Traces of Higher Negative Orders for a String with a Singular Weight

Abstract

We study the linear operator pencil A(λ) = L−λV, λ ∈ ℂ, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ⁻¹₂ [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums $\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}}$, p ≥ 2, where λ_n, n ∈ ℕ, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L⁻¹ are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.