Resolvents of operators with distant perturbations

Planar waveguide with “twisted” boundary conditions: Small width

“…As ε goes to zero, we employ the ideas of [30], [31] this resolvent behaves approximately as a direct sum of two Laplacians in Π (1) subject to the Neumann condition on {x : x 2 = 1} ∪ {x : x 1 < 0, x 2 = 0} and to the Dirichlet condition on {x : x 1 > 0, x 2 = 0}. This operator again has no virtual levels associated with both the spectral points E = 0 and E = π 2 /4, see Lemmas 3.2, 3.3, 3.4.…”

“…Exactly the last description determines the effective boundary condition and the rates of the resolvent convergence. If L is fixed, we again employ the same approach but with the combination of some ideas of [30], [31]. We also mention that one of the effective approaches of studying the asymptotic behavior of the solutions to the problems in the thin domains is the method of matching of asymptotic expansions, see, for instance, [38].…”

Tunneling resonances in systems without a classical trapping

“…Recently we managed to analyze a considerably more general class of of operators with distant perturbations and to prove general convergence theorems and to describe asymptotic behavior of their spectra and the resolvents -cf. [18], [19], [20], [21], [25], [26], [27]. The general approach we have developed is useful here, since the technique employed in this paper is based on the main ideas put forward in the cited work.…”

“…The aim of this section is to rephrase the problem, using the ideas worked out in [18], [19], [20], [21], [25], [26], [27], as an operator equation; analyzing the latter we will be able to derive the leading terms in the resonance asymptotics. Let χ ± ∈ C ∞ (R) be a nonnegative cut-off function satisfying the relations…”