Non-accumulation of the discrete spectrum in the infinite-dimensional case under weaker assumptions

Determine whether the discrete spectrum of the block Jacobi operator J = S* A + B + A S on H = l2(Z, H), with H infinite-dimensional and An, Bn self-adjoint satisfying compactness of An − I and Bn, fails to accumulate at the spectral edges ±2 under weaker hypotheses than those used in Theorem 6.6—specifically, without imposing the third moment condition or the closed-range assumption on the Wronskian W(U+(±1)*, U−(±1))—for example assuming only the first moment condition from Definition 1.1.

Background

The paper establishes non-accumulation (and hence finiteness) of the discrete spectrum of the Jacobi operator J at the spectral edges ±2 under strong conditions: the third moment condition and a closed-range hypothesis on the Wronskian at z = ±1 (Theorem 6.6).

In contrast, when these stronger assumptions are not imposed, the authors are unable to assert non-accumulation and instead provide only quantitative bounds on the rate of possible accumulation of eigenvalues (Theorem 6.7), assuming trace-class perturbations. This leaves unresolved whether non-accumulation can be guaranteed in the infinite-dimensional setting under weaker moment conditions such as the first moment condition.