scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

scholarly journals The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180264 ◽  
Author(s):  
David Krejčiřík ◽  
Vladimir Lotoreichik ◽  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Adjoint Operator ◽  
Inner Product ◽  
Sufficient Condition ◽  
Lagrange Equations ◽  
Entire Class ◽  
Inner Products ◽  
Metric Operator ◽  
A Charge

We propose a unique way to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimizing a ‘Hilbert–Schmidt distance’ to the original inner product among the entire class of admissible inner products. We prove that either the minimizer exists and is unique or it does not exist at all. In the former case, we derive a system of Euler–Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supported by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.

Variations of Integrals in Diffeology

2013 ◽  
Vol 65 (6) ◽  
pp. 1255-1286
Author(s):  
Patrick Iglesias-Zemmour
Keyword(s):  
Differential Forms ◽  
Moment Map ◽  
De Rham Cohomology ◽  
Homotopy Operator ◽  
Chain Homotopy ◽  
De Rham ◽  
A Chain ◽  
The Moment

AbstractWe establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator K, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

scholarly journals An approach to nonlinear elliptic boundary value problems

1983 ◽  
Vol 34 (3) ◽  
pp. 316-335 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Nonlinear Elliptic Boundary ◽  
Image Position

AbstractLet D ⊂ Rn be a bounded domain and L: dom L ⊂ L2 (D) → L2 (D) be a self-adjoint operator of finite dimensional kernel. Let f: D × R → R be a function satisfying the Carathéodory condition. Assume that there are constants λ > 0 and δ ∈ [0, 1] such that and that .Then with the aid of a generalized Krasnosel'skii's theorem it has been proved that under conditions exactly analogous to those of Landesman and Lazer there exists u ∈ L2(D) such that L(u)(x) = f(x, u(x)) for ∀x ∈ D. This result is then used to prove the existence of weak solutions of nonlinesr elliptic boundary value problems.Other abstract results applicable to ordinary and partial differential equations have also been proved.

scholarly journals RECURSION RELATIONS IN SEMIRIGID TOPOLOGICAL GRAVITY

1992 ◽  
Vol 07 (27) ◽  
pp. 6773-6798 ◽  
Author(s):  
EUGENE WONG
Keyword(s):  
Hilbert Space ◽  
Moduli Space ◽  
De Rham Cohomology ◽  
Recursion Relations ◽  
Brst Cohomology ◽  
De Rham ◽  
Topological Gravity

A field theoretical realization of topological gravity is discussed in the semirigid geometry context. In particular, its topological nature is given by the relation between de Rham cohomology on the moduli space and equivariant BRST cohomology on the Hilbert space and the fact that all but one of the physical operators are BRST exact. The puncture equation and the dilaton equation of pure topological gravity are reproduced, following Ref. 1.

scholarly journals On twisted de Rham cohomology

1997 ◽  
Vol 146 ◽  
pp. 55-81 ◽  
Author(s):  
Alan Adolphson ◽  
Steven Sperber
Keyword(s):  
Differential Forms ◽  
Exterior Derivative ◽  
De Rham Cohomology ◽  
De Rham ◽  
Twisted De Rham Cohomology

Abstract.Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø, we compute the cohomology of this complex.

scholarly journals Eigencurves for linear elliptic equations

2019 ◽  
Vol 25 ◽  
pp. 45
Author(s):  
Mauricio A. Rivas ◽  
Stephen B. Robinson
Keyword(s):  
Hilbert Space ◽  
Elliptic Equations ◽  
Separable Hilbert Space ◽  
Elliptic Boundary ◽  
Bilinear Forms ◽  
Asymptotic Results ◽  
Elliptic Boundary Value ◽  
Real Separable Hilbert Space ◽  
Linear Elliptic Boundary ◽  
Two Parameter

This paper provides results forvariational eigencurvesassociated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a,b,m) of continuous symmetric bilinear forms on a real separable Hilbert spaceV.Geometric characterizationsof eigencurves associated with (a,b,m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.

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