Semiclassical Analysis of Low Lying Eigenvalues. III. Width of the Ground State Band in Strongly Coupled Solids*

Double well problems appear to be among the easiest tunneling problems to analyze in a rigorous mathematical manner, in part because it is easier to give precise meaning to an eigenvalue splitting than to a lifetime. The first mathematically rigorous treatment of the precise leading order behavior in a tunneling problem is Harrell’s analysis [ 31 of one dimensional double wells. More recently, Simon [ 8, 9 ] obtained the leading order in certain multidimensional multiwell tunneling problems, and Helffer and Sjostrand [5] have even gone beyond leading order. Harrell realized that widths of bands for strongly coupled periodic potentials are essentially a multiwell problem, and he analyzed such problems in one dimension if the potentials were both periodic and reflection invariant [4]. More recently, Keller and Weinstein [6] have analyzed the one dimensional problem without the reflection symmetry restriction. Our main goal in this note is to obtain the leading asymptotics in the multidimensional case. The proofs are an easy extension of those used in [ 9 ] to handle double wells; indeed, a special case of the situation in [9, Section 6.31, is a large box whose size is a multiple of the basic periods with periodic boundary conditions (discretizing momentum space). Because of the unbounded nature of R” and the infinity of minima, there are some technical issues which we must (and will) handle. We first describe our results and then sketch the proofs drawing heavily on the ideas and results of [9]. We will suppose that the potential V is smooth and has one minimum per unit cell. By adding a constant to V and shifting, we can without loss suppose V> 0 and V(0) = 0. We will also suppose that the minimum is nondegenerate. Thus, our hypotheses are