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

IOP Publishing

Año: 2006 vol. A39 p. 6333 - 6340

0305-4470

We get a generalization of Krein´s formula -which relates the resolvents of different selfadjoint extensions of a differential operator with regular coefficients- to the non-regular case A=-partial_x^2+( u^2-1/4)/x^2+V(x), where 0< u<1 and V(x) is an analytic function of xin {R}^+ 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 t^ u. In particular, these powers are present for those selfadjoint extensions of A which are characterized by boundary conditions that break the local formal scale invariance at the singularity.