Spectral analysis of a~nonselfadjoint differential operator arising in the one-dimensional problem of scattering by a~Brownian particle

S. E. Cheremshantsev

Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 371-390

Résumé

The operator $H=-\partial_{xx}+i\varkappa\partial_{yy}+q(x-y)\cdot$ arising in the averaging of a solution of the Schrödinger equation with a random time-dependent potential is investigated. Analysis of the operator reduces to the study of a family of one-dimensional operators $$ B_p=-\frac{d^2}{dx^2}+2p\frac d{dx}+\frac{q(x)}{1-i\varkappa},\qquad p\in\mathbf R. $$ The distribution of the discrete and continuous spectra of the operators $B_p$ and $H$ is studied. An expansion in eigenfunctions of the operators $B_p$ in $L_2(\mathbf R)$ for almost all $p$ and of the operator $H$ on a set dense in $L_2(\mathbf R^2)$ is obtained. Figures: 1. Bibliography: 4 titles.