Weyl asymptotics for tensor products of operators and Dirichlet divisors

Gramchev, Todor; Pilipović, Stevan; Rodino, Luigi; Vindas, Jasson

doi:10.1007/s10231-014-0400-z

Cited by 3 publications

(2 citation statements)

References 29 publications

(45 reference statements)

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“…Let us suppose that the spectrum of both A 1 and A 2 in (2) is formed by all strictly positive natural numbers, each with multiplicity one. Then, This clear spectral meaning of the Dirichlet divisor problem was one of the main motivation of the papers [BGRP13,GPRVar]. For the connection between Dirichlet divisor problem and standard bisingular operators on the product of closed manifolds see [Bat12].…”

Section: Appendix the Dirichlet Divisors Problemmentioning

confidence: 99%

Sharp Weyl Estimates for Tensor Products of Pseudodifferential Operators

Battisti,

Borsero,

Coriasco

2014

Preprint

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We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on R n , respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.

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Section: Appendix the Dirichlet Divisors Problemmentioning

confidence: 99%

Sharp Weyl Estimates for Tensor Products of Pseudodifferential Operators

Battisti,

Borsero,

Coriasco

2014

Preprint

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show abstract

“…This clear spectral meaning of the Dirichlet divisor problem was one of the main motivation of the papers [BGRP13,GPRVar]. For the connection between Dirichlet divisor problem and standard bisingular operators on the product of closed manifolds see [Bat12].…”

Section: Appendix the Dirichlet Divisors Problemmentioning

confidence: 99%

Sharp Weyl estimates for tensor products of pseudodifferential operators

Battisti

Borsero

Coriasco

2015

Annali di Matematica

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Abstract. We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on R n , respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.

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Unexpected spectral asymptotics for wave equations on certain compact spacetimes

Fox

Strichartz

2018

JAMA

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We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S 1 × S 2 . For the Laplacian on S 1 × S 2 the Weyl asymptotic law gives a growth rate O(s 3/2 ) for the eigenvalue counting function N (s) = #{λ j : 0 ≤ λ j ≤ s}. For the wave operator there are two corresponding eigenvalue counting functions N ± (s) = #{λ j : 0 < ±λ j ≤ s} and they both have a growth rate of O(s 2 ). More precisely there is a leading term π 2 4 s 2 and a correction term of as 3/2 where the constant a is different for N ± . These results are not robust, in that if we include a speed of propagation constant to the wave operator the result depends on number theoretic properties of the constant, and generalizations to S 1 × S q are valid for q even but not q odd. We also examine some related examples. AMS Mathematics subject Classification. Primary 35P20 35L05

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Weyl asymptotics for tensor products of operators and Dirichlet divisors

Cited by 3 publications

References 29 publications

Sharp Weyl Estimates for Tensor Products of Pseudodifferential Operators

Sharp Weyl Estimates for Tensor Products of Pseudodifferential Operators

Sharp Weyl estimates for tensor products of pseudodifferential operators

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

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