EXNER_BORISOV · Spectral properties of perturbed quantum waveguides

Final Activity Report Summary - EXNER-BORISOV (Spectral properties of perturbed quantum waveguides)

The central part of the reported Marie Curie post-doctoral programme was concerned with problems of distant perturbations in multi-dimensional domains. The latter were either whole space or a multi-dimensional quantum 'tube'. A new original scheme was developed that allowed us to consider the general case, with a finite number of distant perturbations modelled using arbitrary operators with a local support. This scheme was applied for the investigation of asymptotic behaviour of the discrete spectrum for the considered class of operators with multiple perturbations. A convergence theorem was established for this situation and we obtained the leading terms in the asymptotic expansions for the eigenvalues which remained isolated in the limit. The systems of a pair of quantum strips or quantum layers coupled laterally by a window in the common boundary were considered and the phenomenon of new eigenvalue emergence from the threshold of the essential spectrum was studied. We proved a necessary and sufficient condition for both the absence and the existence of such emerging eigenvalues and constructed their asymptotic expansions. Moreover, the one-dimensional operators with fast oscillating coefficients' space were also studied and a detailed analysis of the structure of their spectra was carried out. The asymptotic behaviour of the essential spectrum was described and the asymptotic expansions for both the discrete eigenvalues and the associated eigenfunctions were constructed. A one-dimensional periodic operator with a localised and non self-adjoint perturbation was also considered. We established qualitative properties of the continuous, residual and point spectra, such as invariance of the continuous spectrum wrt localised perturbations, absence of the residual spectrum, countability of the point spectrum and criteria for presence of embedded eigenvalues. The asymptotic behaviour of the eigenvalues and the eigenfunctions was finally described.

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