Hermitian boundary conditions at a Dirichlet singularity: the Marletta--Rozenblum model

Research output: Contribution to journalArticle (Academic Journal)peer-review

12 Citations (Scopus)

Abstract

In domains B with smoothly-varying boundary conditions, points where wavefunctions are required to vanish were recently identified as 'Dirichlet singularities' (D points) where the Hamiltonian H does not define discrete eigenvalues and a scattering phase is undetermined (Berry and Dennis 2008 J. Phys. A: Math. Theor. 41 135203). This is explained (Marletta and Rozenblum 2009 J. Phys. A: Math. Theor. 42 125204) by the observation, illustrated with an exactly-solvable separable model, that a D point requires the specification of an additional parameter defining a family of self-adjoint extensions of H. Here the underlying theory is presented in an elementary way, and a D point is identified as a leak, through which current can flow into or out of B. Hermiticity seals the leak, ensuring that no current flows though the D point (as well as across the boundary of B). The solvable model is examined in detail for bound states, where B is a semidisk, and for wave reflections, where B is a half-plane. The quantization condition for a nonseparable billiard is obtained explicitly.
Translated title of the contributionHermitian boundary conditions at a Dirichlet singularity: the Marletta--Rozenblum model
Original languageEnglish
Pages (from-to)16520-1 - 16520-13
Number of pages13
JournalJournal of Physics A: Mathematical and Theoretical
Volume42
DOIs
Publication statusPublished - Apr 2009

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