# Triangular Cross-Section Beam Splitters in Silicon Carbide for Quantum
Information Processing

Sridhar Majety 111Corresponding author<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA  Pranta Saha † Department of Electrical and Computer Engineering,
University of California, Davis, CA 95616, USA  Zbynka Kekula  Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA  Scott Dhuey  The Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Marina Radulaski  Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA

###### Abstract

Triangular cross-section color center photonics in silicon carbide is a
leading candidate for scalable implementation of quantum hardware. Within this
geometry, we model low-loss beam splitters for applications in key quantum
optical operations such as entanglement and single-photon interferometry. We
consider triangular cross-section single-mode waveguides for the design of a
directional coupler. We optimize parameters for a 50:50 beam splitter.
Finally, we test the experimental feasibility of the designs by fabricating
triangular waveguides in an ion beam etching process and identify suitable
designs for short-term implementation.

††footnotetext: † These authors contributed equally to this work

##  1 Introduction

Photonic quantum information processing (QIP) using solid-state single photon
emitters has been extensively explored for applications in quantum
communications and integrated quantum information circuits. Among the quantum
emitters, color centers have been desired solid state qubit candidates due to
their spectral homogeneity, long spin coherence times, and availability of
spin-spin and spin-photon entangling processes, necessary for QIP applications
[1, 2, 3, 4, 5, 6]. Particularly, color centers in silicon carbide (SiC)
attracted attention because of emissions in the telecommunication bands which
are suitable for sending over long distances in fiber optic cable. SiC offers
other advantages like large bandgap, strong second-order non-linearity,
decades-long industry presence, and is CMOS compatible.

The existing QIP algorithms require error correction and heralded measurements
to process information reliably and efficiently. Hence, one of the near-term
goals is to build the on-chip programmable photonic mesh networks to perform
fault-tolerant QIP [7, 8, 9]. Successful implementation of such quantum mesh
photonic hardware requires efficient single photon generation, manipulation,
entanglement, and detection [10, 11, 12]. Beamsplitters (BS) play a crucial
role in on-chip photonic QIP circuits for generating multiphoton correlation
[13, 8] and can be used to realize several important operations like
entanglement generation and single-photon interferometry (Hong-Ou-Mandel and
Hanbury Brown-Twiss). Moreover, a universal quantum computer with linear
optical quantum computing protocols has been proposed using BS, phase
shifters, single photon sources, and photo-detectors [14]. Among the most
studied forms of BS such as Y-branch, directional coupler (DC), and multi mode
interferometer (MMI), DC is the most suited for on-chip QIP as it can function
with single-mode waveguides [15].

Color centers properties are preserved in the bulk of the host material and
for successful nanophotonic integration of these color centers, it is
necessary to be able to grow or attach a high quality thin film on a substrate
with high refractive index contrast in a scalable way. Triangular photonics
has been the top choice in diamond, silicon carbide and rare earth photonics
[16, 17, 18] that provides pristine color centers and high light and matter
interaction. Compared to the other non-traditional processing techniques [19,
20, 21] used for fabricating undercut devices, angle etching method [16, 22,
18] which produces triangular cross-section and undercut photonic devices, is
promising for wafer-scale production [23]. Performance of a variety of such
triangular cross-section photonic structures in 4H-SiC were recently examined
[18, 24, 12, 25], and have proven favorable for QIP applications.

In this paper, we explore triangular cross-section 50:50 BS in 4H-SiC
necessary for performing on-chip quantum interferometry. We construct such a
device that splits an input beam into two output beams of equal intensity
using two concurrent waveguides. In the front and back regions, the waveguides
are far apart (no electromagnetic interaction) on either ends (input, output)
of the device, while in the middle overlap region waveguides propagate in
close proximity to create an electromagnetic interaction between their fields.
When two indistinguishable photons each entangled to a spin of a quantum
emitter arrive at the inputs of a 50:50 BS at the same time, they undergo
bunching to exit together through the same output, creating remote
entanglement between the two spins. Here we specifically study 50:50 BS
designs for nitrogen vacancy (NV) color centers in 4H-SiC, with single photon
emissions in the telecommunication wavelength range (1176 - 1243 nm) [26, 27].

Using Finite-Difference Time-Domain (FDTD) method, we first optimize the width
of triangular cross-section waveguides to achieve single mode propagation
(fundamental TE mode), and high coupling efficiency of the color center (NV in
4H-SiC) emission into that single mode, for different etch angles. Next, we
investigate the formation of supermodes when two identical single mode
waveguides are placed adjacent to each other. We then study the waveguide mode
conversion in bent waveguides and find suitable waveguide bend geometries to
maintain the same waveguide mode in the straight and bent regions of the
waveguide. We use this understanding to design triangular cross-section 50:50
BS in 4H-SiC. Finally, we explore the fabrication of the simulated triangular
cross-section waveguide configurations required for a 50:50 BS.

##  2 Single mode triangular cross-section waveguide

Color center emission is dipole-like and such emission can couple to
fundamental TE (f-TE), fundamental TM (f-TM) and other higher order modes that
are supported in triangular cross-section waveguides [24]. It has been shown
that for each etch angle there exists an optimal width for single mode (f-TE)
propagation in triangular cross-section waveguides [18], which is a necessity
for QIP applications [28]. So, to achieve high coupling efficiencies for color
center emission, the color center should be positioned in a single mode
waveguide at the maximum electric field (E-field) intensity point of the f-TE
mode[18, 12], which is at the centroid of the triangle.

Figure 1: The coupling of a TE oriented dipole emission at the centroid of the
triangular profile into the f-TE mode (($C_{\mathrm{fTE}}$)) as a function of
the waveguide width for three etch angles ($\alpha$). Insets show electric
field intensity profiles of the f-TE mode, TE supermode 1 and TE supermode 2
(top to bottom), at optimal waveguide widths supporting single mode
propagation, for $\alpha=30^{\circ},45^{\circ},60^{\circ}$. The electric field
intensity of the TE supermodes were plotted for adjacent identical waveguides
with 200 nm gap between them.

We use FDTD package in Lumerical software to estimate the coupling efficiency
of color center emission into the f-TE mode ($C_{\mathrm{fTE}}$) for different
widths ($w$) and etch angles ($\alpha$) of a triangular cross-section
waveguide (the angle at the apex of the triangle is $2\alpha$). We position
the dipole emitter at the centroid of the triangle with an emission wavelength
of 1230 nm. We find that for each $\alpha$ there exists a width with highest
coupling ($>80\%$) to the f-TE mode as shown in Figure 1. For our 50:50 BS
simulations, we choose the optimal width values ($w$ = 550 nm, 650 nm, 800 nm
for $\alpha=30^{\circ},45^{\circ},60^{\circ}$ respectively) slightly lower
than the widths with the highest $C_{\mathrm{fTE}}$. The reason being that the
waveguides with lower width have slightly higher evanescent fields (top panels
of inset in Figure 1), necessary to create meaningful coupling between the
adjacent waveguides in the BS, while only slightly reducing $C_{\mathrm{fTE}}$
($<1\%$).

The mode profile of the waveguide f-TE (f-TM) mode has a E-field intensity
maximum closer to the top surface (apex) [24]. As the waveguide width
decreases, the E-field intensity maximum of the f-TM mode moves closer to the
apex and becomes evanascent (not supported in the waveguide). Using MODE
package in Lumerical software, we find that at these optimal widths for
$\alpha=45^{\circ}$ and $60^{\circ}$, the waveguide supports only the f-TE
mode. However, for $\alpha=30^{\circ}$, both f-TE and f-TM modes are supported
in the waveguide at the optimal and smaller widths. The height of the triangle
varies inversely with $\alpha$ and hence the widths needed for the f-TM to
become evanescent are much smaller than the optimal width for
$\alpha=30^{\circ}$.

Figure 2: (a) Top view of the S-bend BS in 4H-SiC and the inset shows the
triangular cross-section geometry. (b) The fraction of the f-TE input mode
coupling to f-TM ($C_{\mathrm{fTM}}$) mode along the bend with variations in
bend y-span ($b_{\mathrm{y}}$) for $\alpha$ = $30^{\circ}$, $w$ = 550 nm, and
$L_{\mathrm{C}}$ = 30 $\mu$m. (c)-(d) Propagation of $E_{\mathrm{z}}$ field in
the $xy$ plane of the structure described in (b) for $b_{\mathrm{y}}$ values
of 20 $\mu$m and 35 $\mu$m, respectively.

In this paper, we design a 50:50 BS using two identical triangular cross-
section waveguides with three regions: 1) input and output - where the
waveguides are far apart with zero electromagnetic interaction, on either ends
of the BS, 2) bent waveguide region, and 3) coupling region - where the
waveguides are close enough to electromagnetically interact, as shown in
Figure 2a. When light is injected into the f-TE mode of one of the waveguides
on the input side (top left), it propagates through the straight and bent
regions of that waveguide. When it reaches the coupling region, some of the
light in the top waveguide is coupled into the bottom waveguide as it
propagates along. Then the light in both the waveguides pass through the bent
and straight regions to reach the output ports (through and drop ports). The
proportion of light coupled into the bottom waveguide is determined by the
coupling strength between the individual waveguide modes and the length of the
waveguides in the coupling region.

When two identical waveguides are close to each other, the individual modes
(f-TE) in each of the waveguides superimpose to form a supermode. The coupling
between the two waveguides can be analyzed in terms of a pair of TE
supermodes, as shown in the inset of Figure 1. For a BS with power
$P_{\mathrm{0}}$ in one of the waveguide at the beginning of the coupling
region, the coupling length ($L_{\mathrm{C}}$) required for a power
$P_{\mathrm{2}}$ to couple into the other waveguide is given by:

$L_{\mathrm{C}}=\frac{\lambda_{0}}{\pi\Delta
n_{\mathrm{eff}}}\sin^{-1}\sqrt{\frac{P_{2}}{P_{0}}}$ (1)

where $\lambda_{0}$ is the free space wavelength, $\Delta n_{\mathrm{eff}}$ is
the difference in effective refractive indicies of the two TE supermodes [29].
The coupling length is inversely proportional to the coupling strength between
the waveguides given by $\Delta n_{\mathrm{eff}}$, which in turn depends on
the confinement of the individual waveguide modes and gap between the
individual waveguides in the coupling region. For a 50:50 BS, where there is
50% light in both the waveguides after the coupling region, the ratio
$P_{\mathrm{2}}$/$P_{\mathrm{0}}$ equals 0.5. We use MODE package in Lumerical
software to estimate the effective index of the two TE supermodes, for
calculating the theoretical values of $L_{\mathrm{C}}$ for various BS
geometries studied in the following section.

##  3 Integrated 4H-SiC beam splitter in triangular geometry

Recent advances in on-chip splitting mostly include rectangular or slab
waveguides [15]. For performing QIP with integrated color centers in 4H-SiC,
we construct the BS structure with symmetric triangular cross-section single
mode S-bend waveguides with an overlap region to enable DC. Figure 2a shows
the top view schematic of the BS with coupling length $L_{\mathrm{C}}$, gap
between the waveguides in the coupling region $g$, S-bend x-span
$b_{\mathrm{x}}$, S-bend y-span $b_{\mathrm{y}}$, and the inset shows the
triangular waveguide cross-section with $w$ and $\alpha$. In this paper, we
investigate 50:50 BS in triangular cross-section waveguides with $\alpha$ =
$30^{\circ}$, $45^{\circ}$, $60^{\circ}$, can be fabricated with the state-of-
the-art processes, by varying the gap $g$ between waveguides and consequently
$L_{\mathrm{C}}$.

In Lumerical FDTD simulations (mesh size = 30 nm), we choose 20 $\mu$m for
$b_{\mathrm{x}}$ and 10 $\mu$m for $b_{\mathrm{y}}$, where $b_{\mathrm{x}}$
corresponds to the terminal points and $b_{\mathrm{y}}$ corresponds to the
curvature control points of the S-bend (Bézier curve), for $\alpha$ =
$45^{\circ}$, $60^{\circ}$. For $\alpha$ = $30^{\circ}$, we need to account
for potential higher-order mode conversion due to the bending curvature as the
$n_{\mathrm{eff}}$, one of the most important parameters for understanding
mode propagation, of the f-TE and the f-TM modes are virtually the same in the
waveguides with $\alpha$ = $30^{\circ}$. Hence, the f-TE mode can transform
into the f-TM due to identical $n_{\mathrm{eff}}$ and inter-mode coupling via
bends [30]. Keeping $b_{\mathrm{x}}$ = 20 $\mu$m constant, we vary
$b_{\mathrm{y}}$ for modulating the bend curvature of the {$\alpha$, $w$,
$L_{\mathrm{C}}$} = {$30^{\circ}$, 550 nm, 30 $\mu$m} BS structure.

To test the inter-mode coupling (cross-talk), we inject via the f-TE mode at
the input port and collect via the f-TM mode at the through port of the BS. It
is observed in Figure 2b that coupling to the f-TM mode ($C_{\mathrm{fTM}}$)
initially increases with the increase in $b_{\mathrm{y}}$ and after reaching a
maximum value of $\sim$ 20% for $b_{\mathrm{y}}$ = 20 $\mu$m,
$C_{\mathrm{fTM}}$ gradually decreases dropping down to 2.7% with
$b_{\mathrm{y}}$ = 35 $\mu$m. This trend occurs due to the expansion of the
curved path with increasing $b_{\mathrm{y}}$, however, after the inversion
point, the bending becomes so gradual that the S-bend mimics a straight path.
For a better understanding of the f-TE and f-TM mode coupling, we also study
the $E_{\mathrm{z}}$ fields of the minimum and maximum $C_{\mathrm{fTM}}$ as
$E_{\mathrm{z}}$ field is a good indicator of the existence of the TE and TM
modes [25]. Figure 2c shows that at the starting point of the bending region
with $b_{\mathrm{y}}$ = 20 $\mu$m, $E_{\mathrm{z}}$ has a nodal plane in the
triangular waveguide which corresponds to the f-TE mode and at the ending
point, $E_{\mathrm{z}}$ field starts to develop in the waveguide, confirming
the coupling to the f-TM mode. On the other hand, for $b_{\mathrm{y}}$ value
of 35 $\mu$m, there is a constant $E_{\mathrm{z}}$ nodal plane indicating the
f-TE mode throughout the S-bend, depicted in Figure 2d, which makes this bend
structure suitable for maintaining the single mode propagation in the BS.

Figure 3: (a) $L_{\mathrm{C}}$ values for 50:50 splitting with an increasing
gap $g$ in $\alpha$ = $30^{\circ}$, $45^{\circ}$, $60^{\circ}$ BS. (b) Top
view of the $E$-field intensity profile for the integrated 50:50 BS SiC
structure with {$\alpha$, $w$, $b_{\mathrm{x}}$, $b_{\mathrm{y}}$, $g$,
$L_{\mathrm{C}}$} = {$30^{\circ}$, 550 nm, 20 $\mu$m, 35 $\mu$m, 300 nm, 44
$\mu$m} indicated by dashed circle in Figure 3a.

The above discussions lead us to the optimal selection of $w$ and
$b_{\mathrm{y}}$ of the 50:50 BS for each $\alpha$. We choose 650 nm and 800
nm as $w$ for $\alpha$ = $45^{\circ}$, $60^{\circ}$ respectively with
$b_{\mathrm{y}}$ = 10 $\mu$m in order to maximize the coupling and maintain
single mode propagation for establishing a suitable environment for QIP with
the triangular cross-section BS. For $\alpha$ = $30^{\circ}$ waveguides, we
pick $w$ = 550 nm and $b_{\mathrm{y}}$ = 35 $\mu$m to achieve similar
outcomes. We vary the gap $g$ from 50 nm to 400 nm for exploring a range of
designs. Figure 3a shows that with increasing $g$, $L_{\mathrm{C}}$ shoots up
rapidly for all three etch angles, experiencing minor losses ($<$ 1.5%) upon
propagation through the entire BS. We observe another interesting phenomenon
that $L_{\mathrm{C}}$ for the 50:50 BS with $\alpha$ = $60^{\circ}$ is
dramatically longer than the $\alpha$ = $30^{\circ}$ and $\alpha$ =
$45^{\circ}$ BS for a particular $g$ when $g\geq$ 200 nm. As a result, owing
to smaller footprints, $\alpha$ = $30^{\circ}$ and $\alpha$ = $45^{\circ}$ BS
appear as better candidates than the $\alpha$ = $60^{\circ}$ BS for integrated
quantum mesh photonic circuitry.

##  4 Fabrication of closely spaced triangular waveguides in 4H-SiC

Triangular cross-section photonics has previously been fabricated in SiC
through Faraday cage assisted etching method [16, 22, 18] suitable for chip-
scale integration of color centers with no degradation in color-center
properties compared to bulk substrate[18]. This approach could be brought to
wafer-scale using the ion beam etching [23]. Therefore, we test a new ion beam
etching process to fabricate triangular cross-section waveguides in 4H-SiC,
with varying gaps between waveguides (100 nm, 200 nm, 300 nm, 500 nm). This
provides an insight into what spacing between the waveguides in the coupling
region of the simulated 50:50 BS are fabrication friendly, using the ion beam
etch process.

We use electron beam lithography to define the patterns on 4H-SiC substrates
and then the pattern is transferred to a 120 nm thick nickel hard mask
deposited using an electron beam evaporation, through a lift-off process.
Triangular cross-section waveguides are then fabricated by ion beam etch using
SF6 and O2 chemistry. We use focused ion beam (FIB) - scanning electron
microscope (SEM) to confirm the triangular cross-section of the fabricated
waveguides, as shown in the inset of Figure 4d. We measure the etch angle
($\alpha$) of these waveguides to be $30^{\circ}$. For adjacent waveguides
with gaps of 100 nm and 200 nm, we observe no substantial etch in the region
between the waveguides, as shown in Figure 4a-b. For waveguide gaps $\geq$ 300
nm (300 nm, 500 nm), the fabricated waveguides have triangular cross-sections,
as shown in Figure 4c-d. The lack of etching in the region between the
waveguides for smaller waveguide gaps may be due to the inability of the ions
which are incident at an angle on the substrate, to reach well below the
surface level of the substrate, and also reduced etching caused by mass-
transport issues in those closely spaced regions.

Figure 4: The SEM images of ion beam etched waveguides in 4H-SiC. (a)-(c) 800
nm wide waveguides with gaps of 100 nm, 200 nm and 300 nm respectively. (d)
1000 nm wide waveguides with a gap of 500 nm. Inset shows a FIB-SEM of the
fabricated 1000 nm waveguide with a triangular cross-section. The scale bar is
2 $\mu$m.

##  5 Discussion

The modeling results presented in this paper offer an approach to build low-
loss triangular cross-section 50:50 BS in 4H-SiC necessary to perform key
quantum interferometry operations for QIP applications. Here, triangular
cross-section photonics provide a scalable route to integrate color centers
into quantum photonic devices and circuits. Moreover, high-performance
triangular cross-section photonic structures in 4H-SiC like waveguides,
waveguides with integrated SNSPDs, photonic crystal mirrors, and photonic
crystal cavities that facilitate efficient generation, collection and
detection of single photon emission from color centers, necessary for
applications in QIP have been demonstrated [18, 24, 12, 25]. Thereby, NV
center in 4H-SiC with emission wavelengths near the telecommunications range
is most suited for building large-scale quantum communication networks.

Our initial fabrication tests show that triangular cross-section waveguides
with gaps greater than 300 nm can be fabricated using the wafer-scale ion beam
etch process. From the simulations, we note that for a waveguide gap $g\geq$
300 nm in the coupling region, $\alpha$ = $30^{\circ}$ and $45^{\circ}$, offer
50:50 BS with fabricatable footprints (e.g. coupling region $L_{C}\leq$ 50
$\mu$m), for applications in QIP. Here, supporting structures for waveguide
suspension would need to be designed. The ion beam etch process conditions
could be further optimized to achieve triangular cross-section waveguides with
waveguide gap $\leq$ 300 nm, by choosing process conditions that result in a
predominantly chemical etch. Under such etch conditions, the byproducts of the
etch are volatile gases (SiF4, CO, CO2), preventing re-deposition, allowing
the etch to continue. When there is a reasonable physical etch component, the
sputtered material has a higher chance of re-deposition, especially when the
gap between the waveguides is small, resulting in a slower etch or sometimes
no etch. Another alternative could be choosing a steeper etch angle, because
such waveguides would not require a deep etch to release the structure.

Triangular cross-section photonics in 4H-SiC provides an avenue to achieve
efficient chip- and wafer-scale integration of color centers, with very little
degradation of color center properties compared to bulk, essential for
applications in QIP. Our simulation results demonstrate that 50:50 BS
necessary for applications in QIP can be implemented using triangular geometry
and our initial etch tests show that ion beam etching is suitable for
achieving this.

##  6 Acknowledgments

M.R., S.M. and P.S. acknowledge support from NSF CAREER (Award 2047564) and
AFOSR Young Investigator Program (Award FA9550-23-1-0266). Z.K. acknowledges
support by CITRIS Workforce Innovation Program. Work at the Molecular Foundry
was supported by the Office of Science, Office of Basic Energy Sciences, of
the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Part of
this study was carried out at the UC Davis Center for Nano and Micro
Manufacturing (CNM2).

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# The numerical range of a periodic tridiagonal operator reduces to the
numerical range of a finite matrix

Benjamín A. Itzá-Ortiz  ,  Rubén A. Martínez-Avendaño  and  Hiroshi Nakazato
Centro de Investigación en Matemáticas, Universidad Autónoma del Estado de
Hidalgo, Pachuca, Hidalgo, Mexico  Departamento Académico de Matemáticas,
Instituto Tecnológico Autónomo de México, Mexico City, Mexico  Department of
Mathematics and Physics, Hirosaki University, Hirosaki City, Japan  Dedicated
to the memory of Rudolf Kippenhahn (1926–2020)

(Date: January 2021)

###### Abstract.

In this paper we show that the closure of the numerical range of an
$n+1$-periodic tridiagonal operator is equal to the numerical range of a
$2(n+1)\times 2(n+1)$ complex matrix.

The second author’s research is partially supported by the Asociación Mexicana
de Cultura A.C.

## Introduction

Consider $\mathcal{A}$ to be a finite set of complex numbers and let
$a=(a_{i})_{i\in{\mathbb{Z}}}$ be a biinfinite sequence in the total shift
space $\mathcal{A}^{\mathbb{Z}}$. In [13], the tridiagonal operator
$A_{a}\colon\ell^{2}({\mathbb{Z}})\to\ell^{2}({\mathbb{Z}})$ associated to $a$
is defined as

(1) $A_{a}=\begin{pmatrix}\ddots&\ddots&&&&&\\\ \ddots&0&1&&&&\\\
&a_{-2}&0&1&&&\\\ &&a_{-1}&\framebox[11.38092pt][l]{0}&1&&\\\ &&&a_{0}&0&1&\\\
&&&&a_{1}&0&\ddots\\\ &&&&&\ddots&\ddots\end{pmatrix}$

where the square marks the matrix entry at $(0,0)$. In the particular case of
the alphabet $\mathcal{A}=\\{-1,1\\}$, the corresponding operator $A_{a}$ is
related to the so called “hopping sign model” introduced in [7] and
subsequently studied in many other works, such as [1, 2, 3, 4, 5, 6, 9, 10,
13], just to name a few. On the other hand, when the alphabet is
$\mathcal{A}=\\{0,1\\}$ some results for computing the numerical range of
$A_{a}$ are presented in [13, 14]. In particular, work in [14] addresses the
case when $a$ is an $n+1$-periodic sequence. Relying on the fact that the
closure of the numerical range of $A_{a}$ may be written as the closure of the
convex hull of an uncountable union of numerical ranges of certain matrices,
in [14] the closure of the numerical range of the 2-periodic case is computed
by substituting such uncountable union of numerical ranges by the convex hull
of the union of the numerical ranges of just two $2\times 2$ matrices. In this
work, we further contribute to the study of the numerical range of $A_{a}$
when $a$ is an $n+1$ periodic biinfinite sequence.

Instead of working with the operators $A_{a}$, we work with the more general
tridiagonal operators $T=T(a,b,c)$ defined in Section 2, since, as can be seen
in [14], the computation of the closure of the numerical range of $A_{a}$ is a
particular case of that of $T$. Using a result of Plaumann and Vinzant [20],
we show that the closure of the numerical range of the $n+1$ periodic
tridiagonal operator $T$ is the numerical range of a $2(n+1)\times 2(n+1)$
matrix (cf. Theorem 2.6).

We divide this work in two sections. In Section 1 we briefly introduce the
notation and terminologies needed in the rest of the paper. In Section 2 we
develop the required machinery, first by computing the Kippenhahn polynomial
of the symbol of $n+1$ periodc tridiagonal operatos $T$ on
$\ell^{2}({\mathbb{N}}_{0})$ and then by combining our computations with
results of Plaumann and Vizant. We will conclude that the closure of the
numerical range of $T$ is equal to the numerical range of a $2(n+1)\times
2(n+1)$ matrix $A$. Furthermore, we provide some examples where $A$ can be
explicitly computed and we show that the size of $A$ is optimal.

##  1\. Preliminaries

In this section we introduce the notation required which will be needed in the
following sections. As usual, the symbols ${\mathbb{N}}$, ${\mathbb{N}}_{0}$,
${\mathbb{Z}}$, ${\mathbb{R}}$ and ${\mathbb{C}}$ will denote the set of
positive integers, the sets of nonnegative integers, the set of integers, the
set of real numbers and the set of complex numbers, respectively.

For a given $n\in{\mathbb{N}}$, let $a$, $b$ and $c$ be $(n+1)$-periodic
infinite sequences in $\mathcal{A}^{{\mathbb{N}}_{0}}$. We will denote by
$T=T(a,b,c)$ the $(n+1)$-periodic tridiagonal operator on
$\ell^{2}({\mathbb{N}}_{0})$ given by

$T=\setcounter{MaxMatrixCols}{11}\begin{pmatrix}b_{0}&c_{0}&&&&&&&&\\\
a_{1}&b_{1}&c_{1}&&&&&&&\\\ &a_{2}&b_{2}&c_{2}&&&&&&\\\
&&\ddots&\ddots&\ddots&&&&&\\\ &&&a_{n}&b_{n}&c_{n}&&&&\\\
&&&&a_{0}&b_{0}&c_{0}&&&\\\ &&&&&\ddots&\ddots&\ddots&&\\\
&&&&&&a_{n-1}&b_{n-1}&c_{n-1}&\\\ &&&&&&&a_{n}&b_{n}&c_{n}&\\\
&&&&&&&&\ddots&\ddots&\ddots\end{pmatrix}.$

We should observe that $T$ is a bounded operator since the sum of the moduli
of the entries in each column (and in each row) is uniformly bounded (see,
e.g., [16, Example 2.3]). The biinfinite matrix $A_{a}$ is also a bounded
operator, as long as the biinfinite sequence $a$ arises from a finite
alphabet.

If $n>1$, for each $\phi\in[0,2\pi)$, following [1, 14] we define the symbol
of $T$, as the following $(n+1)\times(n+1)$ matrix

(2) $T_{\phi}=\begin{pmatrix}b_{0}&c_{0}&0&&&0&a_{0}e^{-i\phi}\\\
a_{1}&b_{1}&c_{1}&0&&&0\\\ 0&a_{2}&b_{2}&c_{2}&0&&\\\
&\ddots&\ddots&\ddots&\ddots&\ddots&\\\ &&0&a_{n-2}&b_{n-2}&c_{n-2}&0\\\
0&&&0&a_{n-1}&b_{n-1}&c_{n-1}\\\
c_{n}e^{i\phi}&0&&&0&a_{n}&b_{n}\end{pmatrix};$

while the symbol of $T$ for $n=1$ is the $2\times 2$ matrix

(3) $T_{\phi}=\begin{pmatrix}b_{0}&c_{0}+a_{0}e^{-i\phi}\\\
a_{1}+c_{1}e^{i\phi}&b_{1}\end{pmatrix}.$

Recall that given a Hilbert space $\mathcal{H}$ and a bounded operator $A$ on
it, the numerical range is defined as the set

$W(A)=\\{\left<Ax,x\right>\,:\,\|x\|=1\\}.$

The Toeplitz-Haussdorf Theorem establishes that $W(A)$ is a bounded convex
subset of ${\mathbb{C}}$ (closed, if the Hilbert space is finite dimensional)
and hence the closure of the numerical range can be seen as the intersection
of the closed half-spaces containing the numerical range.

Kippenhahn [17] (see also [18]) characterized two vertical support lines of
$W(A)$ for a given $n\times n$ matrix as $\mathrm{Re}(z)=\lambda_{1}(A)$ and
$\mathrm{Re}(z)=\lambda_{n}(A)$, where $\lambda_{1}(A)$ and $\lambda_{n}(A)$
are the respective largest and least eigenvalues of $\mathrm{Re}(A)$ (recall
that $\mathrm{Re}(A):=\frac{1}{2}(A+A^{*})$ and
$\mathrm{Im}(A):=\frac{1}{2i}(A-A^{*})$). In fact, if $\alpha\in W(A)$ then
$\lambda_{n}(A)\leq\mathrm{Re}(\alpha)\leq\lambda_{1}(A)$ (and the equalities
hold for some points $\alpha_{1},\alpha_{2}\in W(A)$). Since
$e^{i\theta}W(A)=W(e^{i\theta}A)$ for each $\theta\in[0,2\pi)$, it follows
that if $\alpha\in W(A)$, then $e^{-i\theta}\alpha\in W(e^{-i\theta}A)$ and
hence $\mathrm{Re}(e^{-i\theta}\alpha)\leq\lambda_{1}(e^{-i\theta}A)$. It
follows that the lines $\mathrm{Re}(e^{-i\theta}z)=\lambda_{1}(e^{-i\theta}A)$
are support lines of $W(A)$. Hence the convex set $W(A)$ is uniquely
determined by the numbers $\lambda_{1}(e^{-i\theta}A)$, as $\theta$ varies on
the interval $[0,2\pi)$; i.e. $W(A)$ is determined by the largest eigenvalue
of $\mathrm{Re}(e^{-i\theta}A)$, which equals
$\cos(\theta)\mathrm{Re}(A)+\sin(\theta)\mathrm{Im}(A)$. Thus the numerical
range is determined by the largest roots of the family of characteristic
polynomials

$\det(tI_{n}-\cos(\theta)\mathrm{Re}(A)-\sin(\theta)\mathrm{Im}(A)).$

The homogeneous polynomial
$F_{A}(t,x,y)=\det(tI_{n}+x\mathrm{Re}(A)+y\mathrm{Im}(A))$ is called the
Kippenhahn polynomial of the matrix $A$. It clearly follows that two matrices
have the same numerical range if their Kippenhahn polynomials coincide.
Furthermore,

$\max\\{t\in{\mathbb{R}}\,:\,F_{A}(t,-\cos(\theta),-\sin(\theta))=0\\}=\max\\{\mathrm{Re}(e^{-i\theta}z)\,:\,z\in
W(A)\\}$

for each $\theta\in[0,2\pi)$.

##  2\. The Kippenhahn polynomial of the symbol $T_{\phi}$

In this section, after some preliminary work, we show that the closure of the
numerical range of a $n+1$-periodic tridiagonal operator $T$ is the numerical
range of a $2(n+1)\times 2(n+1)$ matrix.

We will need the following lemma.

######  Lemma 2.1.

Consider the $(n+1)\times(n+1)$ “almost tridiagonal” matrix

$\Lambda=\begin{pmatrix}\lambda_{1,1}&\lambda_{1,2}&0&0&\dots&0&0&\lambda_{1,n+1}\\\
\lambda_{2,1}&\lambda_{2,2}&\lambda_{2,3}&0&\dots&0&0&0\\\
0&\lambda_{3,2}&\lambda_{3,3}&\lambda_{3,4}&\dots&0&0&0\\\
0&0&\lambda_{4,3}&\lambda_{4,4}&\dots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&0&\dots&\lambda_{n-1,n-1}&\lambda_{n-1,n}&0\\\
0&0&0&0&\dots&\lambda_{n,n-1}&\lambda_{n,n}&\lambda_{n,n+1}\\\
\lambda_{n+1,1}&0&0&0&\dots&0&\lambda_{n+1,n}&\lambda_{n+1,n+1}\end{pmatrix},$

where every $\lambda_{i,j}\in{\mathbb{C}}$. Then, $\det(\Lambda)$ equals

$\begin{split}&\det\begin{pmatrix}\lambda_{1,1}&\lambda_{1,2}&0&\dots&0&0\\\
\lambda_{2,1}&\lambda_{2,2}&\lambda_{2,3}&\dots&0&0\\\
0&\lambda_{3,2}&\lambda_{3,3}&\dots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\dots&\lambda_{n-1,n}&0\\\
0&0&0&\dots&\lambda_{n,n}&\lambda_{n,n+1}\\\
0&0&0&\dots&\lambda_{n+1,n}&\lambda_{n+1,n+1}\end{pmatrix}-\lambda_{1,n+1}\lambda_{n+1,1}\det\begin{pmatrix}\lambda_{2,2}&\lambda_{2,3}&\dots&0&0\\\
\lambda_{3,2}&\lambda_{3,3}&\dots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\dots&\lambda_{n-1,n-1}&\lambda_{n-1,n}\\\
0&0&\dots&\lambda_{n,n-1}&\lambda_{n,n}\end{pmatrix}\\\ \\\
&+(-1)^{n}\lambda_{n+1,1}\lambda_{1,2}\lambda_{2,3}\cdots\lambda_{n-1,n}\lambda_{n,n+1}+(-1)^{n}\lambda_{1,n+1}\lambda_{2,1}\lambda_{3,2}\cdots\lambda_{n,n-1}\lambda_{n+1,n}.\end{split}$

###### Proof.

This follows by a long (but straightforward) application of the multilinearity
of the determinant function and the Laplace Expansion Theorem. ∎

Let us set the following notation for the rest of this paper. For $0\leq j<n$
we define

$\alpha_{j}=\frac{c_{j}+\overline{a_{j+1}}}{2},\quad\gamma_{j}=\frac{c_{j}-\overline{a_{j+1}}}{2i}$

and

$\alpha_{n}=\frac{a_{0}+\overline{c_{n}}}{2},\quad\gamma_{n}=\frac{a_{0}-\overline{c_{n}}}{2i}.$

We now find an expression for the Kippenhahn polynomial $F_{T_{\phi}}$ of the
symbol matrix $T_{\phi}$ of an arbitrary $n+1$-periodic tridiagonal matrix $T$
acting on $\ell^{2}({\mathbb{N}}_{0})$, involving the determinants of some
tridiagonal matrices. This expression will be useful in what follows.

######  Proposition 2.2.

Let $n\in{\mathbb{N}}$. Consider the symbol $T_{\phi}$, that is, the
$(n+1)\times(n+1)$ matrix defined as in (2) for $n\geq 2$ and as in (3) for
$n=1$. Then the Kippenhahn polynomial of $T_{\phi}$ is equal to

$\displaystyle F_{T_{\phi}}(t,x,y)={}$ $\displaystyle
G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)+2(-1)^{n}\
\mathrm{Re}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)\cos\phi$
$\displaystyle-2(-1)^{n}\
\mathrm{Im}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)\sin\phi,$

where $G_{n}(t,x,y)$ is the determinant of the tridiagonal $(n+1)\times(n+1)$
matrix

$\begin{pmatrix}\lambda_{1,1}&\lambda_{1,2}&0&0&\ldots&0&0\\\
\lambda_{2,1}&\lambda_{2,2}&\lambda_{2,3}&0&\cdots&0&0\\\
0&\lambda_{3,2}&\lambda_{3,3}&\lambda_{3,3}&\cdots&0&0\\\
0&0&\lambda_{4,3}&\lambda_{4,4}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\cdots&\lambda_{n,n}&\lambda_{n,n+1}\\\
0&0&0&0&\cdots&\lambda_{n+1,n}&\lambda_{n+1,n+1}\end{pmatrix},$

and, where we set $H_{n}(t,x,y)=1$ when $n=1$, and, for $n\geq 2$, we set
$H_{n}(t,x,y)$ to be the determinant of $(n-1)\times(n-1)$ tridiagonal matrix

$\begin{pmatrix}\lambda_{2,2}&\lambda_{2,3}&0&\cdots&0&0\\\
\lambda_{3,2}&\lambda_{3,3}&\lambda_{3,4}&\cdots&0&0\\\
0&\lambda_{4,3}&\lambda_{4,4}&\cdots&0&0\\\
\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\cdots&\lambda_{n-1,n-1}&\lambda_{n-1,n}\\\
0&0&0&\cdots&\lambda_{n,n-1}&\lambda_{n,n}\end{pmatrix}.$

Here we have set, for $1\leq j\leq n+1$,

$\lambda_{j,j}=t+\mathrm{Re}(b_{j-1})x+\mathrm{Im}(b_{j-1})y,$

and for $1\leq j\leq n$,

$\lambda_{j,j+1}=\alpha_{j-1}x+\gamma_{j-i}y\quad\text{
and}\quad\lambda_{j+1,j}=\overline{\alpha_{j-1}}x+\overline{\gamma_{j-1}}y.$

###### Proof.

We divide the proof in two cases. For $n+1=2$, by computing the real and
imaginary parts of the matrix $T_{\phi}$ in (3), we obtain that the $2\times
2$ matrix $tI_{2}+x\mathrm{Re}(T_{\phi})+y\mathrm{Im}(T_{\phi})$ is given by

$\begin{pmatrix}t+\mathrm{Re}(b_{0})x+\mathrm{Im}(b_{0})y&\alpha_{0}x+\gamma_{0}y+(\alpha_{1}x+\gamma_{1}y)e^{-i\phi}\cr(\overline{\alpha_{0}}x+\overline{\gamma_{0}}y)+(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)e^{i\phi}&t+\mathrm{Re}(b_{1})x+\mathrm{Im}(b_{1})y\end{pmatrix},$

where $\alpha_{0}$, $\alpha_{1}$, $\gamma_{0}$ and $\gamma_{1}$ are as defined
above. The determinant of this matrix can be simplified to

$\displaystyle F_{T_{\phi}}(t,x,y)={}$
$\displaystyle(t+\mathrm{Re}(b_{0})x+\mathrm{Im}(b_{0})y)(t+\mathrm{Re}(b_{1})x+\mathrm{Im}(b_{1})y)-|\alpha_{0}x+\gamma_{0}y|^{2}-|\alpha_{1}x+\gamma_{1}y|^{2}$
$\displaystyle\quad-2\mathrm{Re}\left((\alpha_{0}x+\gamma_{0}y)(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)e^{i\phi}\right)$
$\displaystyle={}$
$\displaystyle(t+\mathrm{Re}(b_{0})x+\mathrm{Im}(b_{0})y)(t+\mathrm{Re}(b_{1})x+\mathrm{Im}(b_{1})y)-|\alpha_{0}x+\gamma_{0}y|^{2}-|\alpha_{1}x+\gamma_{1}y|^{2}$
$\displaystyle\quad-2\mathrm{Re}\left((\alpha_{0}x+\gamma_{0}y)(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)\right)\cos\phi+2\mathrm{Im}\left((\alpha_{0}x+\gamma_{0}y)(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)\right)\sin\phi$
$\displaystyle={}$ $\displaystyle
G_{1}(t,x,y)-|\alpha_{1}x+\gamma_{1}t|^{2}H_{1}(t,x,y)$
$\displaystyle\quad-2\mathrm{Re}\left((\alpha_{0}x+\gamma_{0}y)(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)\right)\cos\phi+2\mathrm{Im}\left((\alpha_{0}x+\gamma_{0}y)(\overline{\alpha_{1}}x+\overline{\gamma_{1}}y)\right)\sin\phi,$

as desired.

Now, for the case $n+1\geq 3$, by computing the real and imaginary parts of
the matrix $T_{\phi}$ in (2), we can observe that
$tI_{n+1}+x\mathrm{Re}(T_{\phi})+y\mathrm{Im}(T_{\phi})$ is the matrix

$\begin{pmatrix}\lambda_{1,1}&\lambda_{1,2}&0&0&\ldots&0&\lambda_{1,n+1}\\\
\lambda_{2,1}&\lambda_{2,2}&\lambda_{2,3}&0&\cdots&0&0\\\
0&\lambda_{3,2}&\lambda_{3,3}&\lambda_{3,3}&\cdots&0&0\\\
0&0&\lambda_{4,3}&\lambda_{4,4}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\cdots&\lambda_{n,n}&\lambda_{n,n+1}\\\
\lambda_{n+1,1}&0&0&0&\cdots&\lambda_{n+1,n}&\lambda_{n+1,n+1}\end{pmatrix},$

where we have now set

$\lambda_{1,n+1}=(\alpha_{n}x+\gamma_{n}y)e^{-i\phi}\quad\text{and}\quad\lambda_{n+1,1}=(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y)e^{i\phi}.$

The above matrix is tridiagonal, except for the upper-right and bottom-left
corners.

We can compute the determinant of the matrix polynomial
$tI_{n+1}+x\mathrm{Re}(T_{\phi})+y\mathrm{Im}(T_{\phi})$ by using Lemma 2.1
obtaining

$\displaystyle F_{T_{\phi}}(t,x,y)={}$
$\displaystyle\det(tI_{n+1}+x\mathrm{Re}(T_{\phi})+y\mathrm{Im}(T_{\phi}))$
$\displaystyle={}$ $\displaystyle G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}\
H_{n}(t,x,y)$
$\displaystyle+(-1)^{n}(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y)\prod_{j=0}^{n-1}(\alpha_{j}x+\gamma_{j}y)e^{i\phi}+(-1)^{n}(\alpha_{n}x+\gamma_{n}y)\prod_{j=0}^{n-1}(\overline{\alpha_{j}}x+\overline{\gamma_{j}}y)e^{-i\phi}$
$\displaystyle={}$ $\displaystyle G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}\
H_{n}(t,x,y)+2(-1)^{n}\mathrm{Re}\left((\overline{\alpha_{n}}x+\overline{\gamma_{n}}y)\prod_{j=0}^{n-1}(\alpha_{j}x+\gamma_{j}y)e^{i\phi}\right).$

Computing the real part of the last term above, we obtain the equation

$\displaystyle F_{T_{\phi}}(t,x,y)={}$ $\displaystyle
G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)+2(-1)^{n}\
\mathrm{Re}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)\cos\phi$
$\displaystyle-2(-1)^{n}\
\mathrm{Im}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)\sin\phi,$

which completes the proof. ∎

For every $n\in{\mathbb{N}}$ and for a fixed point $(x,y)\in{\mathbb{R}}^{2}$,
the angle $\phi\in[0,2\pi)$ is involved only in the constant term (with
respect to the variable $t$) of the polynomial $F_{T_{\phi}}(t,x,y)$.
Furthermore, for every $(x,y)\in{\mathbb{R}}^{2}$ and for every
$\phi\in[0,2\pi)$, the polynomial $F_{T_{\phi}}(t,x,y)$, seen as a polynomial
in $t$, has $n+1$ real roots, counting multiplicities, as it is the
characteristic polynomial of the Hermitian matrix
$-x\mathrm{Re}(T_{\phi})-y\mathrm{Im}(T_{\phi})$. The following lemma will be
useful later when applied to the polynomial $F_{T_{\phi}}$.

######  Lemma 2.3.

Let $F(t:\phi)$ be a family of polynomials in ${\mathbb{R}}[t]$ given by the
expression

$F(t:\phi)=t^{n+1}+p_{n}t^{n}+\ldots+p_{1}t+p_{0}-u\cos\phi-v\sin\phi,$

where $\phi\in[0,2\pi)$. Assume that the polynomial $F(t:\phi)$ has $n+1$ real
roots counting multiplicities for any angle $\phi\in[0,2\pi)$. Let $\phi_{0}$,
$\phi_{1}\in[0,2\pi)$ be such that

$u\cos\phi_{0}+v\sin\phi_{0}=-\sqrt{u^{2}+v^{2}}\quad\text{ and }\quad
u\cos\phi_{1}+v\sin\phi_{1}=\sqrt{u^{2}+v^{2}}.$

Then

$\max\left\\{\max\left\\{t\in{\mathbb{R}}\,:\,F(t:\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t\in{\mathbb{R}}\,:\,F(t:\phi_{1})=0\right\\},$

and

$\min\left\\{\max\left\\{t\in{\mathbb{R}}\,:\,F(t:\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t\in{\mathbb{R}}\,:\,F(t:\phi_{0})=0\right\\}.$

###### Proof.

Define $p(t)$ as

$p(t)=t^{n+1}+p_{n}t^{n}+\ldots+p_{1}t+p_{0}.$

Observe that, by assumption, the equation

$p(t)=u\cos\phi+v\sin\phi$

has $n+1$ real solutions (counting multiplicities) for every
$\phi\in[0,2\pi)$. For some $\phi\in[0,2\pi)$, we have
$u\cos\phi+v\sin\phi=0$, and hence $p$ has $n+1$ real roots (counting
multiplicities) and the derivative of $p$ has $n$ real roots (counting
multiplicities). Let $r_{0}$ be the largest root of $p^{\prime}(t)$. Hence,
$p$ is increasing on the interval $[r_{0},\infty)$ and the equations

$p(t)=u\cos\phi+v\sin\phi$

have a unique solution on the interval $[r_{0},\infty)$.

Observe that for every $\phi\in[0,2\pi)$

$-\sqrt{u^{2}+v^{2}}\leq u\cos\phi+v\sin\phi\leq\sqrt{u^{2}+v^{2}};$

equality occurs on the left-hand-side inequality at $\phi_{0}$ while equality
occurs on the right-hand-side inequality at $\phi_{1}$.

For each $\phi\in[0,2\pi)$, consider the number

$\max\\{t\in{\mathbb{R}}\,:\,p(t)=u\cos\phi+v\sin\phi\\}.$

Since the function $p$ is increasing on $[r_{0},\infty)$, the largest of these
numbers, when $\phi$ varies, occurs when $t$ is the largest solution of the
equation

$p(t)=\sqrt{u^{2}+v^{2}}.$

Hence we have

$\max\left\\{\max\left\\{t\in{\mathbb{R}}\,:\,F(t,\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t\in{\mathbb{R}}\,:\,F(t,\phi_{1})=0\right\\}.$

Analogously, the smallest, when $\phi$ varies in $[0,2\pi)$, among the largest
solutions $t$ of the equations

$p(t)=u\cos\phi+v\sin\phi,$

occurs when $t$ is the largest solution of the equation

$p(t)=-\sqrt{u^{2}+v^{2}}.$

Hence we have

$\min\left\\{\max\left\\{t\in{\mathbb{R}}\,:\,F(t,\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t\in{\mathbb{R}}\,:\,F(t,\phi_{0})=0\right\\}.\qed$

In Theorem 2.7, we will show that the closure of the numerical range of $T$ is
the numerical range of a single matrix. One of the key steps in the proof of
said theorem will be to use the following proposition, which computes the
closure of the numerical range of $T$ by using a single homogeneous
polynomial, instead of the uncountable number of Kippenhahn polynomials of the
symbols $T_{\phi}$, which Theorem 2.8 in [14] would suggest: this is achieved
by getting rid of the parameter $\phi$ in the expression of the Kippenhahn
polynomial of the symbol $T_{\phi}$ in Proposition 2.2.

######  Proposition 2.4.

Let $n\in{\mathbb{N}}$. Suppose that $T(a,b,c)$ is an $n+1$-periodic
tridiagonal operator acting on $\ell^{2}({\mathbb{N}}_{0})$. Let $G_{n}$ and
$H_{n}$ be as in Proposition 2.2 and let $P$ be the real homogeneous
polynomial of degree $2(n+1)$ given by

$P(t,x,y)=\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)\right)^{2}-4\prod_{j=0}^{n}\left|\alpha_{j}x+\gamma_{j}y\right|^{2}.$

Then $P(t,0,0)=t^{2(n+1)}$ and

$\sup\left\\{\mathrm{Re}(e^{-i\theta}z)\colon\right.\bigl{.}z\in
W\left(T(a,b,c)\right)\bigr{\\}}=\max\\{t\in{\mathbb{R}}\colon
P(t,-\cos\theta,-\sin\theta)=0\\},$

for each $\theta\in[0,2\pi)$.

###### Proof.

It is trivial to check that $P(t,0,0)=t^{2(n+1)}$. Now, let
$F(t:\phi)=F_{T_{\phi}}(t,x,y)$, where we know by Proposition 2.2 that

$F_{T_{\phi}}(t,x,y)=G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)-u\cos\phi-v\sin\phi,$

where

$u=-2(-1)^{n}\
\mathrm{Re}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)$

and

$v=2(-1)^{n}\
\mathrm{Im}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right).$

Notice that

$\displaystyle u^{2}+v^{2}={}$ $\displaystyle
4\mathrm{Re}^{2}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)+4\mathrm{Im}^{2}\left(\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right)$
$\displaystyle={}$ $\displaystyle
4\left|\left(\overline{\alpha_{n}}x+\overline{\gamma_{n}}y\right){\prod_{j=0}^{n-1}}(\alpha_{j}x+\gamma_{j}y)\right|^{2}$
$\displaystyle=$
$\displaystyle{}4\prod_{j=0}^{n}\left|\alpha_{j}x+\gamma_{j}y\right|^{2}.$

The polynomial $F(t:\phi)$ has the form outlined in Lemma 2.3 and, as was
mentioned before Lemma 2.3, it has $n+1$ real roots, counting multiplicities.
Hence, by Lemma 2.3, for $\phi_{0}$ and $\phi_{1}$ satisfying

$u\cos(\phi_{0})+v\sin(\phi_{0})=-\sqrt{u^{2}+v^{2}},\qquad
u\cos(\phi_{1})+v\sin(\phi_{1})=\sqrt{u^{2}+v^{2}},$

we have that

$\max\left\\{\max\left\\{t:F(t:\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t:F(t:\phi_{1})=0\right\\},$

and

$\min\left\\{\max\left\\{t:F(t:\phi)=0\right\\}:0\leq\phi<2\pi\right\\}=\max\left\\{t:F(t:\phi_{0})=0\right\\}.$

Notice that

$\displaystyle F(t:\phi_{0})\cdot F(t:\phi_{1})={}$
$\displaystyle\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)-\left(u\cos(\phi_{0})+v\sin(\phi_{0})\right)\right)$
$\displaystyle\cdot\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)-\left(u\cos(\phi_{1})+v\sin(\phi_{1})\right)\right)$
$\displaystyle={}$
$\displaystyle\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)\right)^{2}-\left(\sqrt{u^{2}+v^{2}}\right)^{2}$
$\displaystyle={}$
$\displaystyle\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)\right)^{2}-(u^{2}+v^{2})$
$\displaystyle={}$ $\displaystyle P(t,x,y).$

We also have, for each $\theta\in[0,2\pi)$, that

(4) $\begin{split}\max\\{t\in{\mathbb{R}}\colon
F_{T_{\phi}}(t,-\cos\theta,&-\sin\theta)=0,\ 0\leq\phi<2\pi\\}\\\
={}&\max\left\\{\max\left\\{t\in{\mathbb{R}}\colon
F_{T_{\phi}}(t,-\cos\theta,-\sin\theta)=0\right\\}\colon\
0\leq\phi<2\pi\right\\}\\\ ={}&\max\left\\{t\in{\mathbb{R}}\colon
F_{T_{\phi_{1}}}(t,-\cos\theta,-\sin\theta)=0\right\\}\\\
={}&\max\\{t\in{\mathbb{R}}\colon
P(t,-\cos\theta,-\sin\theta)=0\\}.\end{split}$

The last equality follows since the roots of $P(t,-\cos\theta,-\sin\theta)$
are those of $F(t:\phi_{1})=F_{T_{\phi_{1}}}(t,-\cos\theta,-\sin\theta)$ and
$F(t:\phi_{0})=F_{T_{\phi_{0}}}(t,-\cos\theta,-\sin\theta)$, so by the choice
of $\phi_{0}$ and $\phi_{1}$, the largest root of
$P(t,-\cos\theta,-\sin\theta)$ is the largest root of
$F_{T_{\phi_{1}}}(t,-\cos\theta,-\sin\theta)$.

By the definition of the Kippenhahn polynomial, we have

$\max\left\\{t\in{\mathbb{R}}\colon
F_{T_{\phi}}(t,-\cos(\theta),-\sin(\theta))=0\right\\}=\max\left\\{\mathrm{Re}(e^{-i\theta}z):z\in
W(T_{\phi})\right\\}.$

and hence we obtain

(5)
$\begin{split}\max\left\\{t\in{\mathbb{R}}\,:\,F_{T_{\phi}}(t,-\cos(\theta),-\sin(\theta))=0,\
0\leq\phi<2\pi\right\\}\\\ =\max\left\\{\mathrm{Re}(e^{-i\theta}z)\,:\,z\in
W(T_{\phi}),\ 0\leq\phi<2\pi\right\\}.\end{split}$

Lastly, the equality

(6) $\sup\left\\{\mathrm{Re}(e^{-i\theta}z)\,:\,z\in
W\left(T(a,b,c)\right)\right\\}=\max\left\\{\mathrm{Re}(e^{-i\theta}z)\,:\,z\in
W\left(T_{\phi}\right),\ 0\leq\phi<2\pi\right\\}$

follows from Theorem 2.8 in [14]. Putting together equations (4), (5) and (6),
we obtain the desired conclusion. ∎

The following definition will be useful.

######  Definition 2.5.

Suppose that $Q(t,x,y)$ is a real homogeneous polynomial in $3$ variables
$t,x,y$ of degree $m$ with $Q(1,0,0)>0$. If the equation $Q(t,x_{0},y_{0})=0$
in $t$ has $m$ real solutions counting multiplicities for any
$(x_{0},y_{0})\in{\mathbb{R}}^{2}$ with $x_{0}^{2}+y_{0}^{2}>0$, we say that
$Q$ is hyperbolic (with respect to $(1,0,0)$).

The above condition may also be formulated as: “the equation
$Q(t,-\cos\theta,-\sin\theta)=0$ in $t$ has $m$ real solutions for any angle
$0\leq\theta<2\pi$”.

######  Theorem 2.6 (Plaumann and Vinzant [20]).

Suppose that $Q(t,x,y)$ is a real homogeneous hyperbolic polynomial of degree
$m$ with $Q(1,0,0)=1$. Then there exists an $m\times m$ complex matrix $A$
satisfying

$Q(t,x,y)={\rm det}(tI_{m}+x{\rm Re}(A)+y{\rm Im}(A)).$

Remark. Helton and Vinnikov [12] (cf. [11]) proved a result stronger than the
above theorem which guarantees that we can construct an $m\times m$ complex
symmetric matrix $A$ satisfying a similar property. In this paper we do not
use the symmetry of the matrix $A$.

Depending on the above Theorem 2.6, we obtain the main theorem of this paper.

######  Theorem 2.7.

Suppose that $T(a,b,c)$ is an $n+1$-periodic tridiagonal operator acting on
$\ell^{2}({\mathbb{N}}_{0})$. Then there exists a $2(n+1)\times 2(n+1)$
complex matrix $A$ such that

$\overline{W(T(a,b,c))}=W(A)$

where the matrix $A$ is chosen so that it satisfies

$F_{A}(t,x,y)=\left(G_{n}(t,x,y)-|\alpha_{n}x+\gamma_{n}y|^{2}H_{n}(t,x,y)\right)^{2}-4\prod_{j=0}^{n}\left|\alpha_{j}x+\gamma_{j}y\right|^{2},$

where the polynomials $G_{n}$ and $H_{n}$ are as in Proposition 2.2.

###### Proof.

By Theorem 2.6, there exists a $2(n+1)\times 2(n+1)$ matrix $A$ such that
$P(t,x,y)=F_{A}(t,x,y)$, where $P$ is the homogeneous polynomial in
Proposition 2.4. But also, by the same proposition,

$\displaystyle\sup\left\\{\mathrm{Re}(e^{-i\theta}z)\colon z\in
W\left(T(a,b,c)\right)\right\\}={}$
$\displaystyle\max\\{t\in{\mathbb{R}}\colon
F_{A}(t,-\cos\theta,-\sin\theta)=0\\}$ $\displaystyle={}$
$\displaystyle\max\left\\{\mathrm{Re}(e^{-i\theta}z)\colon z\in W(A)\right\\}$

for each $\theta\in[0,2\pi)$, and hence the closure of the numerical range of
$T(a,b,c)$ equals the numerical range of $A$. ∎

It is clear that given the operator $T$, one can compute the polynomial $P$
which, by the Plaumann-Vinzant Theorem, is the Kippenhahn polynomial of some
matrix $A$. The question arises on whether the matrix $A$ can be explicitly
computed. The paper [20] shows a method for constructing such a matrix $A$
(see also [12, 19]).

In some cases, the matrix $A$ can be found explicitly, as the next proposition
shows. The reader should compare our next result to Theorem 4.1 in [1], where
an alternative method for computing the numerical range of the tridiagonal
operator $T(a,b,c)$ is obtained, when $a$, $b$ and $c$ are real $2$-periodic
sequences.

######  Proposition 2.8.

Let $a$ and $c$ be real $2$-periodic sequences and let $b$ the constant $0$
sequence. If

$S=\begin{pmatrix}\alpha_{0}+\alpha_{1}&-\gamma_{0}&-\gamma_{1}&0\\\
-\gamma_{0}&-\alpha_{0}+\alpha_{1}&0&-\gamma_{1}\\\
-\gamma_{1}&0&\alpha_{0}-\alpha_{1}&-\gamma_{0}\\\
0&-\gamma_{1}&-\gamma_{0}&-\alpha_{0}-\alpha_{1}\end{pmatrix}$

then $\overline{W(T(a,b,c))}=W(S)$.

###### Proof.

It is a straightforward computation that the polynomial $P$ in Proposition 2.4
equals

$P(t,x,y)=\left(t^{2}-|\alpha_{0}x+\gamma_{0}y|^{2}-|\alpha_{1}x+\gamma_{1}y|^{2}\right)^{2}-4|\alpha_{0}x+\gamma_{0}y|^{2}|\alpha_{1}x+\gamma_{1}y|^{2}$

But a computation also shows that $F_{S}(t,x,y)=P(t,x,y)$ and hence, by
Theorem 2.7, we have $\overline{W(T(a,b,c))}=W(S)$. ∎

We illustrate the above proposition with some examples.

######  Example 2.9.

Let $a$ be the $2$-periodic sequence with period word $1\ 3$, let $b$ be the
constant $0$ sequence and let $c$ be the $2$-periodic sequence with period
word $4\ 8$. Then, by Proposition 2.8, if

$S=\begin{pmatrix}8&\frac{1}{2}i&-\frac{7}{2}i&0\\\\[5.0pt]
\frac{1}{2}i&1&0&-\frac{7}{2}i\\\\[5.0pt]
-\frac{7}{2}i&0&-1&\frac{1}{2}i\\\\[5.0pt]
0&-\frac{7}{2}i&\frac{1}{2}i&-8\end{pmatrix},$

then $\overline{W(T(a,b,c))}=W(S)$. The boundary of the numerical range of $S$
is shown in Figure 1. The Kippenhahn polynomial of $S$ equals

$P(t,x,y)=t^{4}-65t^{2}x^{2}-25t^{2}y^{2}+64x^{4}+192x^{2}y^{2}+144y^{4}.$

The quartic curve $P(t,x,y)=0$ in the complex projective plane has a pair of
ordinary singular points of multiplicity $2$ at $(t,x,y)=(0,1,\pm
i\sqrt{2/3})$ and there is no other singular point. So the algebraic curve
theory tell us that the homogeneous polynomial $P(t,x,y)$ is irreducible in
the polynomial ring.

Figure 1. Boundary of the numerical range of $S$

Hence, using for example Proposition 2.3 in [8], there cannot be a matrix $R$
of size $m\times m$, with $1\leq m<4$ with $W(R)=W(S)$. Incidentally, this
shows that the size of the matrix $A$ in Theorem 2.7 is optimal.

######  Example 2.10.

Let $a$ and $c$ be real $2$-periodic sequences with period words $a_{0}a_{1}$
and $c_{0}c_{1}$ respectively, and let $b$ be the constant sequence $0$. If
$a_{0}=c_{1}$, then $\gamma_{1}=0$ and then, by Proposition 2.8,
$\overline{W(T(a,b,c))}=W(S)$, where

$S=\begin{pmatrix}\alpha_{0}+\alpha_{1}&-\gamma_{0}&0&0\\\
-\gamma_{0}&-\alpha_{0}+\alpha_{1}&0&0\\\
0&0&\alpha_{0}-\alpha_{1}&-\gamma_{0}\\\
0&0&-\gamma_{0}&-\alpha_{0}-\alpha_{1}\end{pmatrix}.$

But this implies that

$\overline{W(T(a,b,c))}=\mathrm{conv}\left(W(A+\alpha_{1}I)\cup
W(A-\alpha_{1}I)\right),$

where

$A=\begin{pmatrix}\alpha_{0}&-\gamma_{0}\\\
-\gamma_{0}&-\alpha_{0}\end{pmatrix}.$

That is, $\overline{W(T(a,b,c))}$ is the convex hull of two ellipses (possibly
degenerate), each one a translation of a single elllipse (possibly degenarate)
centered at the origin.

######  Example 2.11.

Let $a$ be the $2$-periodic sequence with period word $1,-1$, let $b$ be the
constant $0$ sequence and let $c$ be the constant $1$ sequence. Then, by
Example 2.10, we have that
$\overline{W(T(a,b,c))}=\mathrm{conv}\left(W(A+I)\cup W(A-I)\right)$, where

$A=\begin{pmatrix}0&i\\\\[5.0pt] i&0\end{pmatrix}.$

But it is easy to see that $W(A)$ is the closed line segment joining $-i$ and
$i$. Hence, $\overline{W(T(a,b,c))}$ equals the convex hull of the closed line
segment joining $-1-i$ and $-1+i$ and the closed line segment joining $1-i$
and $1+i$; i.e., the square with vertices $-1-i$, $-1+i$, $1-i$ and $1+i$,
recovering (most of) Theorem 9 in [5].

######  Example 2.12.

Let $a$ and $c$ be real $2$-periodic sequences with period words $a_{0}a_{1}$
and $c_{0}c_{1}$ respectively, and let $b$ be the constant sequence $0$. If
$c_{0}=a_{1}$, then $\gamma_{0}=0$ and then, by Proposition 2.8,
$\overline{W(T(a,b,c))}=W(S)$, where

$S=\begin{pmatrix}\alpha_{0}+\alpha_{1}&0&-\gamma_{1}&0\\\
0&-\alpha_{0}+\alpha_{1}&0&-\gamma_{1}\\\
-\gamma_{1}&0&\alpha_{0}-\alpha_{1}&0\\\
0&-\gamma_{1}&0&-\alpha_{0}-\alpha_{1}\end{pmatrix}.$

But if

$U=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{pmatrix},$

then

$U^{*}SU=\begin{pmatrix}\alpha_{0}+\alpha_{1}&-\gamma_{1}&0&0\\\
-\gamma_{1}&\alpha_{0}-\alpha_{1}&0&0\\\
0&0&-\alpha_{0}+\alpha_{1}&-\gamma_{1}\\\
0&0&-\gamma_{1}&-\alpha_{0}-\alpha_{1}\end{pmatrix}.$

But this implies that

$\overline{W(T(a,b,c))}=\mathrm{conv}\left(W(A+\alpha_{0}I)\cup
W(A-\alpha_{0}I)\right),$

where

$A=\begin{pmatrix}\alpha_{1}&-\gamma_{1}\\\
-\gamma_{1}&-\alpha_{1}\end{pmatrix}.$

That is, $\overline{W(T(a,b,c))}$ is the convex hull of two ellipses (possibly
degenerate), each one a translation of a single elllipse (possibly degenarate)
centered at the origin.

######  Example 2.13.

Let $a$ be the $2$-periodic sequence with period word $01$, let $b$ be the
constant $0$ sequence and let $c$ be the constant $1$ sequence. Then, by
Example 2.12, we have that
$\overline{W(T(a,b,c))}=\mathrm{conv}\left(W(A+I)\cup W(A-I)\right)$, where

$A=\begin{pmatrix}\frac{1}{2}&-\frac{1}{2}i\\\\[5.0pt]
-\frac{1}{2}i&-\frac{1}{2}\end{pmatrix}.$

But, if

$U=\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\\[10.0pt]
-\frac{1}{\sqrt{2}}i&\frac{1}{\sqrt{2}}i\end{pmatrix},$

then $U$ is unitary and $U^{*}AU$ equals

$\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$

Therefore, $\overline{W(T(a,b,c))}=\mathrm{conv}\left(W(B+I)\cup
W(B-I)\right)$, recovering the result in [14, Theorem 3.6].

In the paper [15] we explore some sufficient conditions under which the matrix
$A$ can be explicitly found, namely if $b=0$ and there is some symmetry in the
periodic sequences $a$ and $c$, then the polynomial $P$ can be factored as the
product of the Kippenhahn polynomials of two computable matrices, which
generalizes the previous four examples.

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