Pseudodifferential operators on manifolds, index of elliptic operators

2008 ◽  
pp. 195-215
Author(s):  
Gerd Grubb
Keyword(s):  
Pseudodifferential Operators ◽  
Elliptic Operators

scholarly journals AN INDEX FORMULA FOR THE EXTENDED HEISENBERG ALGEBRA OF EPSTEIN, MELROSE AND MENDOZA

2010 ◽  
Vol 02 (04) ◽  
pp. 527-542 ◽  
Author(s):  
ERIK VAN ERP
Keyword(s):  
Pseudodifferential Operators ◽  
Contact Manifold ◽  
Heisenberg Algebra ◽  
Elliptic Operators ◽  
Index Formula ◽  
Symbolic Calculus ◽  
Fredholm Operators

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah–Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.

scholarly journals Pseudodifferential calculus on noncommutative tori, II. Main properties

2019 ◽  
Vol 30 (08) ◽  
pp. 1950034 ◽  
Author(s):  
Hyunsu Ha ◽  
Gihyun Lee ◽  
Raphaël Ponge
Keyword(s):  
Spectral Theory ◽  
Trace Formula ◽  
Sobolev Spaces ◽  
Pseudodifferential Operators ◽  
Elliptic Operators ◽  
Schatten Class ◽  
Noncommutative Tori ◽  
Dimension Formula ◽  
Pseudodifferential Calculus ◽  
Negative Order

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

A computer-assisted proof method of the invertibility to elliptic operators

2014 ◽  
Vol 1 ◽  
pp. 816-819
Author(s):  
Akitoshi Takayasu ◽  
Shin'ichi Oishi
Keyword(s):  
Elliptic Operators ◽  
Computer Assisted ◽  
Computer Assisted Proof

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Adjoint Operator ◽  
Integral Kernel ◽  
Mathematical Finance ◽  
Elliptic Operators ◽  
Complete Analysis ◽  
Time Behavior ◽  
Degenerate Elliptic Operators ◽  
Regularity Properties ◽  
Degenerate Elliptic ◽  
Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Pseudodifferential operators on differential groupoids

1999 ◽  
Vol 189 (1) ◽  
pp. 117-152 ◽  
Author(s):  
Victor Nistor ◽  
Alan Weinstein ◽  
Ping Xu
Keyword(s):  
Pseudodifferential Operators

scholarly journals Generalized principal eigenvalues of convex nonlinear elliptic operators in ℝN

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anup Biswas ◽  
Prasun Roychowdhury
Keyword(s):  
Eigenvalue Problem ◽  
Principal Eigenvalue ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
Maximum Principles ◽  
Generalized Eigenvalues ◽  
Nonlinear Elliptic ◽  
Generalized Eigenvalue ◽  
General Convex ◽  
Appl Math

AbstractWe study the generalized eigenvalue problem in {\mathbb{R}^{N}} for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

scholarly journals Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

2021 ◽  
Author(s):  
Steve Hofmann ◽  
José María Martell ◽  
Svitlana Mayboroda ◽  
Tatiana Toro ◽  
Zihui Zhao
Keyword(s):  
Carleson Measure ◽  
Elliptic Operators ◽  
Uniform Rectifiability

Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective

2013 ◽  
Vol 174 (1) ◽  
pp. 134-153 ◽  
Author(s):  
G. F. Helminck ◽  
A. G. Helminck ◽  
E. A. Panasenko
Keyword(s):  
Pseudodifferential Operators ◽  
Integrable Deformations
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