A trace formula for higher order ordinary differential operators

Abstract

We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

“…so, y [1] = y , y [2] = y + (σ 0 + τ 1 )y, y [3] = 3 (y). Suppose that p s,0 = s − 1, p s,1 = 3 − s, s = 1, 3, in the linear forms (8). Using the technique of [54], we obtain the eigenvalue asymptotics…”

“…This paper is concerned with the inverse spectral theory for operators generated by the differential expression n (y) :=y (n) + n/2 −1 ∑ k=0 (τ 2k (x)y (k) ) (k) + (n−1)/2 −1 ∑ k=0 (τ 2k+1 (x)y (k) ) (k+1) + (τ 2k+1 (x)y (k+1) ) (k) , x ∈ (0, 1), (1) where the notation a means rounding down, and the functions {τ ν } n−2 ν=0 can be either integrable or distributional. Various aspects of spectral theory for such operators and related issues have been intensively studied in recent years (see, e.g., [1][2][3][4][5][6][7][8][9]). However, the general theory of inverse spectral problems for (1) with arbitrary n > 2 has not been created yet.…”