The essential self-adjointness of Schrödinger-type operators

Abstract

The paper discusses conditions under which the formally self-adjoint elliptic differential operator in R m given by 1 $$\tau {\text{ }}u = \sum\limits_{j,{\text{ }}k = 1}^m {[i\partial _j + b_j (x)]} {\text{ }}a_{jk} (x){\text{ }}[i\partial _k + b_k (x)]{\text{ }}u + q(x){\text{ }}u$$ has a unique self-adjoint extension. The novel feature is that the major conditions on the coefficients have to be imposed only in an increasing sequence of shell-like regions surrounding the origin. On the other hand it is shown that if these shells are broken so as to allow a tube extending to infinity in which the conditions on the coefficients are too weak, then, regardless of the coefficients elsewhere, there may not be a unique self-adjoint extension. The mathematical theorems are linked to the quantum-mechanical interpretation of essential self-adjointness (in the case that τ is the Schrodinger operator), that there is a unique self-adjoint extension if the particle cannot escape to infinity in a finite time.