Numerical results a quantum waveguide with Mixed boundary conditions
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.
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 , or by a magnetic field
[5, 17], or by a second order differential operator
Waveguide with general abstract perturbation of the operator were considered in . 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 . 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 , 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 . For the in , the authors also are proved that in the presence of magnetic field of Aharonov-Bohm 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 , 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. 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 ;
We set . We consider Dirichlet boundary condition on and Neumann boundary condition on .
2.1. The Hamiltonian
Let us define the self-adjoint 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 . We denote the unique self-adjoint operator associated to by and its domain by . It is the hamiltonian describing our system. From  (page 276), we infer that the domain of is
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 min-max principle yields the following
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
This equation is solved by separating variables and considering
We divide the equation (3.3) by , we obtain
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
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
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
The solution of the equation (3.5) is given by , where , and is the Bessel function of first kind of order .
We assume that
yields that , so we get
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
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:
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
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
-  M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York: Dover Publications, (1972).
-  M. Sh. Birman: Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions. Vestnik Leningrad. Univ. 1, no. 1, 22-55 (1962)(in Russian).
-  D. Borisov: Discrete spectrum of a pair of asymmetric window-coupled waveguides. English transl., Sb. Math. 197, no. 3-4, 475-504 (2006).
-  D. Borisov: On the spectrum of two quantum layers coupled by a window. J. Phys. A: Math. Theor. 40, no. 19, 5045-5066 (2007).
-  D. Borisov, T. Ekholm and H. Kovarik: Spectrum of the magnetic Schrodinger operator in a waveguide with combined boundary conditions. Annales Henri Poincaré 6, no. 2, 327-342 (2005).
-  D. Borisov and P. Exner: Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys A: Math and General, 37, no. 10, 3411-3428 (2004).
-  D. Borisov, P. Exner and R. Gadyl’shin, D. Krejcirik: Bound states in weakly deformed strips and layers. Annales Henri Poincaré 2, no. 3, 553-572 (2001).
-  D. Borisov, P. Exner and R. Gadylshin: Geometric coupling thresholds in a two-dimensional strip. J. Math. Phys.43, no. 12, 6265-6278 (2002).
-  Ph. Briet, H. Kovarik, G. Raikov and E. Soccorsi: Eigenvalue asymptotics in a twisted waveguide. Comm. PDE, 34, no. 8, 818-836 (2009).
-  W. Bulla, F. Gesztesy, W. Renger and B. Simon: Weakly coupled bound states inquantum waveguides. Proc. Amer. Math. Soc, 125, no.5, 1487-1495 (1997).
-  W. Bulla and W. Renger: Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys.35, no.1, 1-12 (1995).
-  L. I. Chambers: An upper bound for the first zero of Bessel functions. Math. Comp.38, no. 158, 589-591 (1982).
-  P. Duclos and P. Exner: Curvature-induced bound state in quantum waveguides in two and three dimensions. Rev. Math. Phys.7, no. 1, 73-102 (1995).
-  P. Duclos, P. Exner and P. Stovicek: Curvature-induced resonances in a two-dimensional Dirichlet tube. Annales Henri Poincaré 62, no.1, 81-101 (1995).
-  B. Chenaud, P. Duclos, P. Freitas and D. Krejcirik: Geometrically induced discrete spectrum in curved tubes. Diff. Geom. Appl. 23, no. 2, 95-105 (2005).
-  G. Dunne and R.L. Jaffe: Bound states in twisted Aharonov-Bohm tubes. Ann.Phys. 233, no. 2, 180-196 (1993).
-  T. Ekholm and H. Kovarik: Stability of the magnetic Schrodinger operator in a waveguide. Comm. PDE, 30, no. 4, 539-565 (2005).
-  T. Ekholm, H. Kovarik and D. Krejcirik: A Hardy inequality in twisted waveguides. Arch. Rat. Mech. Anal. 188, no. 2, 245-264 (2008).
-  P. Exner: A Quantum Pipette. J. Phys A: Math and General, 28, no. 18, 5323-5330 (1995).
-  P. Exner, P. Šeba, M. Tater and D. Vaněk: Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys. 37, no. 10, 4867-4887 (1996).
-  P. Exner and S. A. Vugalter: Bound States in a Locally Deformed Waveguide: The Critical Case. Lett. Math. Phys.39, no. 1, 59-68 (1997).
-  P. Exner and S. A. Vugalter: Bound state asymptotic estimate for window-coupled Dirichlet strips and layers. J. Phys. A: Math. Gen. 30, no. 22, 7863-7878, (1997).
-  S. Finch: Bessel Function Zeroes. Unpublished note (2003).
-  R. Gadylshin: On regular and singular perturbations of acoustic and quantum waveguides. C. R. Mécanique, 332, no. 8, 647-652 (2004).
-  V. V. Grushin: On the eigenvalues of finitely perturbed laplace operators in infinite cylindrical domains. Math. Notes, 75, no. 3, 331-340 (2004).
-  L. Lorch: Some inequalities for the first positive zeros of Bessel functions. J. Math. Anal.24, no. 3, 814-823 (1993).
-  H. Najar, S. Ben Hariz and M. Ben Salah: On the Discrete Spectrum of a Spatial Quantum Waveguide with a Disc Window. Math. Phy. Ana. Geom.13, no.1, 19-28 (2010).
-  H. Najar and O. Olendski: Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs. J. Phys. A: Math. Theor. 44, no. 30, 305304-305313 (2011).
-  H. Najar and M. Raissi: A quantum waveguide with Aharonov-Bohm Magnetic Field. Accepted in the Journal Mathematical Methods in the Applied Sciences, (2015).
-  R. Piessens: A series expansion for the first positive zero of the Bessel functions. Math. Comp.42, no. 165, 195-197 (1984).
-  M. Reed and B. Simon: Methods of Modern Mathematical Physics. Vol. IV: Analysis of Operators. Academic, Press (1978).
-  J. C. Robinson: An Introduction to Ordinary Differential Equations. Cambridge University Press, (2004).
-  G. N. Watson: A Treatise On The Theory of Bessel Functions. Cambridge University Press, (1966).
-  D. Zwillinger: Handbook of Differential Equations, Third Edition. Boston, MA: Academic Press, (1997).