Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs

Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $b$ located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip. We prove that such a system exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$. When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is derived. A comparative analysis with the one-window case reveals that due to the additional possibility of the regulating energy spectrum the anticrossing structure builds up as a function of the inner radius with its sharpness increasing for the larger outer radius. Mathematical and physical interpretation of the obtained results is presented; namely, it is derived that the anticrossings are accompanied by the drastic changes of the wave function localization. Parallels are drawn to the other structures exhibiting similar phenomena; in particular, it is proved that, contrary to the two-dimensional geometry, at the critical Neumann radii true bound states exist.Comment: 25 pages, 7 figure

Homogenization for operators with arbitrary perturbations in coefficients

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. This operator is perturbed by a first order differential operator, the coefficients of which depend arbitrarily on a small multi-dimensional parameter. We study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence for such perturbed operator. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. If this is the case, the resolvent of the perturbed operator possesses a complete asymptotic expansion, which converges uniformly to the resolvent. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations, which finally leads to a very wide class of perturbations, for which our results are applicable

Homogenization of the planar waveguide with frequently alternating boundary conditions

We consider Laplacian in a planar strip with Dirichlet boundary condition on the upper boundary and with frequent alternation boundary condition on the lower boundary. The alternation is introduced by the periodic partition of the boundary into small segments on which Dirichlet and Neumann conditions are imposed in turns. We show that under the certain condition the homogenized operator is the Dirichlet Laplacian and prove the uniform resolvent convergence. The spectrum of the perturbed operator consists of its essential part only and has a band structure. We construct the leading terms of the asymptotic expansions for the first band functions. We also construct the complete asymptotic expansion for the bottom of the spectrum

Eigenvalues bifurcating from the continuum in two-dimensional potentials generating non-Hermitian gauge fields

It has been recently shown that complex two-dimensional (2D) potentials $V_\varepsilon(x,y)=V(y+\mathrm{i}\varepsilon\eta(x))$ can be used to emulate non-Hermitian matrix gauge fields in optical waveguides. Here $x$ and $y$ are the transverse coordinates, $V(y)$ and $\eta(x)$ are real functions, $\varepsilon>0$ is a small parameter, and $\mathrm{i}$ is the imaginary unit. The real potential $V(y)$ is required to have at least two discrete eigenvalues in the corresponding 1D Schr\"odinger operator. When both transverse directions are taken into account, these eigenvalues become thresholds embedded in the continuous spectrum of the 2D operator. Small nonzero $\varepsilon$ corresponds to a non-Hermitian perturbation which can result in a bifurcation of each threshold into an eigenvalue. Accurate analysis of these eigenvalues is important for understanding the behavior and stability of optical waves propagating in the artificial non-Hermitian gauge potential. Bifurcations of complex eigenvalues out of the continuum is the main object of the present study. We obtain simple asymptotic expansions in $\varepsilon$ that describe the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate into a single eigenvalue, while every other threshold can bifurcate into a pair of complex eigenvalues. These bifurcations can be controlled by the Fourier transform of function $\eta(x)$ evaluated at certain isolated points of the reciprocal space. When the bifurcation does not occur, the continuous spectrum of 2D operator contains a quasi-bound-state which is characterized by a strongly localized central peak coupled to small-amplitude but nondecaying tails. The analysis is applied to the case examples of parabolic and double-well potentials $V(y)$. In the latter case, the bifurcation of complex eigenvalues can be dampened if the two wells are widely separated.Comment: 24 pages, 6 figures; accepted for Annals of Physic