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Helffer–Sjöstrand Formula
The Helffer–Sjöstrand formula is a mathematical tool used in spectral theory and functional analysis to represent functions of self-adjoint operators. Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without requiring the operator to have a simple or explicitly known spectrum. It is especially useful in quantum mechanics, condensed matter physics, and other areas where understanding the properties of operators related to energy or observables is important. Background If f \in C_0^\infty (\mathbb) , then we can find a function \tilde f \in C_0^\infty (\mathbb) such that \tilde, _ = f , and for each N \ge 0, there exists a C_N > 0 such that , \bar \tilde, \leq C_N , \operatorname z, ^N. Such a function \tilde is called an almost analytic extension of f. The formula If f \in C_0^\infty(\mathbb) and A is a self-adjoint operator on a Hilbert space, then f(A) = \frac \int_ \bar \tilde(z) (z - A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of System of linear equations, systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on Principal axis theorem, principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for all x, y ∊ ''V''. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Helffer
Bernard Helffer (born 8 January 1949, Paris) is a French mathematician, specializing in partial differential equations, spectral theory, and mathematical physics. He is the son of the pianist Claude Helffer and the musicologist Mireille Helffer. Helffer studied from 1968 at the École Polytechnique and received in 1976 from the University of Paris-Sud his doctorate under Charles Goulaouic with dissertation ''Hypoellipticité pour des classes d'opérateurs pseudodifférentiels à caractéristiques multiples''. From 1971 to 1978 he did research at CNRS, from 1978 to 1989 he was a professor at the University of Nantes, and then he was a professor at the University Paris-Sud (and simultaneously taught for five years at the École Normale Superieure). His research in mathematical physics deals with statistical mechanics, liquid crystals, superconductivity, semiclassical approximation, and ground-state nodal lines in Laplace operators and Schrödinger operators. The French Academy o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Sjöstrand
Johannes Sjöstrand (born 1947) is a Swedish mathematician, specializing in partial differential equations and functional analysis. Sjöstrand received his doctorate in 1972 from Lund University under Lars Hörmander. Sjöstrand taught at the University of Paris XI and he is a professor at the University of Burgundy in Dijon. He is a member of the Royal Swedish Academy of Sciences and, since 2017, a member of the American Academy of Arts and Sciences. His research deals with microlocal analysis. He has investigated, ''inter alia'', the Schrödinger equation of an electron in a magnetic field (with a spectrum of the Hofstadter butterfly), Jean Bellissard ''Le papillon de Hofstadter, d'après B. Helffer et J. Sjöstrand'', Séminaire Bourbaki, Nr. 745, 1991/92Online resonances in the semiclassical limit, and quantum tunneling In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
