arXiv subjects

S. A. Vugalter

Publications and source records attributed to S. A. Vugalter.

2 recordsLinked 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.

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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.

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