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

S. E. Cheremshantsev


Abstract: 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.

UDC: 517.4

MSC: Primary 35J10, 35R60, 34B25; Secondary 60J65

Received: 24.12.1984

 English version: Mathematics of the USSR-Sbornik, 1987, 57:2, 371–390

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