Scattering by local deformations of a straight leaky wire

Abstract

We consider a model of a leaky quantum wire with the Hamiltonian $-\Delta -\alpha \delta (x-\Gamma )$ in $L^2({\mathbb{R}}^2)$, where $\Gamma$ is a compact deformation of a straight line. The existence of wave operators is proven and the S-matrix is found for the negative part of the spectrum. Moreover, we conjecture that the scattering at negative energies becomes asymptotically purely one-dimensional, being determined by the local geometry in the leading order, if $\Gamma$ is a smooth curve and $\alpha \to \infty$.

Latex2e, 15 pages

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Cited by in corpus (9)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions
• Leaky Quantum Graphs: A Review
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3
• Scattering from local deformations of a semitransparent plane
• Scattering of particles bounded to an infinite planar curve
• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions