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 ( 2 m ) ( x ) + Ay ( x ) and ℓ ( y ) = ( - 1 ) m y ( 2 m ) ( x ) + Ay ( x ) + Q ( x ) y ( x ) respectively, in the space H 1 = L 2(0, π; > H), with same boundary condition y (2 i−1) (0) = y (2 i−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. lim p → ∞ m q = 1 n p λ q - μ q - 1 π ∫ 0 π ( Q ( x ) φ j q , φ j q ) dx = 1 4 [ trQ ( 0 ) + trQ ( π ) ] - 1 2 π ∫ 0 π 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, μ 1 ⩽ μ 2 ⩽ ⋯and λ 1 ⩽ λ 2 ⩽ ⋯are the eigenvalues of the operators L 0 and L, respectively; and φ 1, φ 2, …is a complete orthonormal sequence consisting of eigenvectors of the operator A.