Weyl asymptotics of bisingular operators and Dirichlet divisor problem

Abstract

We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of J. Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function.

On the η‐function for bisingular pseudodifferential operators

“…Remark The integral for exists by the standard estimate , cf. , Thm. 2.1. Given which admits complex powers we obtain an operator see , Thm.…”

Sharp Weyl estimates for tensor products of pseudodifferential operators

“…Without loss of generality, we can assume m 1 = 1, possibly considering an appropriate power of A, see [Bat12]. Moreover, again without loss of the generality, we can assume that all the eigenvalues are strictly larger than one, so that the Assumptions 1 are fulfilled.…”

“…For the connection between Dirichlet divisor problem and standard bisingular operators on the product of closed manifolds see [Bat12]. Actually, since we deal with the nonsymmetric case, it is not possible to attack directly the traditional Dirichlet divisor problem through the approach described in the previous sections, while our techniques are well suited to treat generalized anisotropic Dirichlet divisors problems like, for instance,…”

“…Similar formulae can be obtained in many other different settings, see [SV97] and [ANPS09] for a detailed analysis and several developments. To mention a few specific situations, see [Shu87,HR81] for the case of the Shubin calculus on R n , [BN03] for the anisotropic Shubin calculus, [BC11,CM13,Nic03] for the SG-operators on R n and the manifolds with ends, [GL02] for operators on conic manifolds, [Mor08] for operators 1 on cusp manifolds, [DD13] for operators on asymptotic hyperbolic manifolds, [Bat12,BGRP13] for bisingular operators.…”

“…In the case r = 2, for classical Hörmander operators on closed manifolds, the operators we consider are a subclass of the so-called bisingular operators, studied by L. Rodino in [Rod75] (see also [NR06]) in connection with the multiplicative property of the Atiyah-Singer index [AS68]. An asymptotic expansion of the counting function of bisingular operators was obtained by the first author in [Bat12]. The basic tool was the spectral ζ-function, in the spirit of Guillemin's so-called soft proof of the Weyl law [Gui85].…”