Modified Krein Formula and Analytic Perturbation Procedure for Scattering on Arbitrary Junction

A Solvable Model for Scattering on a Junction and a Modified Analytic Perturbation Procedure

“…A one-dimensional version of the statement can be found in [9] and a rescription of the classical Krein formula with compensated singularities is given in [44]. In this paper we review the compensation singularities in Theorem 3.1, following [2] for a general thin junction and prove a similar statement, see Theorem 3.3 for the Intermediate ND-map. We also obtain, in course of calculations, an important "byproduct": a version of analytic perturbation procedure for groups of eigen-pairs.…”

“…Presence of these singularities in the conditions of the last theorem of the previous section looks strange. In fact we were able to prove, see [2], that the singularities in the first and second terms of the above Krein formula for M, inherited from L D Γ , compensate each other, so that only the singularities of the denominators of (17) play a role. Similar statement can be proved, see the theorem 3.3 below, for N .…”

“…Hereafter we assume that the junction is thin. The case of an arbitrary junction is considered in [2]. The above definition of thin junctions (and networks) is based on the following motivation.…”

“…The end of the proof Based on some cumbersome calculation, we are able to derive, see [2], that all singularities in the Krein formula, inherited from the eigenvalues λ s of the unperturbed operator L int , are compensated.…”

Current-voltage characteristics for two systems of quantum waveguides with connected quantum resonators

“…В этой ситуации хорошим приближением является модель квантовых волноводов, изучение которых позволяет описывать и предсказывать некоторые интересные явления в наноэлектрон-ных устройствах. Естественный математический аппарат для таких задач дает теория рассеяния [10][11][12][13][14][15].…”