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

摘要

Let L0 and L be operators which are formed by the differential expressions.ℓ0(y)=(-1)my(2m)(x)+Ay(x)andℓ(y)=(-1)my(2m)(x)+Ay(x)+Q(x)y(x)respectively, in the space H1 = L2(0, π; >H), with same boundary condition y(2i−1)(0) = y(2i−1)(π) = 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 ∈ [0, π], 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.limp→∞mq=1npλq-μq-1π∫0π(Q(x)φjq,φjq)dx=14[trQ(0)+trQ(π)]-12π∫0πtrQ(x)dxfor the regularized trace of L. Here, n1 < n2 < ⋯and j1, j2, …are sequences of natural numbers with a particular property. Furthermore, μ1 ⩽ μ2 ⩽ ⋯and λ1 ⩽ λ2 ⩽ ⋯are the eigenvalues of the operators L0 and L, respectively; and φ1, φ2, …is a complete orthonormal sequence consisting of eigenvectors of the operator A.