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Journals and Conferences
The classical Heun equation has the form
We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width ℓ in the common boundary. We show that such a system has at least one bound state for any ℓ > 0. We find the corresponding eigenvalues and eigenfunctions numerically using the mode–matching method, and discuss their behavior in several situations. We… (More)
We study spectral properties of Dirichlet Laplacian on the conical layer of the opening angle π − 2θ and thickness equal to π. We demonstrate that below the continuum threshold which is equal to one there is an infinite sequence of isolated eigenvalues and analyze properties of these geometrically induced bound states. By numerical computation we find… (More)
Recently, PT symmetry of many single-particle non-Hermitian Hamiltonians has been conjectured sufficient for keeping their spectrum real. We show that and how the similar concept of a “weakened Hermiticity” can be extended to some exactly solvable twoand three-particle models. PACS 03.65.Ge, 03.65.Fd February 1, 2008, ptcal.tex file 1 e-mail:… (More)
We discuss differences between the exact S–matrix for scattering on serial structures and a known factorized expression constructed of single–element S–matrices. As an illustration , we use an exactly solvable model of a quantum wire with two point impurities.
We analyze two-dimensional Schrödinger operators with the potential |xy| − λ(x + y) where p ≥ 1 and λ ≥ 0, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant λ. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for λ below the critical value the spectrum is purely… (More)
We study transport in quantum systems consisting of a finite array of N identical single–channel scatterers. A general expression of the S matrix in terms of the individual–element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the… (More)
We present a numerical study of the spectrum of the Laplacian in an unbounded strip with PT-symmetric boundary conditions. We focus on non-Hermitian features of the model reflected in an unusual dependence of the eigenvalues below the continuous spectrum on various boundary-coupling parameters.
We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically. Furthermore, we show that the model has a rich resonance structure.
Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the… (More)