# An Elementary Predictor Obtaining $2\sqrt{T}$ Distance to Calibration

Eshwar Ram Arunachaleswaran  Department of Computer and Information Sciences,
University of Pennsylvania  Natalie Collina  Department of Computer and
Information Sciences, University of Pennsylvania  Aaron Roth  Department of
Computer and Information Sciences, University of Pennsylvania  Mirah Shi
Department of Computer and Information Sciences, University of Pennsylvania

###### Abstract

Błasiok et al. (2023) proposed _distance to calibration_ as a natural measure
of calibration error that unlike expected calibration error (ECE) is
continuous. Recently, Qiao and Zheng (2024) gave a non-constructive argument
establishing the existence of a randomized online predictor that can obtain
$O(\sqrt{T})$ distance to calibration in expectation in the adversarial
setting, which is known to be impossible for ECE. They leave as an open
problem finding an explicit, efficient, deterministic algorithm. We resolve
this problem and give an extremely simple, efficient, deterministic algorithm
that obtains distance to calibration error at most $2\sqrt{T}$.

##  1 Introduction

Probabilistic predictions of binary outcomes are said to be _calibrated_ , if,
informally, they are unbiased conditional on their own predictions. For
predictors that are not perfectly calibrated, there are a variety of ways to
measure calibration error. Perhaps the most popular measure is Expected
Calibration Error (ECE), which measures the average bias of the predictions,
weighted by the frequency of the predictions. ECE has a number of difficulties
as a measure of calibration, not least of which is that it is discontinuous in
the predictions. Motivated by this, Błasiok et al. (2023) propose a different
measure: distance to calibration, which measures how far a predictor is in
$\ell_{1}$ distance from the nearest perfectly calibrated predictor. In the
online adversarial setting, it has been known since Foster and Vohra (1998)
how to make predictions with ECE growing at a rate of $O(T^{2/3})$. Qiao and
Valiant (2021) show that obtaining $O(\sqrt{T})$ rates for ECE is impossible.
Recently, Qiao and Zheng (2024) showed that it was possible to make sequential
predictions against an adversary guaranteeing expected distance to calibration
growing at a rate of $O(\sqrt{T})$. Their algorithm is the solution to a
minimax problem of size doubly-exponential in $T$. They leave as an open
problem finding an explicit, efficient, deterministic algorithm for this
problem. In this note we resolve this problem, by giving an extremely simple
such algorithm with an elementary analysis.

Input: Sequence of outcomes $y^{1:T}\in\\{0,1\\}^{T}$

Output: Sequence of predictions $p^{1:T}\in\\{0,\frac{1}{m},...,1\\}^{T}$ for
some discretization parameter $m>0$

for  _$t=1$ to $T$_ do

Given look-ahead predictions $\tilde{p}^{1:t-1}$, define the look-ahead bias
conditional on a prediction $p$ as:
$\alpha_{\tilde{p}^{1:t-1}}(p):=\sum_{s=1}^{t-1}\mathbbm{1}[\tilde{p}^{s}=p](\tilde{p}^{s}-y^{s})$
Choose two adjacent points $p_{i}=\frac{i}{m},p_{i+1}=\frac{i+1}{m}$
satisfying:  $\alpha_{\tilde{p}^{1:t-1}}(p_{i})\leq 0\text{ and
}\alpha_{\tilde{p}^{1:t-1}}(p_{i+1})\geq 0$ Arbitrarily predict
${p}^{t}=p_{i}$ or ${p}^{t}=p_{i+1}$;

Upon observing the (adversarially chosen) outcome $y^{t}$, set look-ahead
prediction
$\tilde{p}^{t}=\operatorname{\text{argmin}}_{p\in\\{p_{i},p_{i+1}\\}}|p-y^{t}|$

Algorithm 1 Almost-One-Step-Ahead

##  2 Setting

We study a sequential binary prediction setting: at every round $t$, a
forecaster makes a prediction $p^{t}\in[0,1]$, after which an adversary
reveals an outcome $y^{t}\in\\{0,1\\}$. Given a sequence of predictions
$p^{1:T}$ and outcomes $y^{1:T}$, we measure expected calibration error (ECE)
as follows:

$ECE(p^{1:T},y^{1:T})=\sum_{p\in[0,1]}\left|\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p](p^{t}-y^{t})\right|$

Following Qiao and Zheng (2024), we define distance to calibration to be the
minimum $\ell_{1}$ distance between a sequence of predictions produced by a
forecaster and any perfectly calibrated sequence of predictions:

$CalDist(p^{1:T},y^{1:T})=\min_{q^{1:T}\in\mathcal{C}(y^{1:T})}\|p^{1:T}-q^{1:T}\|_{1}$

where $\mathcal{C}(y^{1:T})=\\{q^{1:T}:ECE(q^{1:T},y^{1:T})=0\\}$ is the set
of predictions that are perfectly calibrated against outcomes $y^{1:T}$. First
we observe that distance to calibration is upper bounded by ECE.

######  Lemma 1 (Qiao and Zheng (2024)).

Fix a sequence of predictions $p^{1:T}$ and outcomes $y^{1:T}$. Then,
$CalDist(p^{1:T},y^{1:T})\leq ECE(p^{1:T},y^{1:T})$.

###### Proof.

For any prediction $p\in[0,1]$, define

$\overline{y}^{T}(p)=\sum_{t=1}^{T}\frac{\mathbbm{1}[p^{t}=p]}{\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p]}y^{t}$

to be the average outcome conditioned on the prediction $p$. Consider the
sequence $q^{1:T}$ where $q^{t}=\overline{y}^{T}(p^{t})$. Observe that
$q^{1:T}$ is perfectly calibrated. Thus, we have that

$\displaystyle CalDist(p^{1:T},y^{1:T})$
$\displaystyle\leq\|p^{1:T}-q^{1:T}\|_{1}$
$\displaystyle=\sum_{t=1}^{T}|p^{t}-q^{t}|$
$\displaystyle=\sum_{p\in[0,1]}\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p]|p-\overline{y}^{T}(p)|$
$\displaystyle=\sum_{p\in[0,1]}|p-\overline{y}^{T}(p)|\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p]$
$\displaystyle=\sum_{p\in[0,1]}\left|p\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p]-\overline{y}^{T}(p)\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p]\right|$
$\displaystyle=\sum_{p\in[0,1]}\left|\sum_{t=1}^{T}\mathbbm{1}[p^{t}=p](p-y^{t})\right|$
$\displaystyle=ECE(p^{1:T},y^{1:T})$

∎

The upper bound is not tight, however. The best known sequential prediction
algorithm obtains ECE bounded by $O(T^{\frac{2}{3}})$ (Foster and Vohra,
1998), and it is known that there is no algorithm guaranteeing ECE below
$O(T^{0.528})$ (Qiao and Valiant, 2021). Qiao and Zheng (2024) give an
algorithm that is the solution to a game of size doubly-exponential in $T$
that obtains expected distance to calibration $O(\sqrt{T})$. Here we give an
elementary analysis of a simple efficient deterministic algorithm (Algorithm
1) that obtains distance to calibration $2\sqrt{T}$.

######  Theorem 1.

Algorithm 1 (Almost-One-Step-Ahead) guarantees that against any sequence of
outcomes, $CalDist(p^{1:T},y^{1:T})\leq 2\sqrt{T}$.

##  3 Analysis of Algorithm 1

Before describing the algorithm, we introduce some notation. We will make
predictions that belong to a grid. Let $B_{m}=\\{0,1/m,...,1\\}$ denote a
discretization of the prediction space with discretization parameter $m>0$,
and let $p_{i}=i/m$. For a sequence of predictions
$\tilde{p}^{1},...,\tilde{p}^{t}$ and outcomes $y^{1},...,y^{t}$, we define
the bias conditional on a prediction $p$ as:

$\alpha_{\tilde{p}^{1:t}}(p)=\sum_{s=1}^{t}\mathbbm{1}[\tilde{p}^{s}=p](\tilde{p}^{s}-y^{s})$

To understand our algorithm, it will be helpful to first state and analyze a
hypothetical “lookahead” algorithm that we call “One-Step-Ahead”, which is
closely related to the algorithm and analysis given by Gupta and Ramdas (2022)
in a different model. One-Step-Ahead produces predictions
$\tilde{p}^{1},...,\tilde{p}^{T}$ as follows. At round $t$, before observing
$y^{t}$, the algorithm fixes two predictions $p_{i},p_{i+1}$ satisfying
$\alpha_{\tilde{p}^{1:t-1}}(p_{i})\leq 0$ and
$\alpha_{\tilde{p}^{1:t-1}}(p_{i+1})\geq 0$. Such a pair is guaranteed to
exist, because by construction, it must be that for any history,
$\alpha_{\tilde{p}^{1:t-1}}(0)\leq 0$ and $\alpha_{\tilde{p}^{1:t-1}}(1)\geq
0$. Note that a well known randomized algorithm obtaining diminishing ECE (and
smooth calibration error) uses the same observation to carefully _randomize_
between two such adjacent predictions (Foster, 1999; Foster and Hart, 2018).
Upon observing the outcome $y^{t}$, the algorithm outputs prediction
$\tilde{p}^{t}=\operatorname{\text{argmin}}_{p\in\\{p_{i},p_{i+1}\\}}|p-y^{t}|$.
Naturally, we cannot implement this algorithm, as it chooses its prediction
only after observing the outcome, but our analysis will rely on a key property
this algorithm maintains—namely, that it always produces a sequence of
predictions with ECE upper bounded by $m$, the number of elements in the
discretized prediction space.

######  Theorem 2.

For any sequence of outcomes, One-Step-Ahead achieves
$ECE(\tilde{p}^{1:T},y^{1:T})\leq m$.

###### Proof.

We will show that for any $p_{i}\in B_{m}$, we have
$|\alpha_{\tilde{p}^{1:T}}(p_{i})|\leq 1$, after which the bound on ECE will
follow: $ECE(\tilde{p}^{1:T},y^{1:T})=\sum_{p_{i}\in
B_{m}}|\alpha_{\tilde{p}^{1:T}}(p_{i})|\leq m$. We proceed via an inductive
argument. Fix a prediction $p_{i}\in B_{m}$. At the first round $t_{1}$ in
which $p_{i}$ is output by the algorithm, we have that
$|\alpha_{\tilde{p}^{1:t_{1}}}(p_{i})|=|p^{t_{1}}-y^{t_{1}}|\leq 1$. Now
suppose after round $t-1$, we satisfy $|\alpha_{\tilde{p}^{1:t-1}}(p_{i})|\leq
1$. If $p_{i}$ is the prediction made at round $t$, it must be that either:
$\alpha_{\tilde{p}^{1:t-1}}(p_{i})\leq 0$ and $p_{i}-y^{t}\geq 0$; or
$\alpha_{\tilde{p}^{1:t-1}}(p_{i})\geq 0$ and $p_{i}-y^{t}\leq 0$. Thus, since
$\alpha_{\tilde{p}^{1:t-1}}(p_{i})$ and $p_{i}-y^{t}$ either take value 0 or
differ in sign, we can conclude that

$|\alpha_{\tilde{p}^{1:t}}(p_{i})|=|\alpha_{\tilde{p}^{1:t-1}}(p_{i})+p_{i}-y^{t}|\leq\max\\{|\alpha_{\tilde{p}^{1:t-1}}(p_{i})|,|p_{i}-y^{t}|\\}\leq
1$

which proves the theorem. ∎

Algorithm 1 (Almost-One-Step-Ahead) maintains the same state
$\alpha_{\tilde{p}^{1:t}}(p)$ as One-Step-Ahead (which it can compute at round
$t$ after observing the outcome $y_{t-1}$). In particular, it does not keep
track of the bias of its own predictions, but rather keeps track of the bias
of the predictions that One-Step-Ahead _would have made_. Thus it can
determine the pair $p_{i},p_{i+1}$ that One-Step-Ahead would commit to predict
at round $t$. It cannot make the same prediction as One-Step-Ahead (as it must
fix its prediction before the label is observed) — so instead it
deterministically predicts $p^{t}=p_{i}$ (or $p^{t}=p_{i+1}$ — the choice can
be arbitrary and does not affect the analysis). Since we have that
$|p_{i}-p_{i+1}|\leq\frac{1}{m}$, it must be that for whichever choice One-
Step-Ahead would have made, we have $|\tilde{p}^{t}-p^{t}|\leq\frac{1}{m}$. In
other words, although Almost-One-Step-Ahead does not make the same predictions
as One-Step-Ahead, it makes predictions that are within $\ell_{1}$ distance
$T/m$ after $T$ rounds. The analysis then follows by the ECE bound of One-
Step-Ahead, the triangle inequality, and choosing $m=\sqrt{T}$.

Proof of Theorem 1. Observe that internally, Algorithm 1 maintains the
sequence $\tilde{p}^{1},...,\tilde{p}^{t}$ which corresponds exactly to
predictions made by One-Step-Ahead. Thus, by Lemma 1 and Theorem 2, we have
that $CalDist(\tilde{p}^{1:T},y^{1:T})\leq ECE(\tilde{p}^{1:T},y^{1:T})\leq
m$. Then, we can compute the distance to calibration of the sequence
$p^{1},...,p^{T}$:

$\displaystyle CalDist(p^{1:T},y^{1:T})$
$\displaystyle=\min_{q^{1:T}\in\mathcal{C}(y^{1:T})}\|p^{1:T}-q^{1:T}\|_{1}$
$\displaystyle=\min_{q^{1:T}\in\mathcal{C}(y^{1:T})}\|p^{1:T}-\tilde{p}^{1:T}+\tilde{p}^{1:T}-q^{1:T}\|_{1}$
$\displaystyle\leq\|p^{1:T}-\tilde{p}^{1:T}\|_{1}+\min_{q^{1:T}\in\mathcal{C}(y^{1:T})}\|\tilde{p}^{1:T}-q^{1:T}\|_{1}$
$\displaystyle\leq\frac{T}{m}+m$

where in the last step we use the fact that $|p^{t}-\tilde{p}^{t}|\leq 1/m$
for all $t$ and thus $\|p^{1:T}-\tilde{p}^{1:T}\|_{1}\leq T/m$. The result
then follows by setting $m=\sqrt{T}$. ∎

### Acknowledgements

This work was supported in part by the Simons Collaboration on the Theory of
Algorithmic Fairness, NSF grants FAI-2147212 and CCF-2217062, an AWS AI Gift
for Research on Trustworthy AI, and the Hans Sigrist Prize.

## References

  * Błasiok et al. (2023) Jarosław Błasiok, Parikshit Gopalan, Lunjia Hu, and Preetum Nakkiran.  A unifying theory of distance from calibration.  In _Proceedings of the 55th Annual ACM Symposium on Theory of Computing_ , pages 1727–1740, 2023.
  * Foster (1999) Dean P Foster.  A proof of calibration via blackwell’s approachability theorem.  _Games and Economic Behavior_ , 29(1-2):73–78, 1999.
  * Foster and Hart (2018) Dean P Foster and Sergiu Hart.  Smooth calibration, leaky forecasts, finite recall, and nash dynamics.  _Games and Economic Behavior_ , 109:271–293, 2018.
  * Foster and Vohra (1998) Dean P. Foster and Rakesh V. Vohra.  Asymptotic calibration.  _Biometrika_ , 85(2):379–390, 1998.  ISSN 00063444.  URL http://www.jstor.org/stable/2337364.
  * Gupta and Ramdas (2022) Chirag Gupta and Aaditya Ramdas.  Faster online calibration without randomization: interval forecasts and the power of two choices.  In Po-Ling Loh and Maxim Raginsky, editors, _Proceedings of Thirty Fifth Conference on Learning Theory_ , volume 178 of _Proceedings of Machine Learning Research_ , pages 4283–4309. PMLR, 02–05 Jul 2022.  URL https://proceedings.mlr.press/v178/gupta22b.html.
  * Qiao and Valiant (2021) Mingda Qiao and Gregory Valiant.  Stronger calibration lower bounds via sidestepping.  In _Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing_ , STOC 2021, page 456–466, New York, NY, USA, 2021. Association for Computing Machinery.  ISBN 9781450380539.  doi: 10.1145/3406325.3451050.  URL https://doi.org/10.1145/3406325.3451050.
  * Qiao and Zheng (2024) Mingda Qiao and Letian Zheng.  On the distance from calibration in sequential prediction, 2024.


# Stark effect in quantum waveguide with Neumann window

M. Raissi Département de Mathématiques, Faculté des Sciences de Moanstir,
Avenue de l’environnement 5019 Monastir-TUNISIE.
Laboratoire de recherche: Algèbre G om trie et Théorie Spectrale : LR11ES53.
<EMAIL_ADDRESS>
###### Abstract

In this study, we investigate the bound states of the Hamiltonian describing a
quantum particle living on three dimensional straight strip of width $d$ in
the presence of an the electric field of constant intensity $F$ in the
direction perpendicular to the electron plane. We impose Neumann boundary
condition on a disc window of radius $a$ and Dirichlet boundary conditions on
the remained part of the boundary of the strip. We prove that such system
exhibits discrete eigenvalues below the essential spectrum for any $F,a>0.$
This gives a difference with [19] which the presence of the discrete spectrum
depends on the intensity $F.$ The presence of the electric field is different
from the presence of a magnetic field previously studied [29].

AMS Classification: 81Q10 (47B80, 81Q15)
Keywords: Quantum Waveguide, Shrödinger operator, Electric Field, Bound
states, Dirichlet Laplacians, Stark effect.

##  1 Introduction

The study of quantum waveguide has acquired great interest during last decades
for their important physical consequences. The main reason is that they
represent an interesting physical effect with important applications in
nanophysical devices, but also in flat electromagnetic waveguide. The
waveguides are generally modeled by infinite planar strips and
multidimensional cylinders or layers. In such domains with boundary conditions
are considered, the spectral properties of quantum waveguide with various
perturbations attract a lot of attention. As the examples of possible
perturbations, we indicate local deformation of the boundary condition [7,
11], bending [13, 14, 15, 21] or twisting [9, 18] the waveguide. Perturbation
by adding a potential [13], or by a magnetic field [5, 17], or by a second
order differential operator [25].
Waveguide with general abstract perturbation of the operator were considered
in [24]. The perturbation by changing the type of the boundary condition on
the part of the boundary is more type of the perturbation, which has been
studied. As the examples of the waveguide with a finite Neumann part on the
boundary [6, 8, 24], the two waveguides having a common boundary where a gap
is cut out [3, 4], the Neumann segment or a gap on the boundary were referred
to as window(s) . The existence of the bound state below the essential
spectrum was predicted, for the $2D$ straight Dirichlet waveguide with the
Neumann window [10, 20]. The number of the bound states increases with the
window length $L$ and their energies are monotonically decreasing functions of
$L$ [8]. Recently, this result was extended to the case of the $3D$ spatial
Dirichlet duct with circular Neumann disc(s) [27, 28], for which a proof of
the bound state existence was confirmed. In [27], the number of discrete
eigenvalues as a function of the disc radius $a$ was evaluated and their
asymptotic for the large $a$ was given.
The waveguide with a magnetic field and a window was considered in [5, 29].
For the $2D$ strip, the authors are given an estimate on the maximal length of
the window, for which the discrete spectrum of the considered operator will be
empty. In the case of a compactly supported field, they also are given a
sufficient condition for the presence of eigenvalues below the essential
spectrum [5]. For the $3D$ in [29], the authors also are proved that in the
presence of magnetic field of Aharonov-Bohm type there is some critical values
of $a_{0}>0$, for which we have absence of the discrete spectrum for
$a<a_{0}$. Sufficient condition for the existence of discrete eigenvalues was
established.
Despite numerous investigations of quantum waveguide during last years, many
questions remain to be answered. This concerns, in particular, effects of
external fields. Most attention has been paid to magnetic fields [5, 16, 17,
29], while the influence of an electric field alone remained mostly untreated.
In [19], the authors drew attention to the fact that Stark effect in non-
straight tubes has a rich structure coming from combination of the curvature-
induced attractive interaction and the electrostatic potential which is
nonlinear along the tube even if the field is homogeneous.
The rest of the paper is organized as follows, in Section 2, we define the
model and recall some known results. In sections 3, we present the main result
of this paper followed by a discussion. Section 4 is devoted for numerical
computations.

##  2 The model

The system we are going to study is given in Fig 1. We consider a quantum
particle, this leads to the study of an Hamiltonian which we denote by
$H_{a}(F)$, whose motion is confined to a pair of parallel plans of width $d$.
For simplicity, we assume that they are placed at $z=0$ and $z=d$. We shall
denote this configuration space by $\Omega$

$\Omega=\mathbb{R}^{2}\times[0,d].$

We suppose that the particle is a fermion of a nonzero charge $q$. We also
assume that it is under influence of a homogeneous electric field of an
intensity $E$, we denote $\displaystyle F:=Eq$. Without loss of generality we
shall suppose in the following that $\displaystyle F\geq 0$ and that the
electric field is perpendicular to the electron plane.
Let $\displaystyle\gamma(a)$ be a disc of radius $a$, without loss of
generality we assume that the center of $\displaystyle\gamma(a)$ is the point
$(0,0,0)$;

$\gamma(a)=\big{\\{}(x,y,0)\in\mathbb{R}^{3};\ x^{2}+y^{2}\leq
a^{2}\big{\\}}.$ (2.1)

We set $\displaystyle\Gamma=\partial\Omega\diagdown\gamma(a)$. We consider
Dirichlet boundary condition on $\Gamma$ and Neumann boundary condition on
$\displaystyle\gamma(a)$.

Figure 1:  The waveguide with a disc window and two different boundaries
conditions with orthogonal electric field.

###  2.1 The Hamiltonian

Let us define the self-adjoint operator on
$\displaystyle\mathrm{L}^{2}(\Omega)$ corresponding to the particle
Hamiltonian $\displaystyle H_{a}(F)$. This will be done by the mean of
quadratic forms. Precisely, let $\displaystyle q_{a}$ be the quadratic form

$\displaystyle q_{a}[u,v]$ $\displaystyle=$ $\displaystyle\int_{\Omega}\nabla
u\overline{\nabla v}+Fzu\overline{v}dxdydz\quad u,v\in D(q_{a}),$

where $\displaystyle
D(q_{a}):=\big{\\{}u\in\mathrm{H}^{1}(\Omega),u\lceil\Gamma=0\big{\\}}$ and
$\mathrm{H}^{1}(\Omega)$ is the standard Sobolev space and $u\lceil\Gamma$ is
the trace of the function $u$ on $\Gamma$. It follows that $\displaystyle
q_{a}$ is a densely defined, symmetric, positive and closed quadratic form
[31]. We denote the unique self-adjoint operator associated to $\displaystyle
q_{a}$ by $\displaystyle H_{a}(F)$ and its domain by $\displaystyle D$. It is
the hamiltonian describing our system. From [31] (page 276), we infer that the
domain $\displaystyle D$ of $\displaystyle H_{a}(F)$ is

$\displaystyle D$ $\displaystyle=$
$\displaystyle\big{\\{}u\in\mathrm{H}^{1}(\Omega);\quad-\Delta
u\in\mathrm{L}^{2}(\Omega),u\lceil\Gamma=0,\frac{\partial u}{\partial
z}\lceil\gamma(a)=0\big{\\}},$

and

$H_{a}(F)u=(-\Delta+Fz)u,\quad\forall u\in D.$ (2.2)

###  2.2 Some known facts

Let us start this subsection by considering the particular cases when $a=0$,
we get $H_{0}(F)$ the Dirichlet Laplacian and $a=+\infty$, we get
$H_{\infty}(F)$ the Dirichlet-Neumann Laplacian.
Since, in this two particular cases, we have

$\displaystyle H_{a}(F)$ $\displaystyle=$
$\displaystyle(-\Delta_{\mathbb{R}^{2}})\otimes I_{[0,d]}\oplus
I_{\mathbb{R}^{2}}\otimes h_{a}(F),$ (2.3)

where $\displaystyle I_{\mathbb{R}^{2}}$ and $\displaystyle I_{[0,d]}$ are the
identity operators in $\displaystyle\mathrm{L}^{2}(\mathbb{R}^{2})$ and
$\displaystyle\mathrm{L}^{2}([0,d])$, respectively and $\displaystyle
h_{a}(F):=-\Delta_{[0,d]}+Fz$.
We know that the spectrum of $\displaystyle-\Delta_{\mathbb{R}^{2}}$ is
$[0,+\infty[$ ( see [31]).
The operator $\displaystyle h_{a}(F)$ is a positive self-adjoint elliptic
operator on $\displaystyle\mathrm{L}^{2}([0,d])$ with boundary Dirichlet
conditions at $\\{0\\}$ and $\\{d\\}$ in the case $a=0$ and with Dirichlet and
Neumann boundary conditions at $\\{d\\}$ and $\\{0\\}$, respectively in the
case $a=+\infty$. Thus, we have
$\displaystyle\sigma\big{(}h_{a}(F)\big{)}=\sigma_{d}(h_{a}(F)\big{)}\subset\mathbb{R}_{+}$.
Let $\displaystyle\lambda_{a}$ be an eigenvalue of $\displaystyle h_{a}(F)$,
then

$\displaystyle\big{(}-\partial^{2}_{z}+Fz\big{)}\varphi(z)=\lambda_{a}\varphi(z)$
$\displaystyle\Rightarrow$
$\displaystyle\varphi^{\prime\prime}(z)-F\left(z-\frac{\lambda_{a}}{F}\right)\varphi(z)=0.$
(2.4)

We set $\displaystyle l=z-\frac{\lambda_{a}}{F}\Rightarrow
z=l+\frac{\lambda_{a}}{F}$.
Hence (2.4) becomes

$\displaystyle\psi^{\prime\prime}(l)-Fl\psi(l)=0,\textrm{ where
}\psi(l)=\varphi\left(l+\frac{\lambda_{a}}{F}\right).$ (2.5)

Therefore eigenfunctions of $\displaystyle h_{a}(F)$ are linear combinations
of the fundamental solutions

$u_{\lambda_{a}}(z):=A_{i}\left(F^{1/3}\left(z-\frac{\lambda_{a}}{F}\right)\right);\quad
v_{\lambda_{a}}(z):=B_{i}\left(F^{1/3}\left(z-\frac{\lambda_{a}}{F}\right)\right),$
(2.6)

where $A_{i}$ and $B_{i}$ are the Airy functions of the first kind and of the
second kind, respectively [1].
Consequently, in the case $a=0$, the boundary Dirichlet conditions at
$\\{0\\}$ and $\\{d\\}$ result in two equations:

$\left\\{\begin{array}[]{ll}\alpha u_{\lambda_{a}}(0)+\beta
v_{\lambda_{a}}(0)=0&\\\ \alpha u_{\lambda_{a}}(d)+\beta
v_{\lambda_{a}}(d)=0,\end{array}\right.$

with $\displaystyle\alpha,\beta\in\mathbb{R}$. Or more simply as a matrix
equation:

$\begin{pmatrix}u_{\lambda_{a}}(0)&v_{\lambda_{a}}(0)\\\
u_{\lambda_{a}}(d)&v_{\lambda_{a}}(d)\end{pmatrix}\begin{pmatrix}\alpha\\\
\beta\end{pmatrix}=0.$

The usual condition for the solution to such an equation would be the
following:

$\begin{vmatrix}u_{\lambda_{a}}(0)&v_{\lambda_{a}}(0)\\\
u_{\lambda_{a}}(d)&v_{\lambda_{a}}(d)\end{vmatrix}=0.$

Then, the boundary Dirichlet conditions reads

$u_{\lambda_{0}}(0)v_{\lambda_{0}}(d)-u_{\lambda_{0}}(d)v_{\lambda_{0}}(0)=0.$
(2.7)

Using (2.5) and (2.7), the operator $h_{a}(F)$ has infinitely many simple
eigenvalues ( see [32])

$0<\lambda_{0}^{1}<\lambda_{0}^{2}<...<\lambda_{0}^{n}<.....$

Therefore, we get that the spectrum of $\displaystyle\displaystyle H_{0}(F)$
is $\displaystyle\left[\lambda_{0}^{1},+\infty\right[.$
Of the same, Dirichlet and Neumann boundary conditions at $\\{d\\}$ and
$\\{0\\}$ respectively, in the case $a=\infty$, reads

$u_{\lambda_{\infty}}^{\prime}(0)v_{\lambda_{\infty}}(d)-v_{\lambda_{\infty}}^{\prime}(0)u_{\lambda_{\infty}}(d)=0.$
(2.8)

Using (2.5) and (2.8), the operator $h_{\infty}(F)$ has infinitely many simple
eigenvalues ( see [32])

$0<\lambda_{\infty}^{1}<\lambda_{\infty}^{2}<...<\lambda_{\infty}^{n}<.....$

Then, we obtain that the spectrum of $\displaystyle H_{\infty}(F)$ is
$\displaystyle\left[\lambda_{\infty}^{1},+\infty\right[$.
Using the min-max principle [31], we have

$\sigma\big{(}H_{0}(F)\big{)}\subset\sigma\big{(}H_{a}(F)\big{)}\subset\sigma\big{(}H_{\infty}(F)\big{)}.$
(2.9)

###  2.3 Stability of the essential spectrum

Below, we show the stability of the essential spectrum of $\displaystyle
H_{a}(F)$ with double inclusion.
To prove that
$\displaystyle\left[\lambda_{0}^{1},+\infty\right[\subset\sigma_{ess}\big{(}H_{a}(F)\big{)}$
we construct Weyl sequences. Assume that $k$ is a real number. Let
$\displaystyle\varphi\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{2})$ such that
$\displaystyle supp\varphi\times\\{0\\}\subset\gamma(a)$ and let
$\displaystyle(\varphi_{n})_{n\in\mathbb{N}^{\star}}$ be the sequence in
$\displaystyle\mathcal{C}_{0}^{\infty}(\mathbb{R}^{2})$ given by

$\displaystyle\varphi_{n}(x,y)$ $\displaystyle=$
$\displaystyle\varphi\left(\frac{(x,y)}{n}-(n,n)\right);\quad
n\in\mathbb{N}^{\star}.$ (2.10)

Construct the functions

$\displaystyle\psi_{n}(x,y,z)$ $\displaystyle=$
$\displaystyle\varphi_{n}(x,y)\chi_{1}(z)e^{ik(x+y)/2};\quad
n\in\mathbb{N}^{\star},$ (2.11)

where $\displaystyle\chi_{1}$ is the positive eigenfunction associated to
$\displaystyle\lambda_{0}^{1}$ of the operator $\displaystyle h_{0}(F)$.
We will prove that $(\psi_{n})_{n}$ is a singular sequence for $\displaystyle
H_{a}(F)$ at $\lambda_{0}^{1}+k^{2}$. Clearly $\psi_{n}\in\displaystyle D$ for
every $n$. Using the definition of the sequence
$\displaystyle(\varphi_{n})_{n}$ and $\displaystyle(\psi_{n})_{n}$, we get
$\displaystyle\psi_{n}\xrightarrow[n\rightarrow\infty]{w}0.$
Finally we must show that
$\displaystyle\parallel\big{(}H_{a}(F)-(\lambda_{0}^{1}+k^{2})\big{)}\psi_{n}\parallel_{\mathrm{L}^{2}(\Omega)}\underset{n\rightarrow\infty}{\longrightarrow}0$.

$\displaystyle\parallel\left(H_{a}(F)-\left(\lambda_{0}^{1}+k^{2}\right)\right)\psi_{n}\parallel^{2}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\left|\nabla\left(\varphi_{n}(x,y)\chi_{1}(z)e^{ik(x+y)/2}\right)\right|^{2}dxdydz$
$\displaystyle+\int_{\Omega}Fz\mid\varphi_{n}(x,y)\chi_{1}(z)\mid^{2}dxdydz$
$\displaystyle-\left(\lambda_{0}^{1}+k^{2}\right)\parallel\chi_{1}\parallel^{2}_{\mathrm{L}^{2}([0,d])}\parallel\varphi_{n}\parallel^{2}_{\mathrm{L}^{2}(\mathbb{R}^{2})}$
$\displaystyle=$
$\displaystyle\frac{1}{n^{2}}\parallel\chi_{1}\parallel^{2}_{\mathrm{L}^{2}([0,d])}\parallel\nabla\varphi\parallel^{2}_{\mathrm{L}^{2}(\mathbb{R}^{2})}\quad\underset{n\rightarrow\infty}{\longrightarrow}0.$

We have proved that
$\lambda_{0}^{1}+k^{2}\in\sigma_{ess}\big{(}H_{a}(F)\big{)}$ for all real $k$.
Thus we get

$[\lambda_{0}^{1},+\infty[\subset\sigma_{ess}\big{(}H_{a}(F)\big{)}.$ (2.12)

To prove the inverse inclusion
$\displaystyle\sigma_{ess}\big{(}H_{a}(F)\big{)}\subset[\lambda_{0}^{1},+\infty[$,
it will be enough to prove that
$\displaystyle\inf\sigma_{ess}\big{(}H_{a}(F)\big{)}\geq\lambda_{0}^{1}$.
As in [27, 29], let us split $\displaystyle\mathrm{L}^{2}(\Omega,rdrd\theta
dz)$ as follows:

$\displaystyle\mathrm{L}^{2}(\Omega,rdrd\theta
dz)=\mathrm{L}^{2}(\Omega_{a}^{-},rdrd\theta
dz)\oplus\mathrm{L}^{2}(\Omega_{a}^{+},rdrd\theta dz),$

with

$\displaystyle\Omega_{a}^{-}$ $\displaystyle=$
$\displaystyle\\{(r,\theta,z)\in[0,a]\times[0,2\pi[\times[0,d]\\},$
$\displaystyle\Omega_{a}^{+}$ $\displaystyle=$
$\displaystyle\Omega\setminus\Omega_{a}^{-}.$

Therefore

$H^{-,N}_{a}(F)\oplus H^{+,N}_{a}(F)\leq H_{a}(F)\leq H^{-,D}_{a}(F)\oplus
H^{+,D}_{a}(F).$ (2.13)

Here we index by D and N depending on the boundary conditions added at the
surface $r=a$. The min-max principle leads to

$\sigma_{ess}\big{(}H_{a}(F)\big{)}=\sigma_{ess}\big{(}H^{+,N}_{a}(F)\big{)}=\sigma_{ess}\big{(}H^{+,D}_{a}(F)\big{)}.$
(2.14)

By the inclusion of $\Omega_{a}^{+}$ in $\Omega$ and the min-max principle, we
have

$\lambda_{0}^{1}:=\inf\sigma_{ess}\big{(}H_{0}(F)\big{)}\leq\inf\sigma_{ess}\big{(}H^{+,D}_{a}(F)\big{)}.$
(2.15)

Using the min-max principle and the last two equations, the left hand side
inequality (2.13) gives that

$\lambda_{0}^{1}\leq\inf\sigma_{ess}\Big{(}H^{-,N}_{a}(F)\oplus
H^{+,N}_{a}(F)\Big{)}\leq\inf\sigma_{ess}\big{(}H_{a}(F)\big{)}.$ (2.16)

Consequently, we obtain that

$\sigma_{ess}\big{(}H_{a}(F)\big{)}\subset[\lambda_{0}^{1},+\infty[.$ (2.17)

Using (2.12) and (2.17), we get finally that

$\sigma_{ess}\big{(}H_{a}(F)\big{)}=[\lambda_{0}^{1},+\infty[.$ (2.18)

An immediate consequence of (2.9) and (2.18) is the discrete spectrum lies in
$\displaystyle[\lambda_{\infty}^{1},\lambda_{0}^{1})$.

##  3 The result

We denote by $\displaystyle\sigma_{d}\big{(}H_{a}(F)\big{)}$ the discrete
spectrum of $\displaystyle H_{a}(F)$. The main result of this paper is the
following:

######  Theorem 3.1.

The operator $\displaystyle H_{a}(F)$ has at least one isolated eigenvalue in
$\displaystyle[\lambda_{\infty}^{1},\lambda_{0}^{1})$ for any $a,F>0$.

######  Remark 3.1.

Let $\displaystyle H_{a}(F)$ be the operator defined on (2.2). If $F=0$, for
any $a>0$, $\displaystyle\sigma_{d}\big{(}H_{a}(F)\big{)}\neq\emptyset$, (see
[22, 27]).

######  Remark 3.2.

We remark that this result is different with [19] which the presence of the
discrete spectrum depends on the intensity $F$. Also, the effect of the
presence of the electric field is different on the effect of the presence of a
magnetic field previously studied [29].

Proof. We start by giving some notations that we will use in the rest of this
work: $\langle;\rangle_{r}$ the scaler product in
$\mathrm{L}^{2}(\Omega,dxdydz)=\mathrm{L}^{2}(]0,+\infty[\times[0,2\pi[\times[0,d],rdrd\theta
dz)$ given by

$\displaystyle\langle f,g\rangle_{r}$ $\displaystyle=$
$\displaystyle\int_{]0,+\infty[\times[0,2\pi[\times[0,d]}f\overline{g}rdrd\theta
dz.$

We denote the gradient in cylindrical coordinates by $\nabla_{r}$.
Let us start the proof of the Theorem. To do so, we define the quadratic form
$Q_{r}$

$\displaystyle Q_{r}[f,g]$ $\displaystyle=$
$\displaystyle\int_{]0,+\infty[\times[0,2\pi[\times[0,d]}\nabla_{r}f\overline{\nabla_{r}g}+Fzf\overline{g}rdrd\theta
dz,$ (3.19)

with domain

$\displaystyle D_{r}$ $\displaystyle=$
$\displaystyle\big{\\{}f\in\mathrm{L}^{2}(\Omega,rdrd\theta
dz);\nabla_{r}f\in\mathrm{L}^{2}(\Omega,rdrd\theta
dz);f\lceil\Gamma=0\big{\\}}.$

Consider the functional $Q$ defined by

$\displaystyle Q[\Phi]$ $\displaystyle=$ $\displaystyle
Q_{r}[\Phi]-\lambda_{0}^{1}\|\Phi\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}.$ (3.21)

Since the essential spectrum of $\displaystyle H_{a}(F)$ starts at
$\lambda_{0}^{1}$, if we construct a trial function $\Phi\in D_{r}$ such that
$Q[\Phi]$ has a negative value then the task is achieved. Using the quadratic
form domain, $\Phi$ must be continuous inside but not necessarily smooth. Let
$\chi_{1}$ is the positive eigenfunction associated to $\lambda_{0}^{1}$ of
the operator $h_{0}(F)$, with $\|\chi_{1}\|=1$.
For $\Phi(r,\theta,z)=\varphi(r)\chi_{1}(z)$, we compute

$\displaystyle Q[\Phi]$ $\displaystyle=$ $\displaystyle
Q_{r}[\Phi]-\lambda_{0}^{1}\|\Phi\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$ (3.22) $\displaystyle=$
$\displaystyle\int_{]0,+\infty[\times[0,2\pi[\times[0,d]}\Big{(}|\chi_{1}(z)|^{2}|\varphi^{\prime}(r)|^{2}+|\chi^{\prime}_{1}(z)|^{2}|\varphi(r)|^{2}$
$\displaystyle+Fz|\chi_{1}(z)|^{2}|\varphi(r)|^{2}\Big{)}rdrd\theta
dz-\lambda_{0}^{1}\|\varphi\chi_{1}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$ $\displaystyle=$ $\displaystyle
2\pi\|\varphi^{\prime}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta dz)}.$

Now, let us consider an interval $I=[0,b]$ for a positive $b>a$ and a function
$\varphi\in\mathcal{S}([0,+\infty[)$ such that $\varphi(r)=1$ for $r\in I$. We
also define a family $\\{\varphi_{\tau}:\tau>0\\}$ by

$\varphi_{\tau}(r)=\left\\{\begin{array}[]{ll}\varphi(r)&\mbox{if
}r\in(0,b)\\\ \varphi(b+\tau(\ln r-\ln b))&\mbox{if }r\geq
b.\end{array}\right.$ (3.23)

Let us write

$\displaystyle\|\varphi^{\prime}_{\tau}\|_{\mathrm{L}^{2}([0,+\infty[,rdr)}$
$\displaystyle=$
$\displaystyle\int_{]0,+\infty[}|\varphi^{\prime}_{\tau}(r)|^{2}rdr$ (3.24)
$\displaystyle=$
$\displaystyle\int_{[b,+\infty[}\frac{\tau^{2}}{r^{2}}|\varphi^{\prime}(b+\tau(\ln
r-\ln b))|^{2}rdr$ $\displaystyle=$
$\displaystyle\tau\int_{[b,+\infty[}\frac{\tau}{r}|\varphi^{\prime}(b+\tau(\ln
r-\ln b))|^{2}dr$ $\displaystyle=$
$\displaystyle\tau\int_{[0,+\infty[}|\varphi^{\prime}(s)|^{2}ds$
$\displaystyle=$
$\displaystyle\tau\|\varphi^{\prime}\|^{2}_{\mathrm{L}^{2}([0,+\infty[)}.$

Let $\phi$ be a localization function from $\mathcal{C}^{\infty}_{0}(0,a)$ and
for $\tau,\varepsilon>$, we define

$\displaystyle\Phi_{\tau,\varepsilon}(r,z)$ $\displaystyle=$
$\displaystyle\varphi_{\tau}(r)[\chi_{1}(z)+\varepsilon\phi(r)^{2}]$ (3.25)
$\displaystyle=$
$\displaystyle\varphi_{\tau}(r)\chi_{1}(z)+\varepsilon\varphi_{\tau}(r)\phi(r)^{2}$
$\displaystyle=$
$\displaystyle\Phi_{1,\tau,\varepsilon}(r,z)+\Phi_{2,\tau,\varepsilon}(r,z).$

we compute

$\displaystyle Q[\Phi_{\tau,\varepsilon}]$ $\displaystyle=$ $\displaystyle
Q[\Phi_{1,\tau,\varepsilon}+\Phi_{2,\tau,\varepsilon}]$ $\displaystyle=$
$\displaystyle
Q_{r}[\Phi_{1,\tau,\varepsilon}+\Phi_{2,\tau,\varepsilon}]-\lambda_{0}^{1}\|\Phi_{1,\tau,\varepsilon}+\Phi_{2,\tau,\varepsilon}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$ $\displaystyle=$ $\displaystyle
Q_{r}[\Phi_{1,\tau,\varepsilon}]-\lambda_{0}^{1}\|\Phi_{1,\tau,\varepsilon}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}+Q_{r}[\Phi_{2,\tau,\varepsilon}]-\lambda_{0}^{1}\|\Phi_{2,\tau,\varepsilon}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$
$\displaystyle+2\langle\nabla_{r}\Phi_{1,\tau,\varepsilon},\nabla_{r}\Phi_{2,\tau,\varepsilon}\rangle_{r}+2\langle
Fz\Phi_{1,\tau,\varepsilon},\Phi_{2,\tau,\varepsilon}\rangle_{r}-2\lambda_{0}^{1}\langle\Phi_{1,\tau,\varepsilon},\Phi_{2,\tau,\varepsilon}\rangle.$

Using the properties of $\chi_{1}$ and taking into account equations (3.22)
and (3.24), we have

$\displaystyle
Q_{r}[\Phi_{1,\tau,\varepsilon}]-\lambda_{0}^{1}\|\Phi_{1,\tau,\varepsilon}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$ $\displaystyle=$ $\displaystyle
2\pi\tau\|\varphi^{\prime}\|^{2}_{\mathrm{L}^{2}([0,+\infty[)}.$

Noting that the supports of $\varphi$ and $\phi$ are disjoints, we get

$\displaystyle
Q_{r}[\Phi_{2,\tau,\varepsilon}]-\lambda_{0}^{1}\|\Phi_{2,\tau,\varepsilon}\|^{2}_{\mathrm{L}^{2}(\Omega,rdrd\theta
dz)}$ $\displaystyle=$
$\displaystyle\varepsilon^{2}d\pi\Big{(}8\|\phi\phi^{\prime}\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}+Fd\|\phi\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}$
$\displaystyle-2\lambda_{0}^{1}\|\phi^{2}\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}\Big{)}.$

Using the properties of $\chi_{1}$, we obtain

$\displaystyle\langle\nabla_{r}\Phi_{1,\tau,\varepsilon},\nabla_{r}\Phi_{2,\tau,\varepsilon}\rangle_{r}$
$\displaystyle=$ $\displaystyle 0,$

and

$\displaystyle\langle(Fz-\lambda_{0}^{1})\Phi_{1,\tau,\varepsilon},\Phi_{2,\tau,\varepsilon}\rangle_{r}$
$\displaystyle=$
$\displaystyle\varepsilon\langle(Fz-\lambda_{0}^{1})\chi_{1},\phi^{2}\rangle_{r}$
$\displaystyle=$
$\displaystyle-\varepsilon\langle-\chi^{\prime\prime}_{1},\phi^{2}\rangle_{r}<0.$

Therefore, we have

$\displaystyle Q[\Phi_{\tau,\varepsilon}]$ $\displaystyle=$ $\displaystyle
2\pi\tau\|\varphi^{\prime}\|^{2}_{\mathrm{L}^{2}([0,+\infty[)}+\varepsilon^{2}d\pi\Big{(}8\|\phi\phi^{\prime}\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}+Fd\|\phi\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}$
(3.26)
$\displaystyle-2\lambda_{0}^{1}\|\phi^{2}\|^{2}_{\mathrm{L}^{2}([0,+\infty[,rdr)}\Big{)}-\varepsilon\langle-\chi^{\prime\prime}_{1},\phi^{2}\rangle_{r}.$

Firstly, we notice that only the first term of the last equation depends on
$\tau$. Secondly, the linear term in $\varepsilon$ is negative and could be
chosen sufficiently small so that it dominates over the quadratic one. Fixing
this $\varepsilon$ and then choosing $\tau$ sufficiently small the right hand
side of (3.26) is negative. This ends the proof of the Theorem 3.1.

$\blacksquare$

##  4 Numerical computations

This section is devoted to some numerical computations. Let us start this
section by giving some notations that we will use in the rest of this work:
$\displaystyle\lambda_{k}(H^{-,N}_{a}(F))$,
$\displaystyle\lambda_{k}(H^{-,D}_{a}(F))$ and
$\displaystyle\lambda_{k}(H_{a}(F))$, the $k-$th eigenvalue of $\displaystyle
H^{-,N}_{a}(F)$, $\displaystyle H^{-,D}_{a}(F)$ and $H_{a}(F)$, respectively.
Then, the min-max principle yields the following

$\lambda_{k}(H^{-,N}_{a}(F))\leq\lambda_{k}(H_{a}(F))\leq\lambda_{k}(H^{-,D}_{a}(F))$
(4.27)

and for $2\geq k$

$\lambda_{k-1}(H^{-,D}_{a}(F))\leq\lambda_{k}(H_{a}(F))\leq\lambda_{k}(H^{-,D}_{a}(F)).$
(4.28)

Thus, if $\displaystyle H^{-,D}_{a}(F)$ exhibits a discrete spectrum below
$\displaystyle\lambda_{0}^{1}$, then $\displaystyle H_{a}(F)$ do as well. We
mention that its a sufficient condition.
Let us consider the eigenvalue equation is given by

$H^{-,D}_{a}(F)f(r,\theta,z)=\lambda f(r,\theta,z).$ (4.29)

This equation is solved by separating variables and considering
$\displaystyle f(r,\theta,z)=R(r)P(\theta)Z(z)$.
We divide the equation (4.29) by $f$, we obtain

$\frac{1}{R}(R^{\prime\prime}+\frac{1}{r}R^{\prime})+\frac{1}{r^{2}}\frac{P^{\prime\prime}}{P}+\frac{Z^{\prime\prime}}{Z}-Fz=-\lambda.$
(4.30)

Plugging the last expression in equation (4.30) and first separate the term
$\displaystyle\frac{P^{\prime\prime}}{P}$ which has all the $\theta$
dependance. Using the fact that the problem has an axial symmetry and the
solution has to be $2\pi$ periodic and single value in $\theta$, we obtain
$\displaystyle\frac{P^{\prime\prime}}{P}$ should be a constant
$\displaystyle-m^{2}$ for $m\in\mathbb{Z}$.
Second, we separate $Z$ by putting all the $z$ dependence in one term so that
$\displaystyle\frac{Z^{\prime\prime}}{Z}-Fz$ can only be constant. The
constant is taken as $\displaystyle\lambda_{\infty}^{n}$ for $n\in\mathbb{N}$.
Finally, we write the equation (4.30) as a function of $R$

$R^{\prime\prime}(r)+\frac{1}{r}R^{\prime}(r)+[\lambda-\lambda_{\infty}^{n}-\frac{m^{2}}{r^{2}}]R(r)=0.$
(4.31)

We notice that the equation (4.31), is the Bessel equation and its solutions
could be expressed in terms of Bessel functions. More explicit solutions could
be given by considering boundary conditions.
The solution of the equation (4.31) is given by $\displaystyle
R(r)=cJ_{m}(\eta r)$, where $\displaystyle c\in\mathbb{R}^{\star}$,
$\displaystyle\eta^{2}=\lambda-\lambda_{\infty}^{n}$ and $\displaystyle J_{m}$
is the Bessel function of first kind of order $m$.
We assume that

$\displaystyle R(a)=0$ $\displaystyle\Leftrightarrow$ $\displaystyle
J_{m}(\eta a)=0$ (4.32) $\displaystyle\Leftrightarrow$ $\displaystyle
a\eta=x_{m,k}.$

Where $\displaystyle x_{m,k}$ is the $k-$th positive zero of the Bessel
function $\displaystyle J_{m}$ (see [27]).
Then $\displaystyle H^{-,D}_{a}(F)$ has a sequence of eigenvalues [27, 33],
given by

$\displaystyle\lambda_{n,m,k}$ $\displaystyle=$
$\displaystyle\left(\frac{x_{m,k}}{a}\right)^{2}+\lambda_{\infty}^{n}.$

the condition

$\lambda_{n,m,k}<\lambda_{0}^{1},$ (4.33)

yields that $n=0$, so we get

$\lambda_{1,m,k}=\left(\frac{x_{m,k}}{a}\right)^{2}+\lambda_{\infty}^{1}.$
(4.34)

This yields that the condition (4.33) to be fulfilled, will depends on the
value of $\displaystyle\left(\frac{x_{m,k}}{a}\right)^{2}$. We recall that
$\displaystyle x_{m,k}$ are the positive zeros of the Bessel function
$\displaystyle J_{m}$. So, for any $\lambda_{a}$ eigenvalue of $\displaystyle
H_{a}(F)$, there exists $\displaystyle
m,k,m^{\prime},k^{\prime}\in\mathbb{N}$, such that

$\left(\frac{x_{m^{\prime},k^{\prime}}}{a}\right)^{2}+\lambda_{\infty}^{1}\leq\lambda_{a}\leq\left(\frac{x_{m,k}}{a}\right)^{2}+\lambda_{\infty}^{1}.$

Using the boundary conditions, we obtain that the operators $h_{0}(F)$ and
$h_{\infty}(F)$ have a sequence of eigenvalues

  * •

in the case of weak electric field respectively given by:

$\displaystyle\lambda_{0}^{n}$ $\displaystyle=$
$\displaystyle\left(\frac{n\pi+\sqrt{n^{2}\pi^{2}+d^{3}F}}{2d}\right)^{2}+o(F);\quad
n\in\mathbb{N}^{*}.$ $\displaystyle\lambda_{\infty}^{n}$ $\displaystyle=$
$\displaystyle\left(\frac{(2n+1)\frac{\pi}{2}+\sqrt{(2n+1)^{2}(\frac{\pi}{2})^{2}+d^{3}F}}{2d}\right)^{2}+o(F);\quad
n\in\mathbb{N}.$

  * •

in the case of strong electric field respectively given by:

$\displaystyle\lambda_{0}^{n}$ $\displaystyle=$
$\displaystyle\left[\frac{3F\pi}{2}\left(2n-\frac{1}{4}\right)\right]^{\frac{2}{3}};\quad
n\in\mathbb{N}^{*}.$ $\displaystyle\lambda_{\infty}^{n}$ $\displaystyle=$
$\displaystyle\left[\frac{3F\pi}{2}\left(2n+\frac{3}{4}\right)\right]^{\frac{2}{3}};\quad
n\in\mathbb{N}.$

In the following of this section, we represent the area of existence of the
first three eigenvalues of $H_{a}(F)$ $\lambda_{a}^{1}$, $\lambda_{a}^{2}$ and
$\lambda_{a}^{3}$ and the threshold of appearance of eigenvalues, for the
electric field of constant weak intensity $F$ in Figure 2, and for $F$ strong
enough in Figure 3.
We observe that the area of existence of the eigenvalues of $H_{a}(F)$ is
proportional to the intensity $F$.

Figure 2: We represent $\left(\frac{x(i)}{a}\right)^{2}+\lambda_{\infty}^{1}$
where $x(1),x(2),x(3)$ are the first three zeros of the bessel functions
increasingly ordered. Figure 3: We represent
$\left(\frac{x(i)}{a}\right)^{2}+\lambda_{\infty}^{1}$ where $x(1),x(2),x(3)$
are the first three zeros of the bessel functions increasingly ordered.

Acknowledgements: The author is grateful to professors Hatem NAJAR under whom
this work is supervised and Philippe BRIET for the hospitality in CPT.

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