Various Manifestations of Wood Anomalies in Locally Distorted Quantum Waveguides
- 18 Downloads
Anomalies of the diffraction pattern at near-threshold frequencies of the continuous spectrum of a cylindrical quantum waveguide with regular (smooth gentle) or singular (small cavities and bumps) perturbations of the boundary are studied. Wood anomalies are characterized by rapid variations in the scattering matrix near the thresholds. Conditions under which a Wood anomaly is absent, appears, and enhances are obtained by constructing asymptotics of solutions to the Dirichlet problem for the Helmholtz equation. The results are obtained by analyzing an artificial object—the augmented scattering matrix—and involve only operations with real values of the spectral parameter, but the relation between Wood anomalies and complex resonance points is also considered. Generated by almost standing waves, threshold resonances that cause near-threshold anomalies of other types are discussed.
Keywords:quantum waveguide regular and singular perturbations of the boundary asymptotics resonances near-threshold frequencies Wood anomaly
This work was supported by the Russian Science Foundation, project no. 17-11-01003.
- 6.L. A. Vainshtein, Diffraction Theory and Factorization Method (Sovetskoe Radio, Moscow, 1966) [in Russian].Google Scholar
- 18.L. I. Mandelstam, Lectures on Optics, Relativity Theory, and Quantum Mechanics (Akad. Nauk SSSR, Moscow, 1947), Vol. 2 [in Russian].Google Scholar
- 21.N. A. Umov, Equations of Energy Transfer in Bodies (Tipogr. Ul’rikha i Shul’tse, Odessa, 1874) [in Russian].Google Scholar
- 25.A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Am. Math. Soc., RI, Providence, 1992).Google Scholar
- 27.W. G. Mazja, S. A. Nasarow, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten (Akademie-Verlag, Berlin, 1991), Vol. 1 (English translation: V. Maz’ya, S. Nazarov, and B. Plamenevsky, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains (Birkhäuser, Basel, 2000), Vol. 1).Google Scholar
- 28.V. A. Kondrat’ev, “Boundary value problems for elliptic equations in domains with conical or corner points,” Tr. Mosk. Mat. O–va 16, 219–292 (1963).Google Scholar
- 30.S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain,” in Sobolev Spaces in Mathematics, Vol. 2, Ed. by V. Maz’ya, International Mathematical Series (Springer, New York, 2008), Vol. 9, pp. 261–309.Google Scholar