Boundary conditions on thin manifolds and the semiboundedness of the three-particle Schr\"odinger operator with pointwise potential

B. S. Pavlov

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 161-175

Résumé

The purpose of this article is to describe the formulation of a selfadjoint spectral problem with boundary conditions on a sufficiently thin manifold. Namely, let $\mathscr L$ be a selfadjoint operator in $L_2(\mathbf R^n)$, let $L$ be a smooth manifold, let $\mathscr L_0$ be the restriction of $\mathscr L$ to the lineal in $\mathscr D(\mathscr L_0)$ consisting of all functions which vanish in a neighborhood of $L$. It is shown that the deficiency elements of this restriction can be represented as “tensor layers” with densities of a definite class of smoothness, concentrated on the “boundary” of $L$. If $L$ is sufficiently thin, there is only one family of deficiency elements, and it is analogous to the single-layer potentials. In this case, calculation of the boundary form and the description of the selfadjoint extensions appears to be quite simple. This case is studied in detail because the investigation of the simplest model of the three-particle problem of quantum mechanics reduces to it. Bibliography: 16 titles.