Comparison Theorems for the Gap of Schrtidinger Operators

There are two cases where it is well known that Schriidinger operators have non-degenerate eigenvalues: The lowest eigenvalue in general dimension and all one-dimensional eigenvalues. One can ask about making this quantitative, i.e., obtain explicit lower bounds on the distance to the nearest eigenvalues. Obviously, one cannot hope to do this without any restrictions on V since, for example, if 1 is the characteristic function of (1, 1 ), one can show that, for I large, &/dx’ x(x) x(x Z) has at least two eigenvalues and E, -I$, 40 as I--+ co (see, e.g., Harrell [7]). Thus, we ask the following: Can one obtain lower bounds on eigenvalue splittings only in terms of geometric properties of the set with V(x) < E (E at or near the eigenvalues in question) and the size of V on this set? We will do precisely this for the two lowest eigenvalues in general dimension in this paper, and we have proven results on any one-dimensional eigenvalue in [ll]. This is not the first paper to try to estimate the gap E, -E. for -A + V; see, e.g., [8, 16, 9, 193. Here we will present a very elementary device which is also quite powerful. It depends on the fact that many SchrGdinger operators can be realized as Dirichlet forms. This subject has been studied