The regularized trace of a self adjoint differential operator of higher order with unbounded operator coefficient

Abstract

Let L-0 and L be operators which are formed by the differential expressions. l(0)(y) = (-1)(m)y((2m)) (x) + Ay(x) and l(y) = (-1)(m)y((2m))(x) + Ay(x) + Q(x)y(x) respectively, in the space H-1 = L-2(0, pi; > H), with same boundary condition y((2i - 1))(0) = y((2i - 1))(pi) = 0, (i = 1,2, ... , m) where H is an infinite dimensional separable Hilbert space. Here, A is an unbounded self adjoint operator in H and, for every x is an element of [0, pi], Q(x) is a self-adjoint trace class operator in H. Assuming the operator A and the operator function Q(x) satisfy some additional conditions, the following formula has been found. lim(p ->infinity) um(q=1)(np) [lambda(q) - mu(q) - 1/pi integral(pi)(0) (Q(x)phi(jq), phi(jq))dx] = 1/4 [trQ(0) + trQ(pi)] -1/2 pi integral(pi)(0) trQ(x)dx for the regularized trace of L. Here, n(1) < n(2) < ... and j(1), j(2), ... are sequences of natural numbers with a particular property. Furthermore, mu(1) <= mu(2) <= ... and lambda(1) <= lambda(2) <= ... are the eigen-values of the operators L-0 and L, respectively; and phi(1), phi(2), ... is a complete orthonormal sequence consisting of eigenvectors of the operator A. Crown Copyright (c) 2011 Published by Elsevier Inc. All rights reserved.