On Generalized Spectral Shift Functions and a Relation between the Krein and Gel'fand-Levitan Trace Formulas

摘要

In their celebrated 1953 paper [1], GePfand and Levitan obtained a regularized trace formula for the Sturm-Liouville operator -y" + q(x) y = λy, y'(0) = 0, y'(π) = 0 (1) under the additional condition ∫_0~πq(x)dt = ; the formula is ∑ from n=0 to ∞ of (μ_n - λ_n) = 1/4(q(0) + q(π)), (2) where the μ_n are the eigenvalues of operator (1) and λ_n = n~2 are the eigenvalues of the same operator with q(x) ≡ 0.