arXiv subjects

P. Exner

Publications and source records attributed to P. Exner.

39 records · Page 3Linked to original sources

arXiv1996

Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window

Consider the Laplacian in a straight planar strip of width $\,d\,$, with the Neumann boundary condition at a segment of length $\,2a\,$ of one of the boundaries, and Dirichlet otherwise. For small enough $\,a\,$ this operator has a single eigenvalue $\,ε(a)\,$; we show that there are positive $\,c_1,c_2\,$ such that $\,-c_1 a^4 \le ε(a)- \left(π/ d\right)^2 \le -c_2 a^4\,$. An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length $\,2a\,$ in the common boundary.

funct-an↗

arXiv1996

Bound states in a locally deformed waveguide: the critical case

We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one straight boundary and a width $\,a(1+λf(x))\,$, where $\,f\,$ is a smooth function of a compact support with a length $\,2b\,$. We show that in the critical case, $\,\int_{-b}^b f(x)\, dx=0\,$, the operator has no bound states for small $\,|λ|\,$ if $\,b<(\sqrt{3}/4)a\,$. On the other hand, a weakly bound state exists provided $\,\|f'\|< 1.56 a^{-1}\|f\|\,$; in that case there are positive $\,c_1, c_2\,$ such that the corresponding eigenvalue satisfies $\,-c_1λ^4\le ε(λ)- (π/a)^2 \le -c_2λ^4\,$ for all $\,|λ|\,$ sufficiently small.

funct-an↗

arXiv1996

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width $ \ell $ in the common boundary. We show that such a system has at least one bound state for any $ \ell>0 $. We find the corresponding eigenvalues and eigenfunctions numerically using the mode--matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase--shift spacing statistics shows that in distinction to closed pseudo--integrable billiards, the present system is essentially non--chaotic. Finally, we illustrate time evolution of wave packets in the present model.

cond-mat↗