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On the discrete spectrum of a spatial quantum waveguide with a disc window

S. Ben Hariz, M. Ben Salah, H. Najar

TL;DR

This work analyzes bound states for a Schrödinger particle confined to a three-dimensional straight waveguide with a disc Neumann window of radius $a$ on the boundary and Dirichlet conditions elsewhere. The Hamiltonian is constructed via a quadratic form on $L^2(Ω)$ with mixed boundary conditions, and the problem is reduced using cylindrical coordinates to a radial-Bessel framework. The authors prove that at least one discrete eigenvalue exists below the essential spectrum for any $a>0$, and derive its asymptotic behavior $\lambda(a)= (\frac{\pi}{2d})^{2} + o(1/a^{2})$ as $a\to\infty$; they also provide numerical estimates for the number of bound states in terms of $a/d$ using zeros of Bessel functions. The results highlight geometry- and boundary-condition-induced bound states in quantum waveguides and supply quantitative tools for estimating bound-state counts in related disc-window configurations.

Abstract

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$. We impose the 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 $a>0$. We give also a numeric estimation of the number of discrete eigenvalue as a function of $\displaystyle \frac{a}{d}$. When $a$ tends to the infinity, the asymptotic of the eigenvalue is given.

On the discrete spectrum of a spatial quantum waveguide with a disc window

TL;DR

This work analyzes bound states for a Schrödinger particle confined to a three-dimensional straight waveguide with a disc Neumann window of radius on the boundary and Dirichlet conditions elsewhere. The Hamiltonian is constructed via a quadratic form on with mixed boundary conditions, and the problem is reduced using cylindrical coordinates to a radial-Bessel framework. The authors prove that at least one discrete eigenvalue exists below the essential spectrum for any , and derive its asymptotic behavior as ; they also provide numerical estimates for the number of bound states in terms of using zeros of Bessel functions. The results highlight geometry- and boundary-condition-induced bound states in quantum waveguides and supply quantitative tools for estimating bound-state counts in related disc-window configurations.

Abstract

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width . We impose the Neumann boundary condition on a disc window of radius 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 . We give also a numeric estimation of the number of discrete eigenvalue as a function of . When tends to the infinity, the asymptotic of the eigenvalue is given.

Paper Structure

This paper contains 7 sections, 1 theorem, 34 equations, 4 figures.

Table of Contents

  1. Itroduction
  2. The model
  3. The Hamiltonian
  4. Some known facts
  5. Preliminary: Cylindrical coordinates
  6. The result
  7. Numerical computations

Key Result

Theorem 3.1

The operator $H$ has at least one isolated eigenvalue in $\left[ (\frac{\pi }{2d})^{2},(\frac{\pi }{d})^{2}\right]$ for any $a>0$. Moreover for $a$ big enough, if $\lambda (a)$ is an eigenvalue of $H$ less then $\frac{\pi ^{2}}{d^{2}}$, then we have.

Figures (4)

  • Figure 1: The waveguide with a disc window and two different boundaries conditions
  • Figure 2: We represent $a \mapsto (\frac{\pi}{2d})^2+(\frac{x(i)}{a})^{2}$ where $x(1), x(2), x(3)$ are the first three zeros of the bessel functions increasingly ordered.
  • Figure 3: The number of the eigenvalues of the operator $H^D$ function of $\lambda\equiv a/d.$
  • Figure 4: The number of the eigenvalues of the operator $H^D$ function of $d$ and $a.$

Theorems & Definitions (1)

  • Theorem 3.1