A general trace formula for a differential operator on a segment with zero order coefficient perturbed by a finite signed measure

Abstract

A first order trace formula is obtained for a regular differential operator perturbed by a finite signed measure multiplication operator. ∞ N=1 . In what follows

On the trace formula for higher-order ODO

“…The constant c is defined in (30) and depends only on the leading coefficients of polynomials P j and Q j in boundary conditions (2). The Log sign in (12) stands for the branch of logarithm with ℑLog ∈ (−π, π).…”

“…This new term corresponds to the case where q has an atom in the midpoint a+b 2 . These results were partially announced in [2].…”

“…For n = 2 and arbitrary regular boundary conditions, the formula similar to (6) was obtained in [3]. Also it was shown in [3] that for n ≥ 3 a nonlinear terms does not appear, and it was conjectured that for high order operators formula (4) holds for q ∈ M[a, b].…”

“…Levitan in 1953. In [4] they considered the problem − y ′′ + q(x)y = λy; y(0) = y(π) = 0 (3) and showed that for a real-valued function q(x) ∈ C 1 [0, π] the following relation holds:…”

“…Let us recall some notation. We denote by L 0 the operator generated by the differential expression L 0 = (−i) n D n and boundary conditions (2). The eigenvalues of L 0 are denoted by {λ 0 N } ∞ N=1 .…”