS(\mathscr{F})+SS(\mathscr{G})$.

  2. (2)

$SS(\mathscr{F})\cap SS(\mathscr{G})\subset T_{X}^{*}X\Rightarrow
SS(\underline{Rhom}(\mathscr{F};\mathscr{G}))\subset
SS(\mathscr{F})^{a}+SS(\mathscr{G})$.

When it comes to the products and projections, a sharper control of
microsupports is available. Let $E$ be a nontrivial finite-dimensional real
vector space and $\pi:X\times E\rightarrow X$ be the projection map.

######  Proposition 2.4.

[7, Corollary 3.4] Let $\kappa:T^{*}X\times E\times E^{\star}\rightarrow
T^{*}X\times E^{\star}$ be the projection and let $i:T^{*}X\rightarrow
T^{*}X\times E^{\star}$ be the closed embedding
$(x,\omega)\mapsto(x,\omega,0)$ of zero-section. Then for $\mathscr{F}\in
D(X\times E)$ we have

$SS(R\pi_{!}\mathscr{F}),SS(R\pi_{*}\mathscr{F})\subset
i^{-1}\overline{\kappa(SS(\mathscr{F}))}.$

When $E$ is replaced by a smooth manifold $Y$, the same argument shows the
following: If an object $\mathscr{F}\in D(X\times Y)$ is nonsingular in a
neighborhood of $\\{(x,\omega)\\}\times T_{Y}^{*}Y$ in $T^{*}(X\times Y)$,
then its projection along $Y$ will remain nonsingular at $(x,\omega)$.

###  2.2. The Tamarkin categories $\mathcal{D}$

It was shown in [5] that for any $\mathscr{F}$ in $D(Q)$ the correpsonding
microsupport $SS(\mathscr{F})$ is always a closed $\mathds{R}^{+}$-conic
coisotropic subset of $T^{*}Q$. To tame the conic structure, Tamarkin [7]
considered certain subcategories of the category $D(X\times\mathds{R})$ as
follows. Denote the coordinate of $T^{*}(Q\times\mathds{R})$ by
$(q,p,z,\zeta)$ where $(q,p)\in T^{*}Q$, $z\in\mathds{R}$, and $\zeta\in
T^{*}_{z}\mathds{R}$. Following [7], we consider the following full
subcategory of $D(Q\times\mathds{R})$:

$D_{\zeta\leq 0}=\\{\mathcal{F}\in
D(Q\times\mathds{R})|SS(\mathscr{F})\subset\\{\zeta\leq 0\\}\\}.$

######  Definition 2.5.

The Tamarkin category is defined to be

(2) $\mathcal{D}(Q\times\mathds{R}):=D_{\zeta\leq
0}(Q\times\mathds{R})^{\perp,l}$

Here the left orthogonal complement is taken with respect to $Rhom$ in
$D(Q\times\mathds{R})$.

A remarkable feature of about Tamarkin category
$\mathcal{D}(Q\times\mathds{R})$ is that there exists a quotient functor
$D(Q\times\mathds{R})\rightarrow\mathcal{D}(Q\times\mathds{R})$. Moreover,
this quotient functor can be represented by a kernel convolution.

Let $\mathscr{F}\in D(Q_{1}\times Q_{2}\times\mathds{R})$ and $\mathscr{G}\in
D(Q_{2}\times Q_{3}\times\mathds{R})$. We define their convolution to be

(3)
$\mathscr{F}\bullet_{Q_{2}}\mathscr{G}=R\pi_{13\sigma!}({\pi_{Q12R1}}^{-1}\mathscr{F}\otimes{\pi_{Q23R2}}^{-1}\mathscr{G})\in
D(Q_{1}\times Q_{3}\times\mathds{R})$

where $\pi_{Q12R1}:Q_{1}\times Q_{2}\times
Q_{3}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow Q_{1}\times
Q_{2}\times\mathds{R}_{1}$ and $\pi_{Q23R2}:Q_{1}\times Q_{2}\times
Q_{3}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow Q_{2}\times
Q_{3}\times\mathds{R}_{2}$ are projections and $\pi_{13\sigma}:Q_{1}\times
Q_{2}\times Q_{3}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow
Q_{1}\times Q_{3}\times\mathds{R}$ maps $(q_{1},q_{2},q_{3},z_{1},z_{2})$ to
$(q_{1},q_{3},z_{1}+z_{2})$.

When $Q_{2}$ and $Q_{3}$ are single points, by abuse of notation we denote by
$\mathscr{F}\bullet\mathscr{G}$ the convolution product
$\mathscr{F}\bullet_{Q_{2}}\mathscr{G}$ for $\mathscr{F}\in
D(Q\times\mathds{R})$ and $\mathscr{G}\in D(\mathds{R})$. Therefore
convoluting with a given object of $D(\mathds{R})$ becomes a functor
$D(Q\times\mathds{R})\rightarrow D(Q\times\mathds{R})$.

For any subset $A\subset\mathds{R}$, let $\mathds{k}_{A}\in D(\mathds{R})$ be
the constant sheaf supported by $A$. We also denote
$\mathds{k}_{0}=\mathds{k}_{\\{z=0\\}}$, $\mathds{k}_{\geq
c}=\mathds{k}_{[c,\infty)}$ and $\mathds{k}_{>c}=\mathds{k}_{(c,\infty)}$. For
any $\mathscr{F}\in D(Q\times\mathds{R})$ we have
$\mathscr{F}\bullet\mathds{k}_{0}=\mathscr{F}$. Moreover:

######  Proposition 2.6.

[7, Proposition 2.1, Proposition 2.2] Convolution with the exact triangle

(4) $\mathds{k}_{\geq
0}\rightarrow\mathds{k}_{0}\rightarrow\mathds{k}_{>0}[1]\xrightarrow{+1}$

gives rise to the semiorthogonal decomposition with respect to the triple
$(\mathcal{D}(Q\times\mathds{R}),D(Q\times\mathds{R}),D_{\zeta\leq
0}(Q\times\mathds{R}))$.

Another important fact about the Tamarkin category theory is that the category
$\mathcal{D}(Q\times\mathds{R})$, originally defined to be the left orthogonal
complement of $D_{\zeta\leq 0}(Q\times\mathds{R})$, is also the right
orthogonal complement of the same category $D_{\zeta\leq
0}(Q\times\mathds{R})$. The quotient functor from $D(Q\times\mathds{R})$ to
$D_{\zeta\leq 0}(Q\times\mathds{R})^{\perp,r}$ is represented by
$\hom^{\star}(\mathds{k}_{\geq 0},-)$, the adjoint functor of the convolution
with $\mathds{k}_{\geq 0}$.

To summarize, the situation of the Tamarkin category
$\mathcal{D}(Q\times\mathds{R})$ inside $D(Q\times\mathds{R})$ is illustrated
by the following diagram:

(5)
$\scriptstyle{\bigcup}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\bullet\mathds{k}_{\geq
0}}$$\scriptstyle{\hom^{\star}(\mathds{k}_{\geq
0},-)}$$\textstyle{{D_{\zeta\leq
0}}^{\perp,l}}$$\textstyle{\mathcal{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{D_{\zeta\leq
0}}^{\perp,r}}$
---

Now we introduce the notion of Tamarkin subsest categories:

######  Definition 2.7 (Tamarkin Subset Category).

Consider the de-homogenization map
$\rho:T^{*}Q\times\dot{T}^{*}\mathds{R}\rightarrow T^{*}Q$,
$\rho(q,p,z,\zeta)=(q,\frac{p}{\zeta})$.

  1. (1)

For $A\overset{cls}{\subset}T^{*}Q$, we define
$\mathcal{D}_{A}(Q\times\mathds{R}):=\\{\mathscr{F}\in\mathcal{D}|SS(\mathscr{F})\subset\rho^{-1}(A)\\}$.

  2. (2)

For $U\overset{open}{\subset}T^{*}Q$, we define
$\mathcal{D}_{U}(Q\times\mathds{R}):={D_{T^{*}Q\setminus U}}^{\perp,l}$ w.r.t
$Rhom$ in $\mathcal{D}$.

###  2.3. Sheaf quantization

Let $I\subset\mathds{R}$ be an open interval (connected subset of $\mathds{R}$
with non-empty interior) containing the origin and let $h:I\times
T^{*}Q\rightarrow T^{*}Q$ be the Hamiltonian diffeomorphism generated by an
autonomous Hamiltonian function $H:T^{*}Q\rightarrow\mathds{R}$.

It is convenient to construct such sheaf quantization using the actional
functional (which is implicitly carried out in Guillermou-Kashiwara-Schapira’s
[3, Proposition A.6]). Define the action functional $S:I\times
T^{*}Q\rightarrow\mathds{R}$ by $S(s_{1};q_{1},p_{1})=\int_{\gamma}pdq-Hds$
where $\gamma$ is the Hamiltonian trajectory $\gamma(s)=h_{s}(q_{1},p_{1})$
connecting $\gamma(0)=(q_{1},p_{1})$ and $\gamma(s_{1})=(q_{2},p_{2})$. The
Hamiltonian diffeomorphism $h$ lifts to a Hamiltonian diffeomorphism
$\tilde{h}:I\times T^{*}Q\times T^{*}_{>0}\mathds{R}\rightarrow T^{*}Q\times
T^{*}_{>0}\mathds{R}$ by

(6)
$\tilde{h}_{s}(q_{1},p_{1};z,\zeta)=(q_{2},p_{2};z+S(s;q_{1},\frac{p_{1}}{\zeta}),\zeta),$

where $(q_{2},p_{2})=h_{s}(q_{1},p_{1})$. The map $\tilde{h}$ is homogeneous
in the sense that $h\circ(id_{I}\times\rho)=\rho\circ\tilde{h}$ as maps from
$I\times T^{*}Q\times T^{*}_{>0}\mathds{R}$ to $T^{*}Q$. In other words, the
following diagram commutes:

(7) $\textstyle{{{T_{\zeta>0}^{*}}(Q\times\mathds{R})\times
I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho\times
id_{I}}$$\scriptstyle{\tilde{h}}$$\textstyle{{{T_{\zeta>0}^{*}}(Q\times\mathds{R})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{{T^{*}Q\times
I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{T^{*}Q}.}$
---

Let $\Lambda\subset T^{*}(I\times Q\times Q)\times\mathds{R}$ be the
Legendrian suspension of the graph of $h$, defined by

$\Lambda:=\\{(-Hds;q_{1},-p_{1};q_{2},p_{2};-S(s;q_{1},p_{1}))|(q_{2},p_{2})=h_{s}(q_{1},p_{1})\\}.$

Its homogenization $\tilde{\Lambda}\subset T^{*}(I\times Q\times
Q\times\mathds{R})$ is defined by

(8) $\tilde{\Lambda}=\\{(\zeta x;z,\zeta)|(x,z)\in\Lambda,\zeta>0\\},$

which is just the Lagrangian suspension of the graph of $\tilde{h}$ in the
sense of convolution product.

######  Proposition 2.8.

Suppose that the flow of $h$ satisfies the condition ($\star$) and we take
$I=\mathds{R}$. Then there exists an object $\mathscr{S}\in\mathcal{D}(I\times
Q\times Q\times\mathds{R})$ such that $\mathscr{S}|_{0\times Q\times
Q\times\mathds{R}}$ is isomorphic to $\mathds{k}_{\Delta}$ in $\mathcal{D}$,
where $\Delta=\\{(q,q,z)|z\geq 0\\}$ and
$SS(\mathscr{S})\cap\\{\zeta>0\\}\subset\tilde{\Lambda}$. The object
$\mathscr{S}$ is called the sheaf quantization of the Hamiltonian
diffeomorphism $h$.

###### Proof.

Recall from Definition 1.1 there is a connected open interval $J$ such that
for any given $0<s\in J$ and $q_{1},q_{2}\in Q$ there exists at most one
Hamiltonian trajectory $\gamma$ from $T^{*}_{q_{1}}Q$ to $T^{*}_{q_{2}}Q$ with
time $s$. For such $s\in J$ and such $q_{1},q_{2}$ we can write
$p_{1}=p_{1}(q_{1},q_{2})$ and rewrite the action functional
$S(s;q_{1},p_{1})$ by $S(s;q_{1},q_{2})$. The local sheaf quantization is
defined by

(9) $\mathscr{S}_{J}:=\mathds{k}_{\\{(s,q_{1},q_{2},z)|z+S(s;q_{1},q_{2})\geq
0\\}}\in\mathcal{D}(J\times Q\times Q\times\mathds{R}).$

We then adapt the strategy from [3, Lemma 3.5(B)] to glue to the global sheaf
$\mathscr{S}\in\mathcal{D}(I\times Q\times Q\times\mathds{R})$.

∎

Let us point out that our notion of sheaf quantization is slightly different
from that originally established in Guillermou-Kashiwara-Schapira’s [3].
First, in [3] the sheaf quantization is defined for homogeneous Hamiltonian on
$\dot{T}^{*}M$ with its zero section cut off. In the scenario of homogenizing
$h$ on $T^{*}Q$ to $\tilde{h}$ on $\dot{T}^{*}M$ where $M=Q\times\mathds{R}$,
it is required by [3] that the pre-homogenized Hamiltonian $h$ satisfies
compact support condition for (6) to be defined at $\zeta=0$. Second, in [3]
it is shown that the sheaf quantization $\mathscr{S}$ satisfies
$\mathscr{S}|_{s=0}\cong\mathds{k}_{\Delta}$ in the ambient category
$D(I\times M\times M)=D(I\times Q\times Q\times\mathds{R}\times\mathds{R})$.
In constrast to [3] we only require the sheaf quantization at time $s=0$ to be
isomorphic to $\mathds{k}_{\Delta}$ within the Tamarkin category
$\mathcal{D}(I\times Q\times Q\times\mathds{R})$, and instead of the
composition product we use the convolution product which is more compatible
with $\mathcal{D}$. This slightly weaker isomorphism allows us to treat
certain $h$ not necessarily extendable to $\dot{T}^{*}(Q\times\mathds{R})$ and
drop the compact support condition.

For example let us take $H=\frac{1}{2}p^{2}$ then one has
$h_{s}(q_{1},p_{1})=(q_{1}+sp_{1},p_{1})$ and
$S(s;q_{1},p_{1})=\frac{-1}{2}s{p_{1}}^{2}$. Therefore
$S(s;q_{1},q_{2})=\frac{-1}{2s}(q_{2}-q_{1})^{2}$ for $s\neq 0$. Let
$\mathscr{S}:=\mathds{k}_{\\{(s,q_{1},q_{2},z)|z+S(s;q_{1},q_{2})\geq 0\\}}$
be the associated sheaf quantization. We see that when $q_{1}\neq q_{2}$, let
$s\rightarrow 0^{+}$ then $z$ has no support at all. On the other hand let
$s\rightarrow 0^{-}$ then we see that $z$ is supported on the whole
$\mathds{R}$. These lead to $\mathscr{S}|_{s=0}\cong\mathds{k}_{\Delta}$ in
the Tamarkin category $\mathcal{D}$ rather than in the ambient category $D$.

##  3\. Microlocal Projector

###  3.1. Projectors associated to sublevel sets

This subsection is devoted to the construction of the projection kernel
associated with the open sublevel set. We assume

$H:T^{*}Q\rightarrow\mathds{R}$

is an autonomous Hamiltonian function satisfying the condition ($\star$) (See
Definition 1.1).

For this Hamiltonian $H$ we take the object $\mathscr{S}\in\mathcal{D}(I\times
Q\times Q\times\mathds{R})$ obtained in Proposition 2.8. By the completeness
of $H$ we choose the time interval $I$ to be the whole $\mathds{R}$. We stick
to the notation $I$ in order to distinguish the time (in $s$) with the extra
action variable $\mathds{R}$ (in $z$). We denote the dual of $I$ by $I^{*}$
(in $t$).

Let us perform the Fourier-Sato-Tamarkin transform on $\mathscr{S}$ with
respect to the time variable $I\rightsquigarrow I^{*}$ :

(10)
$\widehat{\mathscr{S}}:=\mathscr{S}\underset{s}{\bullet}\mathds{k}_{\\{(s,t,z)|z+st\geq
0\\}}[1]\in\mathcal{D}(I^{*}\times Q\times Q\times\mathds{R})$

In the realm of functional analysis, the Fourier transform is expected to
intertwine uniform distributions and delta functions. Similarly in algebraic
microlocal analysis we have the following property for the Fourier-Sato-
Tamarkin transform:

(11)
$\widehat{\mathscr{S}}\underset{t}{\circ}{\mathds{k}_{\mathds{R}}}=\mathscr{S}\underset{s}{\bullet}\mathds{k}_{\\{(s,t,z)|z+st\geq
0\\}}[1]\underset{t}{\circ}{\mathds{k}_{\mathds{R}}}\cong\mathscr{S}\underset{s}{\bullet}\mathds{k}_{\\{s=0,z\geq
0\\}}=\mathscr{S}|_{s=0}$

which is, by Theorem 2.8,

(12)
$\widehat{\mathscr{S}}\underset{t}{\circ}{\mathds{k}_{\mathds{R}}}\cong\mathds{k}_{\Delta}.$

Note that in the above formulation we write the composition product
"$\underset{t}{\circ}$" instead of the convolution product
"$\underset{t}{\bullet}$" under the convention that
$\mathds{k}_{\mathds{R}}=\mathds{k}_{\mathds{R}_{t}}$ lacks the $z$-component.
It is easy to verify that the composition
$(-)\underset{t}{\circ}{\mathds{k}_{\mathds{R}}}$ is equivalent to the
convolution
$(-)\underset{t}{\bullet}{\mathds{k}_{\\{(t,z)|t\in\mathds{R},z\geq 0\\}}}$ in
$\mathcal{D}$.

Let $C$ be a real number and let $B_{C}=\\{(q,p)\in T^{*}Q|H(q,p)<C\\}$ be the
open sublevel set of $H$ of level $C$.

######  Definition 3.1 (Microlocal Projectors).

We define the microlocal projectors associated to $B_{C}$ and $T^{*}Q\setminus
B_{C}$ respectively by

(13)
$\mathscr{P}_{C}:=\widehat{\mathscr{S}}\underset{t}{\circ}\mathds{k}_{\\{t<C\\}}$

and

(14)
$\mathscr{Q}_{C}:=\widehat{\mathscr{S}}\underset{t}{\circ}\mathds{k}_{\\{t\geq
C\\}}.$

respectively.

Together with $\mathds{k}_{\Delta}$ they form the following exact triangle

(15)
$\mathscr{P}_{C}\rightarrow\mathds{k}_{\Delta}\rightarrow\mathscr{Q}_{C}\xrightarrow{+1}$

which can be obtained by composing $\widehat{\mathscr{S}}$ with the
fundamental triangle

(16)
$\mathds{k}_{\\{t<C\\}}\rightarrow\mathds{k}_{\mathds{R}}\rightarrow\mathds{k}_{\\{t\geq
C\\}}\xrightarrow{+1}.$

Since $\mathscr{P}_{C}$, $\mathscr{Q}_{C}$, and $\mathds{k}_{\Delta}$ are
objects of $\mathcal{D}(Q\times Q\times\mathds{R})$, convolutions with them
give rise to functors from $\mathcal{D}(Q\times\mathds{R})$ to
$\mathcal{D}(Q\times\mathds{R})$. Moreover we have the following theorem:

######  Theorem 3.2.

The convolution with the exact triangle

(17)
$\mathscr{P}_{C}\rightarrow\mathds{k}_{\Delta}\rightarrow\mathscr{Q}_{C}\xrightarrow{+1}$

gives rise to the semiorthogonal decomposition with respect to the triple of
subset categories

(18)
$(\mathcal{D}_{B_{C}}(Q\times\mathds{R}),\mathcal{D}(Q\times\mathds{R}),\mathcal{D}_{T^{*}Q\setminus
B_{C}}(Q\times\mathds{R})).$

In particular, we have idempotent functors
$\mathscr{P}_{C}\bullet\mathscr{P}_{C}\cong\mathscr{P}_{C}$ and
$\mathscr{Q}_{C}\bullet\mathscr{Q}_{C}\cong\mathscr{Q}_{C}$ (this is where the
term projector comes from).

###  3.2. Proof of Theorem 3.2

Before proving the Theorem, let us briefly explain how to deduce the
idempotency of the projector $\mathscr{P}_{C}$ (similarly for
$\mathscr{Q}_{C}$) from the desired semiorthogonal decomposition. For any
$\mathscr{F}\in\mathcal{D}(Q\times\mathds{R})$ we have its semidecomposition
represented by the following exact triangle

(19)
$\mathscr{P}_{C}\bullet\mathscr{F}\rightarrow\mathscr{F}\rightarrow\mathscr{Q}_{C}\bullet\mathscr{F}\xrightarrow{+1}.$

We then perform the operation $\mathscr{P}_{C}\bullet(-)$ and get the
following exact triangle

(20)
$\mathscr{P}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}\rightarrow\mathscr{P}_{C}\bullet\mathscr{F}\rightarrow\mathscr{P}_{C}\bullet\mathscr{Q}_{C}\bullet\mathscr{F}\xrightarrow{+1}.$

On the other hand, by the semiorthogonal decomposition of the object
$\mathscr{P}_{C}\bullet\mathscr{F}$ there is an exact triangle

(21)
$\mathscr{P}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}\rightarrow\mathscr{P}_{C}\bullet\mathscr{F}\rightarrow\mathscr{Q}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}\xrightarrow{+1}.$

It is easy to see that the first morphisms in both triangles are identical. By
the uniqueness of the cone one has the equality
$\mathscr{P}_{C}\bullet\mathscr{Q}_{C}\bullet\mathscr{F}\cong\mathscr{Q}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}$.
This object belongs to both the category $\mathcal{D}_{T^{*}Q\setminus
B_{C}}(Q\times\mathds{R})$ and its left orthogonal complement
$\mathcal{D}_{T^{*}Q\setminus B_{C}}(Q\times\mathds{R})^{\perp,l}$, hence
$\mathscr{P}_{C}\bullet\mathscr{Q}_{C}\bullet\mathscr{F}\cong\mathscr{Q}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}\cong
0$ is isomorphic to the zero object of $\mathcal{D}(Q\times\mathds{R})$. So we
have the isomorphism
$\mathscr{P}_{C}\bullet\mathscr{P}_{C}\bullet\mathscr{F}\cong\mathscr{P}_{C}\bullet\mathscr{F}$
functorially for any $\mathscr{F}\in\mathcal{D}(Q\times\mathds{R})$ and we
deduce the idempotency of $\mathscr{P}_{C}$.

Now we turn to the main body of the proof. First we verify that the
convolution with $\mathscr{Q}_{C}$ projects to the subcategory
$\mathcal{D}_{T^{*}Q\setminus B_{C}}(Q\times\mathds{R})$.

######  Lemma 3.3.

For any object $\mathscr{G}\in\mathcal{D}(Q\times\mathds{R})$, one has
$\mathscr{G}\underset{Q}{\bullet}\mathscr{Q}_{C}\in\mathcal{D}_{T^{*}Q\setminus
B_{C}}(Q\times\mathds{R})$.

###### Proof of Lemma 3.3.

The proof is word for word valid for the Step 1 of the proof of [2, Theorem
3.11]. For the reader’s convenience we reproduce the argument loc. cit. as
follows:

Denote by $Q_{1}=Q_{2}=Q$ the base manifold and denote by
$\mathds{R}_{1}=\mathds{R}_{2}=\mathds{R}$ the extra action variable. We take
$\mathscr{G}\in\mathcal{D}(Q_{1}\times\mathds{R}_{1})$ and
$\widehat{\mathscr{S}}\in\mathcal{D}(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{2})$. Let $\pi_{Q1R1}:I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow
Q_{1}\times\mathds{R}_{1}$ and $\pi_{IQ12R2}:I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow Q_{1}\times
Q_{2}\times\mathds{R}_{2}$ be projection maps. Let $\pi_{\sigma}:I^{*}\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow
Q_{2}\times\mathds{R}$ be the product of maps that take sum
$\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow\mathds{R}$ and project
$I^{*}\times Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}$ onto
$Q_{2}$.

By the definition of microlocal projector we have

(22)
$\mathscr{G}\underset{Q}{\bullet}\mathscr{Q}_{C}=\mathscr{G}\underset{Q}{\bullet}\widehat{\mathscr{S}}\underset{t}{\circ}\mathds{k}_{\\{t\geq
C\\}}=R{\pi_{\sigma}}_{!}({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}\otimes\mathds{k}_{[R,\infty)\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}})$

In the cotangent bundle $T^{*}(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2})$ one obtains the following
estimates using Proposition 2.2 and Proposition 2.8:

(23)
$\begin{split}SS({\pi_{Q1R1}}^{-1}\mathscr{G})\subset\\{(t,0;q_{1},\zeta_{1}p_{1};q_{2},0;z_{1},\zeta_{1};z_{2},0)|\zeta_{1}\geq
0\\}\\\
SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\subset\\{(t,\zeta_{2}s;q_{1},\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},0;z_{2},\zeta_{2})|\zeta_{2}\geq
0,\\\ \quad\quad
H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t,(q_{2},p_{2}^{\prime})=h_{s}(q_{1},p_{1}^{\prime})\\}.\end{split}$

From (23) it is clear that

$SS({\pi_{Q1R1}}^{-1}\mathscr{G})\cap
SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})^{a}\subset T^{*}_{I^{*}\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}(I^{*}\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2})$

Therefore by Proposition 2.3 we have

(24)
$\begin{split}SS({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\subset
SS({\pi_{Q1R1}}^{-1}\mathscr{G})+SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\\\
\subset\\{(t,\zeta_{2}s;q_{1},\zeta_{1}p_{1}+\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},\zeta_{1};z_{2},\zeta_{2})|\zeta_{1}\geq
0,\zeta_{2}\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t\\}\end{split}$

On the other hand, we know that

(25) $\begin{split}SS(\mathds{k}_{[R,\infty)\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}})&=\\{(C,\zeta)|\zeta\geq
0\\}\times T^{*}_{Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}(Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2})\\\ &\cup\\{(t,0)|t>C\\}\times
T^{*}_{Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}(Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}).\end{split}$

Thus from (24) and (25) we have

(26)
$\begin{split}SS({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\cap
SS(\mathds{k}_{[R,\infty)\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}})^{a}\\\ \subset
T^{*}_{I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2})\end{split}$

Apply Proposition 2.3 again we obtain

(27)
$\begin{split}SS({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}\otimes\mathds{k}_{[R,\infty)\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}})\\\ \subset
SS({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})+SS(\mathds{k}_{[R,\infty)\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}})\\\
\subset\\{(C,\zeta_{2}s+\zeta;q_{1},\zeta_{1}p_{1}+\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},\zeta_{1};z_{2},\zeta_{2})\\\
|\zeta_{1}\geq 0,\zeta_{2}\geq 0,\zeta\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=C\\}\\\
\bigcup\\{(t,\zeta_{2}s;q_{1},\zeta_{1}p_{1}+\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},\zeta_{1};z_{2},\zeta_{2})\\\
|\zeta_{1}\geq 0,\zeta_{2}\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t>C\\}.\end{split}$

We factor the map $\pi_{\sigma}:I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow
Q_{2}\times\mathds{R}$ into the composition of
$g:(t,q_{1},q_{2},z_{1},z_{2})\mapsto(t,q_{1},q_{2},z_{1},z_{1}+z_{2})$ and
$\pi_{Q2R}:(t,q_{1},q_{2},z_{1},z)\mapsto(q_{2},z)$. Then by applying
Proposition 2.1 on (27) we get

(28)
$\begin{split}SS(Rg_{!}({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}\otimes\mathds{k}_{[R,\infty)\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}))\\\
\subset\\{(C,\zeta_{2}s+\zeta;q_{1},\zeta_{1}p_{1}+\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},\zeta_{1}-\zeta_{2};z_{2},\zeta_{2})\\\
|\zeta_{1}\geq 0,\zeta_{2}\geq 0,\zeta\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=C\\}\\\
\bigcup\\{(t,\zeta_{2}s;q_{1},\zeta_{1}p_{1}+\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},\zeta_{1}-\zeta_{2};z_{2},\zeta_{2})\\\
|\zeta_{1}\geq 0,\zeta_{2}\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t>C\\}.\end{split}$

Recall from (22) we have

(29)
$\mathscr{G}\underset{Q}{\bullet}\mathscr{Q}_{C}=R{\pi_{Q2R}}_{!}Rg_{!}({\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}\otimes\mathds{k}_{[R,\infty)\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}}).$

Suppose $(q_{2},p,z_{2},1)\in
SS(\mathscr{G}\underset{Q}{\bullet}\mathscr{Q}_{C})$ is a vector with nonzero
component in $\zeta$. By applying Proposition 2.4 to the map
$\pi_{Q2R}:(t,q_{1},q_{2},z_{1},z)\mapsto(q_{2},z)$ and the estimate (28) we
deduce that there are sequences $\\{\zeta_{2}\\}$ and $\\{p_{2}^{\prime}\\}$
subject to the constraint of (28) that satisfy $\\{\zeta_{2}\\}\rightarrow 1$
and $\\{\zeta_{2}p_{2}^{\prime}\\}\rightarrow p$. Therefore
$\\{p_{2}^{\prime}\\}\rightarrow p$ and we obtain $H(q_{2},p)=\lim
H(q_{2},p_{2}^{\prime})\geq C$.

∎

Next, we need to prove that convolution with $\mathscr{P}_{C}$ projects to the
left semiorthogonal complement of $\mathcal{D}_{T^{*}Q\setminus
B}(Q\times\mathds{R})$.

######  Lemma 3.4.

For any object $\mathscr{F}$ of $\mathcal{D}_{T^{*}Q\setminus
B_{C}}(Q\times\mathds{R})$ and object $\mathscr{G}$ of
$\mathcal{D}(Q\times\mathds{R})$, we have
$Rhom(\mathscr{G}\underset{Q}{\bullet}\mathscr{P}_{C},\mathscr{F})\cong 0$.

###### Proof of Lemma 3.4.

Again let $Q_{1}=Q_{2}=Q$ be our base manifold and let
$\mathds{R}_{1}=\mathds{R}_{2}=\mathds{R}$ be the extra action variable. Let

${I^{*}}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\xrightarrow{\pi_{IQ1R1}}{I^{*}}\times
Q_{1}\times\mathds{R}_{1}\xrightarrow{\pi_{Q1R1}}Q_{1}\times\mathds{R}_{1}$

be successive projections onto the domain of $\mathscr{G}$, and

$\pi_{\sigma}:{I^{*}}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\xrightarrow{\sigma}{I^{*}}\times
Q_{2}\times\mathds{R}\xrightarrow{\pi_{Q2R}}Q_{2}\times\mathds{R}$

onto the domain of $\mathscr{F}$. Here the map $\sigma$ performs summation
$\mathds{R}_{1}\times\mathds{R}_{2}\rightarrow\mathds{R}$ and projects the
other components to a point. The map $\pi_{Q2R}$ is the projection along
${I^{*}}$.

In order to describe the restriction to $(-\infty,C)$, let us define

(30) $\begin{split}(-\infty,C)\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}&\xrightarrow{j}{I^{*}}\times
Q_{1}\times Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\\\ (-\infty,C)\times
Q_{1}\times\mathds{R}_{1}&\xrightarrow{j_{1}}{I^{*}}\times
Q_{1}\times\mathds{R}_{1}\\\ (-\infty,C)\times
Q_{2}\times\mathds{R}&\xrightarrow{j_{2}}{I^{*}}\times
Q_{2}\times\mathds{R}\end{split}$

to be the corresponding open embeddings of $(-\infty,C)$ into ${I^{*}}$.
Define the projection map ${I^{*}}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\xrightarrow{\pi_{IQ12R2}}{I^{*}}\times
Q_{1}\times Q_{2}\times\mathds{R}_{2}$ onto the domain of
$\widehat{\mathscr{S}}$.

(31)

|
---|---
$\textstyle{{I^{*}}\times Q_{1}\times
Q_{2}\times\mathds{R}_{2}}$$\textstyle{{I^{*}}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{IQ1R1}}$$\scriptstyle{\sigma}$$\scriptstyle{\pi_{\sigma}}$$\scriptstyle{\pi_{IQ12R2}}$$\textstyle{(-\infty,C)\times
Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{\sigma}$$\scriptstyle{\pi_{IQ1R1}}$$\textstyle{{I^{*}}\times
Q_{1}\times\mathds{R}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Q1R1}}$$\textstyle{(-\infty,C)\times
Q_{1}\times\mathds{R}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\textstyle{(-\infty,C)\times
Q_{2}\times\mathds{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{2}}$$\textstyle{{I^{*}}\times
Q_{2}\times\mathds{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{Q2R}}$$\textstyle{Q_{1}\times\mathds{R}_{1}}$$\textstyle{Q_{2}\times\mathds{R}}$

Recall that $\mathscr{F}\in\mathcal{D}(Q_{2}\times\mathds{R})$ and
$\widehat{\mathscr{S}}\in\mathcal{D}(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{2})$. We define a functor
$\mathcal{E}_{C}:\mathcal{D}(Q\times\mathds{R})\rightarrow\mathcal{D}(Q\times\mathds{R})$
that sends any sheaves $\mathscr{F}$ to

(32)
$\mathcal{E}_{C}(\mathscr{F}):=R{\pi_{Q1R1}}_{*}R{j_{1}}_{*}R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F})$

######  Lemma 3.5.

The functor $\mathcal{E}_{C}$ is the right adjoint functor of
$(-)\bullet\mathscr{P}_{C}$.

###### Proof.

Let us unwrap $\mathscr{P}_{C}$ all the way using the maps in the above
diagram and apply the adjunction relations between their corresponding
Grothendieck six functors. Let $\mathscr{G}$ be an object of
$\mathcal{D}(Q_{1}\times\mathds{R}_{1})$. One has by definition,

(33)
$\mathscr{G}\bullet\mathscr{P}_{C}=\mathscr{G}\underset{Q}{\bullet}(\widehat{\mathscr{S}}\underset{t}{\circ}\mathds{k}_{\\{t<R\\}})=(\mathscr{G}\underset{Q}{\bullet}\widehat{\mathscr{S}})\underset{t}{\circ}\mathds{k}_{\\{t<R\\}}=R{\pi_{Q2R}}_{!}R{j_{2}}_{!}{j_{2}}^{-1}(\mathscr{G}\underset{Q}{\bullet}\widehat{\mathscr{S}}).$

By unwrapping $\mathscr{G}\underset{Q}{\bullet}\widehat{\mathscr{S}}$ one
obtain

(34)
$\mathscr{G}\bullet\mathscr{P}_{C}=R{\pi_{Q2R}}_{!}R{j_{2}}_{!}{j_{2}}^{-1}[R\sigma_{!}({\pi_{IQ1R1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})].$

Apply commutation relations ${j_{2}}^{-1}R\sigma_{!}=R\sigma_{!}j^{-1}$ and
$R{j_{2}}_{!}R\sigma_{!}=R\sigma_{!}Rj_{!}$ and the composition
$R{\pi_{Q2R}}_{!}R\sigma_{!}=R{\pi_{\sigma}}_{!}$ to (34) one get

(35)
$\begin{split}\mathscr{G}\bullet\mathscr{P}_{C}=R{\pi_{\sigma}}_{!}Rj_{!}j^{-1}({\pi_{IQ1R1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\\\
=R{\pi_{\sigma}}_{!}(Rj_{!}j^{-1}{\pi_{IQ1R1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\\\
=R{\pi_{\sigma}}_{!}(Rj_{!}{\pi_{IQ1R1}}^{-1}{j_{1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}).\end{split}$

Therefore

(36)
$\begin{split}Rhom(\mathscr{G}\bullet\mathscr{P}_{C},\mathscr{F})=Rhom(R{\pi_{\sigma}}_{!}(Rj_{!}{\pi_{IQ1R1}}^{-1}{j_{1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}}),\mathscr{F})\\\
\cong
Rhom(Rj_{!}{\pi_{IQ1R1}}^{-1}{j_{1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G}\otimes{\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}},{\pi_{\sigma}}^{!}\mathscr{F})\\\
\cong
Rhom(Rj_{!}{\pi_{IQ1R1}}^{-1}{j_{1}}^{-1}{\pi_{Q1R1}}^{-1}\mathscr{G},\underline{Rhom}({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}},{\pi_{\sigma}}^{!}\mathscr{F}))\\\
\cong
Rhom(\mathscr{G},R{\pi_{Q1R1}}_{*}R{j_{1}}_{*}R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}},{\pi_{\sigma}}^{!}\mathscr{F})).\end{split}$

That is,

(37) $Rhom(\mathscr{G}\bullet\mathscr{P}_{C},\mathscr{F})\cong
Rhom(\mathscr{G},\mathcal{E}_{C}(\mathscr{F})).$

∎

We check that the abovementioned operations are compatible with $\mathcal{D}$.

######  Lemma 3.6.

For any $\mathscr{F}\in D(Q\times\mathds{R})$, one has
$\mathcal{E}_{C}(\mathscr{F})\in\mathcal{D}(Q\times\mathds{R})$.

###### Proof.

Pick any $\mathscr{G}\in D_{\leq 0}(Q\times\mathds{R})$ then in
$D(Q\times\mathds{R})$ one has

(38) $Rhom(\mathscr{G},\mathcal{E}_{C}(\mathscr{F}))\cong
Rhom(\mathscr{G}\bullet\mathscr{P}_{C},\mathscr{F})\cong
Rhom(0,\mathscr{F})=0.$

Therefore $\mathcal{E}_{C}(\mathscr{F})\in{D_{\leq 0}}^{\perp,r}$. Apply the
fact (5) that $\mathcal{D}$ is both left and right orthogonal complement of
$D_{\leq 0}$.

∎

Now we prove that $\mathcal{E}_{C}$ is microlocally vanishing outside the open
sublevel set $B_{C}$.

######  Lemma 3.7.

For all $\mathscr{F}\in\mathcal{D}_{T^{*}Q\setminus B}(Q\times\mathds{R})$,
one has $\mathcal{E}_{C}(\mathscr{F})\cong 0$.

###### Proof.

Recall that

(39)
$\mathcal{E}_{C}(\mathscr{F}):=R{\pi_{Q1R1}}_{*}R{j_{1}}_{*}R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F}).$

By Lemma 3.6, it suffices to show that

(40)
$R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F})\in
D_{\leq 0}((-\infty,C)\times Q_{1}\times\mathds{R}_{1}).$

By Proposition 2.2 one has, up to the zero section $\\{\zeta=0\\}$, the
following estimate:

(41) $SS({\pi_{\sigma}}^{!}\mathscr{F})\subset\\{(t,0;q_{1},0;q_{2},\zeta
p_{2};z_{1},\zeta;z_{2},\zeta)|\zeta>0,H(q_{2},p_{2})\geq C\\}.$

On the other hand, recall from $(\ref{akira})$ one has

(42)
$\begin{split}SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\subset\\{(t,\zeta_{2}s;q_{1},\zeta_{2}p_{1}^{\prime};q_{2},\zeta_{2}p_{2}^{\prime};z_{1},0;z_{2},\zeta_{2})|\\\
\zeta_{2}\geq
0,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t,(q_{2},p_{2}^{\prime})=h_{s}(q_{1},p_{1}^{\prime})\\}\end{split}$

It is clear that

(43) $SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})\bigcap
SS({\pi_{\sigma}}^{!}\mathscr{F})\subset T^{*}_{(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2})}(I^{*}\times Q_{1}\times
Q_{2}\times\mathds{R}_{1}\times\mathds{R}_{2}).$

Thus by Proposition 2.3, one obtain

(44)
$\begin{split}SS(\underline{Rhom}({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}},{\pi_{\sigma}}^{!}\mathscr{F}))\subset
SS({\pi_{IQ12R2}}^{-1}\widehat{\mathscr{S}})^{a}+SS({\pi_{\sigma}}^{!}\mathscr{F})\\\
\subset\\{(t,-\zeta_{2}s;q_{1},-\zeta_{2}p_{1}^{\prime};q_{2},\zeta
p_{2}-\zeta_{2}p_{2}^{\prime};z_{1},\zeta;z_{2},\zeta-\zeta_{2})|\zeta>0,\zeta_{2}\geq
0,\\\ H(q_{2},p_{2})\geq
C,H(q_{1},p_{1}^{\prime})=H(q_{2},p_{2}^{\prime})=t,(q_{2},p_{2}^{\prime})=h_{s}(q_{1},p_{1}^{\prime})\\}.\end{split}$

Note that
$R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F})$
is an object of $D((-\infty,C)\times Q_{1}\times\mathds{R}_{1})$. We pick an
arbitrary vector $v$ in $T^{*}_{\zeta_{1}>0}((-\infty,C)\times
Q_{1}\times\mathds{R}_{1})$ that satisfies

(45) $v=(t,\tau;q_{1},p_{1};z_{1},1)\in
SS(R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F}).$

By Proposition 2.4, there exist sequences
$\\{-\zeta_{2}s\\}$,$\\{-\zeta_{2}p_{1}^{\prime}\\}$,$\\{\zeta
p_{2}-\zeta_{2}p_{2}^{\prime}\\}$, $\\{\zeta\\}$, $\\{\zeta-\zeta_{2}\\}$
subject to the conditions of (44) such that they and their base coordinates
approach to the corresponding entries of
$(t,\tau;q_{1},p_{1};-,0;z_{1},1;-,0)$, the extension of $v$ by the zero
section, as follows:

(46) $\begin{split}\\{-\zeta_{2}s\\}&\rightarrow\tau\\\
\\{-\zeta_{2}p_{1}^{\prime}\\}&\rightarrow p_{1}\\\ \\{\zeta
p_{2}-\zeta_{2}p_{2}^{\prime}\\}&\rightarrow 0\\\ \\{\zeta\\}&\rightarrow 1\\\
\\{\zeta-\zeta_{2}\\}&\rightarrow 0\end{split}$

From (46) we deduce that $\\{s\\}\rightarrow-\tau$,
$\\{p_{1}^{\prime}\\}\rightarrow-p_{1}$ and
$\\{p_{2}-p_{2}^{\prime}\\}\rightarrow 0$. Since the sequences are subject to
the condition $(q_{2},p_{2}^{\prime})=h_{s}(q_{1},p_{1}^{\prime})$, we deduce
that $(q_{2},\lim\\{p_{2}^{\prime}\\})=h_{-\tau}(q_{1},-p_{1})$ is finite from
the assumption of $H$ generating complete Hamiltonian flow. Therefore
$\lim\\{p_{2}\\}$ is finite, since $\lim\\{p_{2}-p_{2}^{\prime}\\}=0$ and
$\lim\\{p_{2}^{\prime}\\}$ is finite. The other conditions from (44) then
imply that

(47) $t=H(q_{2},\lim\\{p_{2}^{\prime}\\})=H(q_{2},\lim\\{p_{2}\\})\geq C,$

which contradicts to the assumption

(48) $(t,\tau;q_{1},p_{1};z_{1},1)=v\in T^{*}_{\zeta_{1}>0}((-\infty,C)\times
Q_{1}\times\mathds{R}_{1}),$

that is $t<C$.

∎

This finishes the proof of Lemma 3.4.

∎

By the general property $SS(\mathscr{F}\oplus\mathscr{G})=SS(\mathscr{F})\cup
SS(\mathscr{G})$, all the subcategories mentioned above are thick (closed
under direct summands). Therefore Lemma 3.3 and Lemma 3.4 together imply
Theorem 3.2.

######  Remark 3.8.

In fact Lemma 3.3 does not involve the assumption of complete flow.
Nevertheless, such assumption is essential in Lemma 3.4, where the
semiorthogonality needs an additional proof. Note that our argument in proving
Lemma 3.7 is of $C^{0}$ nature. We use only the continuity of the function $H$
and its flow $h$.

##  4\. A Non-example

What if the Hamiltonian flow fails to be complete?

Let us consider the famous Cherry Hamiltonian [1] on $T^{*}\mathds{R}^{2}$
with parameter $\mu$,

(49)
$H_{ch}(q,p)=\frac{1}{2}({q_{1}}^{2}+{p_{1}}^{2})-({q_{2}}^{2}+{p_{2}}^{2})+\mu[q_{2}({q_{1}}^{2}-{p_{1}}^{2})-2q_{1}p_{1}p_{2}]$

The Hamiltonian equations of the flow are

(50) $\begin{split}\dot{q_{1}}&=p_{1}-2\mu q_{2}p_{1}-2\mu q_{1}p_{2}\\\
\dot{p_{1}}&=-q_{1}-2\mu q_{1}q_{2}+2\mu p_{1}p_{2}\\\
\dot{q_{2}}&=-2p_{2}-2\mu q_{1}p_{1}\\\
\dot{p_{2}}&=2q_{2}+\mu{p_{1}}^{2}-\mu{q_{1}}^{2}\end{split}$

The Cherry Hamiltonian $H_{ch}$ is an example of unstable equilibrium which
linearization is stable. In fact, the linear part (corresponding to $\mu=0$)
of the Cherry Hamiltonian (49) is just the superposition of two simple
harmonic oscillators.

For $\mu>0$, on the non-compact level surface $H_{ch}^{-1}(0)$ there is a
family of solutions of (50), parametrized by two parameters $a$ and $b$:

(51) $\begin{split}q_{1}(s)=\frac{-\sqrt{2}}{2\mu s+a}\sin(s+b)\quad,&\quad
p_{1}(s)=\frac{\sqrt{2}}{2\mu s+a}\cos(s+b)\\\ q_{2}(s)=\frac{1}{2\mu
s+a}\sin(2s+2b)\quad,&\quad p_{2}(s)=\frac{1}{2\mu
s+a}\cos(2s+2b).\end{split}$

The flow on this $H_{ch}^{-1}(0)$ cannot be made complete. For example when
$(a,b)=(0,0)$, the solution (51) is a curve defined for $s\neq 0$:

(52) $\begin{split}q_{1}(s)=\frac{-\sin(s)}{\sqrt{2}\mu s}\qquad,&\qquad
p_{1}(s)=\frac{\cos(s)}{\sqrt{2}\mu s}\\\ q_{2}(s)=\frac{\sin(2s)}{2\mu
s}\qquad,&\qquad p_{2}(s)=\frac{\cos(2s)}{2\mu s}\end{split}$

When $s$ approaches to $\pm\infty$, the solution curve (51) swirls around and
approaches to the origin. When $s$ approaches to $0$, its $q$-components
become stable but the $p$-components totally blow up, in a way depending on
the sign of $s$.

Let us show that $H_{ch}$ violates the projector property by violating the
middle equality in (47). For each $n\in\mathbb{N}$, consider two sequences
$y_{n}=(\sqrt{n},\sqrt{n})$ and $y^{\prime}_{n}=(\sqrt{n+1},\sqrt{n})$ in
$\mathds{R}^{2}$ satisfying

(53)
$\begin{split}\lim\\{y_{n}\\}=\lim\\{y^{\prime}_{n}\\}&=(+\infty,+\infty),\\\
\lim\\{y^{\prime}_{n}-y_{n}\\}&=(0,0).\end{split}$

We choose $\mu>0$ and $x=(q_{1},q_{2})\in\mathds{R}^{2}$ such that
$\frac{1}{2}+2\mu q_{1}+\mu q_{2}=0$. Then from (50) one has

(54) $\begin{split}H_{ch}(x,y_{n})&=\frac{1}{2}q_{1}^{2}-q_{2}^{2}+\mu
q_{1}^{2}q_{2}-(\frac{1}{2}+2\mu q_{1}+\mu q_{2})n\\\ &=-2\mu
q_{1}^{3}-\frac{2}{\mu}q_{1}-\frac{1}{4\mu^{2}}\end{split}$

and

(55)
$\begin{split}H_{ch}(x,y^{\prime}_{n})&=\frac{1}{2}q_{1}^{2}-q_{2}^{2}+\mu
q_{1}^{2}q_{2}+1+(2\mu q_{1})[(n+1)-\sqrt{n(n+1)}]\\\ &=-2\mu
q_{1}^{3}-\frac{2}{\mu}q_{1}-\frac{1}{4\mu^{2}}+1+(2\mu
q_{1})[(n+1)-\sqrt{n(n+1)}]\end{split}$

It follows that when $n\rightarrow\infty$, one has

(56) $\lim H_{ch}(x,y_{n})=-2\mu
q_{1}^{3}-\frac{2}{\mu}q_{1}-\frac{1}{4\mu^{2}},$

while on the other hand

(57) $\lim H_{ch}(x,y^{\prime}_{n})=-2\mu
q_{1}^{3}-\frac{2}{\mu}q_{1}-\frac{1}{4\mu^{2}}+1+\mu q_{1}.$

The term $H_{ch}(x,y_{n})$ is independent of $n\in\mathbb{N}$ and takes every
real value as $x$ varies, and the term $\lim H_{ch}(x,y^{\prime}_{n})$ is
always different from $\lim H_{ch}(x,y_{n})$ unless
$x=(\frac{-1}{\mu},\frac{3}{2\mu})$. This allows us to choose a sheaf
$\mathscr{F}\in\mathcal{D}_{T^{*}\mathds{R}^{2}\setminus
B_{C}}(\mathds{R}^{2}\times\mathds{R})$ supported on $\\{\frac{1}{2}+2\mu
q_{1}+\mu q_{2}=0\\}$, and a vector

(58) $v\in T^{*}_{\zeta_{1}>0}\cap
SS(R{\pi_{IQ1R1}}_{*}j^{!}\underline{Rhom}({\pi_{IQ12R2}}^{-1}(\widehat{\mathscr{S}}),{\pi_{\sigma}}^{!}\mathscr{F})$

as in (45), which causes $\mathcal{E}_{C}(\mathscr{F})\neq 0$.

## References

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