Numerical results a quantum waveguide with Mixed boundary conditions
Abstract.
This article is devoted to the numerical study of the existence of the eigenvalues of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width in the presence of an electric field of constant intensity in the direction perpendicular to the electron plane. We impose Neumann boundary conditions on a disc window of radius and Dirichlet boundary conditions on the remaining part of the boundary of the strip.
Key words and phrases:
Quantum Waveguide, Shrödinger operator, Electric Field, Bound states, Stark effect.2010 Mathematics Subject Classification:
Primary 81Q10; Secondary 47B80, 81Q15.1. 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
straight Dirichlet waveguide with the Neumann window
[10, 20]. The number of the bound states increases with the
window length and their energies are monotonically decreasing
functions of [8]. Recently, this result was extended to
the case of the 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 was
evaluated and their asymptotic for the large was given.
The waveguide with a magnetic field and a window was considered in
[5, 29]. For the 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 in [29], the authors also are proved that in the
presence of magnetic field of AharonovBohm type there is some
critical values of , for which we have absence of the
discrete spectrum for . 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 nonstraight
tubes has a rich structure coming from combination of the
curvatureinduced 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. 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 Figure 1. We consider a quantum particle, this leads to the study of an Hamiltonian which we denote by , whose motion is confined to a pair of parallel plans of width . For simplicity, we assume that they are placed at and . We shall denote this configuration space by
We suppose that the particle is a
fermion of a nonzero charge . We also assume that it is under
influence of a homogeneous electric field of an intensity , we
denote . Without loss of generality we shall
suppose in the following that and that the electric field is perpendicular to the electron plane.
Let be a disc of radius , without loss
of generality we assume that the center of
is the point ;
(2.1) 
We set . We consider Dirichlet boundary condition on and Neumann boundary condition on .
2.1. The Hamiltonian
Let us define the selfadjoint operator on corresponding to the particle Hamiltonian . This will be done by the mean of quadratic forms. Precisely, let be the quadratic form
where and is the standard Sobolev space and is the trace of the function on . It follows that is a densely defined, symmetric, positive and closed quadratic form [31]. We denote the unique selfadjoint operator associated to by and its domain by . It is the hamiltonian describing our system. From [31] (page 276), we infer that the domain of is
and
(2.2) 
3. 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: , and , the th eigenvalue of , and , respectively. Then, the minmax principle yields the following
(3.1) 
and for
(3.2) 
Thus, if exhibits a discrete spectrum
below , then do as well. We mention that its a sufficient condition.
Let us consider the eigenvalue equation is given by
(3.3) 
This equation is solved by separating variables and considering
.
We divide the
equation (3.3) by , we obtain
(3.4) 
Plugging the last expression in equation (3.4) and first
separate the term which has all the
dependance. Using the fact that the problem has an axial
symmetry and the solution has to be periodic and single value
in , we obtain should be a
constant for
.
Second, we separate by putting all the dependence in one
term so that can only be constant.
The constant is taken as for
.
Finally, we write the equation (3.4) as a function of
(3.5) 
We notice that the equation (3.5), 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 (3.5) is given by , where , and is the
Bessel function of first kind of order .
We assume that
(3.6)  
Where is the th positive zero of the Bessel function
(see [27]).
Then has a sequence of eigenvalues
[27, 33], given by
the condition
(3.7) 
yields that , so we get
(3.8) 
This yields that the condition (3.7) to be fulfilled, will depends on the value of . We recall that are the positive zeros of the Bessel function . So, for any eigenvalue of , there exists , such that
(3.9) 
Using the boundary conditions, we obtain that the operators and have a sequence of eigenvalues

in the case of weak electric field respectively given by:

in the case of strong electric field respectively given by:
Where and are the th negative zeros of the Airy functions and respectively. Consequently, we have

in the case of weak electric field respectively given by:

in the case of strong electric field respectively given by:
Remark 3.1.
Using the inequality (3.9), for big enough, if is an eigenvalue of the operator less then then we have
In the following of this section, we represent the
area of existence of the first three eigenvalues of
, and and the threshold of
appearance of eigenvalues, for the electric field of constant weak
intensity
in Figure 2, and for strong enough in Figure 3.
We observe that the area of existence
of the eigenvalues of is proportional to the intensity .
In the Figure 4, we set the intensity of the electric field by a low value . We represent the curve of the number of eigenvalues of the operator a function of the quotient of the radius value by the width of the strip
In the Figure 5, Similarly we set the intensity of the electric field l’intensité du champ électriqueby a great value . We represent the curve of the number of eigenvalues of the operator a function of the quotient of the radius value by the width of the strip
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