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Weyl formula for quasi-elliptic pseudo-differential operators. (English) Zbl 1072.58018

The paper deals with the asymptotic behaviour for large \(\lambda\) of the counting function \(N(\lambda)\) associated to quasi-elliptic anisotropic pseudo-differential operators on compact manifolds. The technique employed makes use of the observation that the quasi-homogeneous principal symbol of an anisotropic operator takes invariant meaning on a new vector bundle \(T^*_qM\) which replaces the cotangent bundle \(T^*M\) in the homogeneous case. The Weyl formula then is reached by means of the heat method.

MSC:

58J37 Perturbations of PDEs on manifolds; asymptotics
58J40 Pseudodifferential and Fourier integral operators on manifolds
35H30 Quasielliptic equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
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