Paul Loya

The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2∕dr2−1∕(4r2)

In this paper we analyze the resolvent, the heat kernel and the spectral zeta function of the operator −d2∕dr2−1∕(4r2) over the finite interval. The structural properties of these spectral functions depend strongly on the chosen self-adjoint realization of the operator, a choice being made necessary because of the singular potential present. Only for the Friedrichs realization standard properties are reproduced, for all other realizations highly nonstandard properties are observed. In particular, for k∊N we find terms like (logt)−k in the small-t asymptotic expansion of the heat kernel. Furthermore, the zeta function has s=0 as a logarithmic branch point.

TEMPERED OPERATORS AND THE HEAT KERNEL AND COMPLEX POWERS OF ELLIPTIC PSEUDODIFFERENTIAL OPERATORS

The resolvent (A – λ)−1 of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is shown to lie in a calculus of operators, tempered in the parameter λ in a special way. We show that the Laplace and Mellin transforms, with respect to λ, of these tempered operators can be defined and that they have Schwartz kernels which can be described geometrically. As a corollary, we obtain the structures of the kernels of the heat operator and complex powers of b-pseudodifferential operators, as the heat operator and complex powers are the Laplace and Mellin transforms, respectively, of the resolvent. The heat operator and complex powers are then used to generalize the index formula of Atiyah, Patodi, and Singer for Dirac operators on manifolds with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners.

On the gluing problem for Dirac operators on manifolds with cylindrical ends

Combining elements of the b-calculus and the theory of elliptic boundary value problems, we solve the gluing problem for b-determinants of Dirac type operators on manifolds with cylindrical ends. As a corollary of our proof, we derive a gluing formula for the b-eta invariant and also a relative invariant formula relating the b-spectral invariants on a manifold with cylindrical end to the spectral invariants with the augmented APS boundary condition on the corresponding compact manifold with boundary.

The ζ-determinant of generalized APS boundary problems over the cylinder

In this paper, we explicitly compute the ζ-determinant of a Dirac Laplacian with Atiyah–Patodi–Singer (APS) boundary conditions over a finite cylinder. Using this exact result, we illustrate the gluing and comparison formulae for the ζ-determinants of Dirac Laplacians proved by Loya and Park.

Complex powers of differential operators on manifolds with conical singularities

We construct the complex powersAz for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatAz exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofAz, both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(Az).

Index theory of Dirac operators on manifolds with corners up to codimension two

In this expository article, we survey index theory of Dirac operators using the Gauss-Bonnet formula as the catalyst to discuss index formulas on manifolds with and without boundary. Considered in detail are the Atiyah-Singer and Atiyah-Patodi-Singer index theorems, their heat kernel proofs, and their generalizations to manifolds with corners of codimension two via the method of `attaching cylindrical ends’.

ON THE SINGULARITIES OF THE ZETA AND ETA FUNCTIONS OF AN ELLIPTIC OPERATOR

Let P be a self-adjoint elliptic operator of order m > 0 acting on the sections of a Hermitian vector bundle over a compact Riemannian manifold of dimension n. General arguments show that its zeta and eta functions may have poles only at points of the form , where k ranges over all nonzero integers ≤ n. In this paper, we construct elementary and explicit examples of perturbations of P which make the zeta and eta functions become singular at all points at which they are allowed to have singularities. We proceed within three classes of operators: Dirac-type operators, self-adjoint first-order differential operators and self-adjoint elliptic pseudodifferential operators. As consequences, we obtain genericity results for the singularities of the zeta and eta functions in those settings. In particular, in the setting of Dirac-type operators we obtain a purely analytical proof of a well-known result of Branson–Gilkey [Residues of the eta function for an operator of Dirac type, J. Funct. Anal. 108(1) (1992) 47–87], which was obtained by invoking Riemannian invariant theory. As it turns out, the results of this paper contradict Theorem 6.3 of [R. Ponge, Spectral asymmetry, zeta functions and the noncommutative residue, Int. J. Math. 17 (2006) 1065–1090]. Corrections to that statement are given in this paper.

Asymptotic properties of the heat kernel on conic manifolds

We derive asymptotic properties for the heat kernel of elliptic cone (or Fuchs type) differential operators on compact manifolds with boundary. Applications include asymptotic formulas for the heat trace, counting function, spectral function, and zeta function of cone operators.

Spectral functions for the Schrödinger operator on R + with a singular potential

In this article we analyze the spectral zeta function, the heat kernel, and the resolvent of the operator −d2/dr2+κ/r2+r2 over the interval (0,∞) for κ≥−1/4. Depending on the self-adjoint extension chosen, nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed. In particular, for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s=−k, k∊N0, do occur. This implies that the small-t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk/(ln t)l+1 for k,l∊N0. The corresponding statements for the resolvent are also shown. Furthermore, we evaluate the zeta determinant of the operator for all values of κ and any self-adjoint extension.

ζ-determinants of Laplacians with Neumann and Dirichlet boundary conditions

In this paper, we derive a formula for the ratio of the ζ-determinants of the Laplacian with Neumann and Dirichlet boundary conditions over a noncompact manifold with an infinite cylindrical end and a compact boundary in terms of the ζ-determinant of the Dirichlet to Neumann map.