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Spectral theory of SG pseudo-differential operators on L p ( )

Aparajita Dasgupta, M. W. Wong (2008)

Studia Mathematica

To every elliptic SG pseudo-differential operator with positive orders, we associate the minimal and maximal operators on L p ( ) , 1 < p < ∞, and prove that they are equal. The domain of the minimal ( = maximal) operator is explicitly computed in terms of a Sobolev space. We prove that an elliptic SG pseudo-differential operator is Fredholm. The essential spectra of elliptic SG pseudo-differential operators with positive orders and bounded SG pseudo-differential operators with orders 0,0 are computed....

Spectral theta series of operators with periodic bicharacteristic flow

Jens Marklof (2007)

Annales de l’institut Fourier

The theta series ϑ ( z ) = exp ( 2 π i n 2 z ) is a classical example of a modular form. In this article we argue that the trace ϑ P ( z ) = Tr exp ( 2 π i P 2 z ) , where P is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of ϑ P ( z ) near the real axis, and the proof of logarithm laws and limit theorems for its value...

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