Krein Formula and Heat-Kernel Expansion for some Singular Operators

H. FALOMIR AND P.A.G. PISANI

Barcelona

Congreso; Quantum Field Theory under the Influence of External Conditions 05; 2005

We get a generalization of Krein's formula?which relates the resolvents of different self-adjoint extensions of a differential operator with regular coefficients?to the non-regular case A = −∂2x + (ν2 − 1/4)/x2 + V(x), where 0 < ν < 1, and V(x) is an analytic function of bounded from below. We show that the trace of the heat kernel e−tA admits a non-standard small-t asymptotic expansion which contains, in general, integer powers of tν. In particular, these powers are present for those self-adjoint extensions of A which are characterized by boundary conditions that break the local formal scale invariance at the singularity.