General two-electron exponential type orbital integrals in polyatomics without orbital translations

2009 ◽  
Vol 109 (13) ◽  
pp. 2926-2932 ◽  
Author(s):  
Philip E. Hoggan
Keyword(s):  
Exponential Type ◽  
Orbital Integrals ◽  
Type Orbital ◽  
Exponential Type Orbital

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Functions in the Schwartz algebra that are invertible in the sense of Ehrenpreis

2019 ◽  
Vol 484 (1) ◽  
pp. 7-11
Author(s):  
N. F. Abuzyarova
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Zero Set ◽  
Schwartz Algebra

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

scholarly journals Semi-Exponential Operators

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 637
Author(s):  
Monika Herzog
Keyword(s):  
Exponential Type ◽  
Probabilistic Approach ◽  
Weighted Spaces ◽  
Approximation Properties

In this paper we study approximation properties of exponential-type operators for functions from exponential weighted spaces. We focus on some modifications of these operators and we derive a new example of such operators. A probabilistic approach for these modifications is also demonstrated.

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