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A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. (English) Zbl 0559.58025
Let p be a classical observable and P a corresponding quantum observable. By comparing classical and quantum partition functions one can prove that for large $$\lambda$$ (1) $$N(\lambda)\sim \gamma volume$$ $$(p\leq 1)$$. Here $$\gamma$$ is a thermodynamical constant (independent of p and P) and $$N(\lambda)$$ is the number of eigenvalues of P less than $$\lambda$$. The author’s purpose is to give a ”soft” proof of (1). For this reason he introduces a Weyl algebra W associated with a symplectic cone y, and considers self-adjoint positive elliptic operators of order one belonging to W. He then examines the function $$\zeta (P,\mu):=\int_{0}\lambda^{\mu}dN(\lambda)$$, $$\mu\in {\mathbb{C}}$$, and proves that $$\zeta(P,\mu)$$ converges in the half-plane Re$$\mu<-d$$, $$2d=\dim y$$, and has a meromorphic continuation to the whole $$\mu$$-plane with poles at $$\mu =-d,-d-1$$, etc. The pole at $$\mu =-d$$ is simple with residue equal to a constant $$\gamma$$, depending only on W, times the symplectic volume of the set where the symbol of P is less than one. From this (1) is deduced by a Tauberian argument.
Reviewer: N.Jacob

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 81S99 General quantum mechanics and problems of quantization 35P20 Asymptotic distributions of eigenvalues in context of PDEs 58J40 Pseudodifferential and Fourier integral operators on manifolds 53D50 Geometric quantization
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