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Spectral Analysis of a Quantum Waveguide with Elliptical Window

H. Najar, F. Chogle

Abstract

We investigate the Dirichlet Laplacian in two spatial waveguides coupled through an elliptic window. The elliptic geometry breaks rotational symmetry and introduces anisotropy through the semi-axes of the aperture, which modifies the coupling of transverse modes and the low-lying spectrum. We prove that the operator has a finite number of discrete eigenvalues below the threshold of the essential spectrum and study their dependence on the geometric parameters of the ellipse. In contrast to the circular case, the elliptic setting gives rise to spectral effects such as eigenvalue splitting. Numerical simulations illustrate the variation of the first eigenvalues and the ground state with the window geometry.

Spectral Analysis of a Quantum Waveguide with Elliptical Window

Abstract

We investigate the Dirichlet Laplacian in two spatial waveguides coupled through an elliptic window. The elliptic geometry breaks rotational symmetry and introduces anisotropy through the semi-axes of the aperture, which modifies the coupling of transverse modes and the low-lying spectrum. We prove that the operator has a finite number of discrete eigenvalues below the threshold of the essential spectrum and study their dependence on the geometric parameters of the ellipse. In contrast to the circular case, the elliptic setting gives rise to spectral effects such as eigenvalue splitting. Numerical simulations illustrate the variation of the first eigenvalues and the ground state with the window geometry.
Paper Structure (12 sections, 1 theorem, 55 equations, 9 figures)

This paper contains 12 sections, 1 theorem, 55 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Short description of the problem
  3. Quantum waveguides
  4. The result
  5. Model and Formulation
  6. The Hamiltonian
  7. Some known facts
  8. Preliminary: Elliptic coordinates system
  9. The Eigenvalue problem
  10. Analytical results
  11. Numerical computations
  12. Conclusion

Key Result

Theorem 3.1

The operator $\hat{H}$ has at least one isolated eigenvalue in $\left[(\frac{\pi}{2d})^2,(\frac{\pi}{d})^2\right]$ for any nonzero $a$ and $b$. Moreover, for $b$ big enough and $E(a,b)$ being an eigenvalue of $\hat{H}$ less then $(\frac{\pi}{d})^2$, there exist positive constants $C_a$ and $C_b$ suc

Figures (9)

  • Figure 1: Dirichlet wave guide with elliptic Neumann window of radii $a$ and $b$.
  • Figure 2: Energy spectrum of the ground state wavefunction $(m=0)$ as a function of the elliptical axes $a$ and $b$.
  • Figure 3: The energy curve when $a$ is fixed and the eccentricity is decreased
  • Figure 4: The hyperbolic behavior of the curves for $a=1$ and $a=1.5$.
  • Figure 5: In (a), $a$ is fixed and we decrease the eccentricity (we increase $b$) of the ellipse. In (b), $b$ is fixed and we increase the eccentricity (we increase $a$) of the ellipse.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • Remark 3.25