Asymptotic analysis of the spectrum of a quantum waveguide with a wide Neumann “window” in the light of mechanics of cracks

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 176-237

Résumé

Various asymptotic expansions are derived for eigenvalues in the discrete spectrum of the boundary-value problem for the Laplace operator in the unit strip with the Dirichlet condition on its lateral sides everywhere with exception of an interval with length $2\ell>0$ where the Neumann condition is imposed (a planar quantum waveguide with the “window”). Since the total multiplicity of the discrete spectrum grows indefinitely as $\ell\rightarrow+\infty$, there exists a sequence of the critical lengths $\{\ell^\ast_m\}$, for which the problem operator enjoys the threshold resonance. This phenomenon is characterized by the existence of a nontrivial bounded solution, that is, either trapped, or almost standing wave, and provides miscellaneous near-threshold spectral anomalies. The quality of the threshold resonances is examined and asymptotic formulas for the values $\ell^\ast_m$ are obtained for large numbers $m$. The analysis is systematically performed by means of methods from fracture mechanics.

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