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

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 ( ℝ ⁿ ) , 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...

Spectre d'ordre supérieur et problèmes aux limites quasi-linéaires

Aomar Anane, Omar Chakrone, Jean-Pierre Gossez (2001)

Bollettino dell'Unione Matematica Italiana

Nello studio dei problemi del tipo - Δ ⁢ u = f ⁢ x , u + h ⁢ x , si impongono generalmente delle condizione sul comportamento asintotico di f ⁢ x , u rispetto allo spettro di - Δ . Avendo in vista dei problemi quasilineari del tipo - Δ ⁢ u = f ⁢ x , u , ∇ ⁡ u + h ⁢ x , sembra naturale introdurre una nozione di spettro per - Δ che tenga conto della dipendenza del membro di destra rispetto al gradiende ∇ ⁡ u . L'oggetto di questo lavoro è di definire, studiare e applicare questa nuova nozione di spettro.

Spectre négatif d'un opérateur elliptique avec des conditions au bord de Robin.

Yuri V. Egorov, Mohammed El Aidi (2001)

Publicacions Matemàtiques

In this article we discuss some estimates of the number of the negative eigenvalues and their moments of energy for an elliptic operator L = L0 - V(x) defined in Hm(R+n) with the Robin boundary conditions containing a potential W(x), in terms of some integrals of V and W.

Spectres et groupes cristallographiques. II : domaines sphériques

Pierre Bérard, Gérard Besson (1980)

Annales de l'institut Fourier

Dans cet article, nous donnons une description des spectres du laplacien dans certains domaines sphériques. Les représentations des groupes de Coxeter cristallographiques y jouent un rôle fondamental.

Spectrum of the Laplace operator and periodic geodesics: thirty years after

Yves Colin de Verdière (2007)

Annales de l’institut Fourier

What is called the “Semi-classical trace formula” is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of...

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