Geodesic
On occurrence of resonances from multiple eigenvalues of the Schr\"odinger operator in a cylinder with scattering perturbations
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 3-14

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In this paper, the Schrödinger operator with a localized potential in a multidimensional cylinder is considered. The boundary of the cylinder is split into three parts, two of which are “sleeves” going to infinity, and the third (central) part is located between them. On the sleeves and the central part, respectively, the Neumann and Dirichlet boundary conditions are posed. We examine the situation where the distance between the sleeves increases. We assume that the same Schrödinger operator in the same cylinder endowed with the Dirichlet condition on the whole boundary has an isolated double eigenvalue. We show that for a sufficiently large distance between the sleeves, this double eigenvalue splits into a pair of resonances of the original operator. For these resonances, we explicitly obtain the first terms of their asymptotic expansions and describe the behavior of the imaginary part of the resonances.
Keywords: Schrödinger operator, perturbation of continuous spectrum, resonance, scattering perturbation. 
@article{INTO_2019_163_a0,
     author = {D. I. Borisov and A. M. Golovina},
     title = {On occurrence of resonances from multiple eigenvalues of the {Schr\"odinger} operator in a cylinder with scattering perturbations},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--14},
     publisher = {mathdoc},
     volume = {163},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_163_a0/}
}
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D. I. Borisov; A. M. Golovina. On occurrence of resonances from multiple eigenvalues of the Schr\"odinger operator in a cylinder with scattering perturbations. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 3-14. http://geodesic.mathdoc.fr/item/INTO_2019_163_a0/