Locally Compact Operators and Their Application to Schrödinger Operators

Abstract

The resolvent R _H(Z), Im z ≠0, of a Schrödinger operator H = -△ + V on L ²(ℝⁿ) is typically not compact (however, it usually is on L ²(X), when X is compact). If R _H(Z) is compact, then σ(R _H(Z)) is discrete with zero the only possible point in the essential spectrum. Hence, one would expect that H has discrete spectrum with the only possible accumulation point at infinity (i.e., σ_ess(H)) = 0). In this way, the σ(H) reflects the compactness of R _H(Z)- It turns out that these properties are basically preserved if, instead of R _H(Z) being compact, it is compact only when restricted to any compact subset of ℝⁿ. This is the notion of local compactness. From an analysis of this notion we will see that the discrete spectrum of H is determined by the behavior of H on bounded subsets of ℝⁿ and the essential spectrum of H is determined by the behavior of H (in particular, V) in a neighborhood of infinity.