Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications

1993 ◽  
Vol 17 (4) ◽  
pp. 451-463 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Clifford Algebra ◽  
Adjoint Operators

scholarly journals A characterization of singular packing subspaces with an application to limit-periodic operators

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Silas L. Carvalho ◽  
César R. de Oliveira
Keyword(s):  
Packing Dimension ◽  
Spectral Measures ◽  
Adjoint Operators ◽  
Periodic Operators

AbstractA new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a

Monotone convergence theorems for semi-bounded operators and forms with applications

2010 ◽  
Vol 140 (5) ◽  
pp. 927-951 ◽  
Author(s):  
Jussi Behrndt ◽  
Seppo Hassi ◽  
Henk de Snoo ◽  
Rudi Wietsma
Keyword(s):  
Differential Operators ◽  
Multiplication Operators ◽  
Monotone Convergence ◽  
Monotone Sequence ◽  
Bounded Operators ◽  
Von Neumann ◽  
Sturm Liouville ◽  
Adjoint Operators ◽  
Liouville Operators

AbstractLet Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

scholarly journals A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 385-404 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Inner Product ◽  
Type I ◽  
Weak Formulation ◽  
Friedrichs Extension ◽  
Sesquilinear Form ◽  
Adjoint Operators ◽  
Special Case

AbstractIn this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

scholarly journals On the Self-Adjointness of H+A∗+A

2020 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrea Posilicano
Keyword(s):  
Field Theory ◽  
Annihilation Operator ◽  
The Self ◽  
Quantum Field ◽  
Bounded Operators ◽  
Nelson Model ◽  
Adjoint Operators ◽  
Nonperturbative Theory

AbstractLet $H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}$ H : dom ( H ) ⊆ F → F be self-adjoint and let $A:\text {dom}(H)\to \mathfrak {F}$ A : dom ( H ) → F (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations $\widehat H$ H ̂ of the formal Hamiltonian H + A∗ + A with $\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}$ dom ( H ) ∩ dom ( H ̂ ) = { 0 } . We give an explicit characterization of $\text {dom}(\widehat H)$ dom ( H ̂ ) and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$ ( − H ̂ + z ) − 1 − ( − H + z ) − 1 . Moreover, we consider the problem of the description of $\widehat H$ H ̂ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$ H + A n ∗ + A n + E n , where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.

scholarly journals The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

2021 ◽  
Vol 93 (2) ◽  
Author(s):  
Robert Denk ◽  
Markus Kunze ◽  
David Ploß
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Boundary Conditions ◽  
Regularity Of Solutions ◽  
Lipschitz Domains ◽  
Lipschitz Boundary ◽  
Full Characterization ◽  
Wentzell Boundary Conditions ◽  
Adjoint Operators

AbstractWe investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d with Lipschitz boundary $$\Gamma $$ Γ . More precisely, using form methods, we show that the associated operator on the ground space $$L^2(\Omega )\times L^2(\Gamma )$$ L 2 ( Ω ) × L 2 ( Γ ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

Some shaper uncertainty principles for multivector-valued functions

2016 ◽  
Vol 14 (06) ◽  
pp. 1650043
Author(s):  
Minggang Fei ◽  
Yubin Pan ◽  
Yuan Xu
Keyword(s):  
Fourier Transform ◽  
Clifford Algebra ◽  
Uncertainty Principle ◽  
Dunkl Transform ◽  
Uncertainty Principles ◽  
Heisenberg Uncertainty Principle ◽  
Clifford Fourier Transform ◽  
Heisenberg Uncertainty ◽  
The Fourier Transform ◽  
Adjoint Operators

The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. In this paper, in the framework of Clifford algebra, we establish a stronger Heisenberg–Pauli–Wely type uncertainty principle for the Fourier transform of multivector-valued functions, which generalizes the recent results about uncertainty principles of Clifford–Fourier transform. At the end, we consider another stronger uncertainty principle for the Dunkl transform of multivector-valued functions.

The characterization of quantum-mechanical operators

1950 ◽  
Vol 46 (4) ◽  
pp. 614-619 ◽  
Author(s):  
J. L. B. Cooper
Keyword(s):  
Quantum Theory ◽  
Representation Theorem ◽  
Mathematical Theory ◽  
Physical Significance ◽  
The Self ◽  
Quantum Mechanical ◽  
Resolution Of The Identity ◽  
Adjoint Operators

It is well known that the operators mainly employed in quantum theory are hermitian; it is less well known amongst physicists that they are required, in addition, to be self-adjoint. This is essential for the validity of the result known in quantum theory as the representation theorem and in the mathematical theory as the resolution of the identity. The purpose of this paper is to show that the self-adjoint operators can be characterized by a condition which is nearer to having a physical significance than those given in the literature.

On the rigorous ergodic theorem for a class of nonlinear Klein–Gordon wave propagations

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Luiz C. L. Botelho
Keyword(s):  
Continuous Spectrum ◽  
Ergodic Theorem ◽  
Path Integrals ◽  
Classical Wave ◽  
Complete Study ◽  
Klein Gordon ◽  
Adjoint Operators ◽  
Rage Theorem

AbstractWe present a complete study of the ergodic theorem for the problem of nonlinear Klein–Gordon classical wave propagations through cylindrical measures, rigorous mathematical path integrals and the well-known Ruelle–Amrein–Georgescu–Enss (RAGE) theorem on the characterization of the continuous spectrum of self-adjoint operators.

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