Krein's formula and heat-kernel expansion for some differential operators with a regular singularity

H. FALOMIR; P.A.G. PISANI

Journal of Physics A-Mathematical and General

Lugar: Institute of Physics; Año: 2006 vol. 39 p. 6333 - 6340

0305-4470

We get a generalization of Krein’s formula—which relates the resolventsof different self-adjoint extensions of a differential operator with regularcoefficients—to the non-regular case A = −∂^2_x + (ν^2 −1/4)/x^2 + V (x), where0 < ν < 1, and V (x) is an analytic function of x ∈ R+ bounded frombelow. We show that the trace of the heat kernel e^{−tA} admits a non-standardsmall-t asymptotic expansion which contains, in general, integer powers of t^ν .In particular, these powers are present for those self-adjoint extensions of Awhich are characterized by boundary conditions that break the local formalscale invariance at the singularity.Online at stacks.iop.org/JPhysA/39/6333PACS numbers: 02.30.Gp, 03.70.+k