Traces for a class of singular differential operators

Abstract

In the space L₂[0, ∞) we consider the operator generated by the expression

$$l(y) \equiv ( - 1)^n \frac{{d^{2n} y(x)}}{{dx^{2n} }} + xy(x)$$

and boundary conditions at the point x=0 that fix a self-adjoint extension. For such operators we compute the regularized traces of all orders, i.e., series of the form

$$\sum\limits_{k = 1}^\infty {\left[ {\lambda _k^m - A_m (k)} \right],}$$

where m is an arbitrary positive integer, λ_k are the eigenvalues of the operator, and A_m(k) are certain specific numbers depending on k and guaranteeing that the series converges. Bibliography: 7 titles.