On the Right Hamiltonian for Singular Perturbations: General Theory

1997 ◽  
Vol 09 (05) ◽  
pp. 609-633 ◽  
Author(s):  
Hagen Neidhardt ◽  
Valentin Zagrebnov
Keyword(s):  
General Theory ◽  
Symmetric Operator ◽  
Stability Domain ◽  
Bounded Operators ◽  
Friedrichs Extension ◽  
Abstract Problem ◽  
Dense Domain ◽  
Adjoint Extension ◽  
The Right ◽  
Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

TOWARDS THE RIGHT HAMILTONIAN FOR SINGULAR PERTURBATIONS VIA REGULARIZATION AND EXTENSION THEORY

1996 ◽  
Vol 08 (05) ◽  
pp. 715-740 ◽  
Author(s):  
HAGEN NEIDHARDT ◽  
VALENTIN ZAGREBNOV
Keyword(s):  
Operator Method ◽  
Stability Domain ◽  
Extension Theory ◽  
Friedrichs Extension ◽  
Natural Domain ◽  
Abstract Operator ◽  
Adjoint Extension ◽  
Ill Posed ◽  
The Stability ◽  
The Right

For singular potentials in quantum mechanics it can happen that the Schrödinger operator is not esssentially self-adjoint on a natural domain, i.e., each self-adjoint extension is a candidate for the right physical Hamiltonian. Traditional way to single out this Hamiltonian is the removing cut-offs for regularizing potential. Connecting regularization and extension theory we develop an abstract operator method to treat the problem of the right Hamiltonian. We show that, using the notion of the maximal (with respect to the perturbation) Friedrichs extension of unperturbed operator, one can classify the above problem as wellposed or ill-posed depending on intersection of the quadratic form domain of perturbation and deficiency subspace corresponding to restriction of unperturbed operator to stability domain. If this intersection is trivial, then the right Hamiltonian is unique: it coincides with the form sum of perturbation and the Friedrich extension of the unperturbed operator restricted to the stability domain. Otherwise it is not unique: the family of “right Hamiltonians” can be described in terms of symmetric extensions reducing the ill-posed problem to the well-posed problem.

The Stieltjes Moment Problem

1981 ◽  
Vol 24 (3) ◽  
pp. 279-282
Author(s):  
G. Klambauer
Keyword(s):  
Hilbert Space ◽  
Moment Problem ◽  
Symmetric Operator ◽  
Extension Theorem ◽  
Spectral Theorem ◽  
Friedrichs Extension ◽  
Stieltjes Moment Problem ◽  
Operator Version ◽  
Adjoint Extension ◽  
Adjoint Operators

We shall apply the spectral theorem for self adjoint operators in Hilbert space to study an operator version of the Stieltjes moment problem [1]. In the course of the work we shall make use of the Friedrichs extension theorem which states that any non-negative symmetric operator in Hilbert space has a non-negative self adjoint extension.

ON THE PROBLEM OF THE RIGHT HAMILTONIAN UNDER SINGULAR FORM-SUM PERTURBATIONS

2000 ◽  
Vol 12 (01) ◽  
pp. 1-24 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
VOLODYMYR KOSHMANENKO
Keyword(s):  
Quadratic Forms ◽  
Symmetric Operator ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Singular Form ◽  
Approximating Sequence ◽  
Sum Formula ◽  
Definition Of ◽  
The Right ◽  
Adjoint Operators

Let a perturbation of the self-adjoint operator H0>0 in the Hilbert space ℋ be given by an operator V (or by a quadratic form ν) which is possibly singular and in general nonpositive, so H0+V on [Formula: see text] is only a symmetric operator with nontrivial deficiency indices. The definition of the sum [Formula: see text] in the sense of quadratic forms is extended to cases which are not covered by the well-known KLMN-theorem and conditions are found which ensure the unique self-adjoint realization of H in ℋ. It is also shown that ℋ coincides with the strong resolvent limit of the approximating sequence Hn = H0+Vn, where Vn are bounded self-adjoint operators such that Vn → V in a suitable sense. Essentially that operator V might be strongly singular and acts in the H0-scale of spaces, V:ℋ+→ℋ-.

On an open problem of Weidmann: essential spectra and square-integrable solutions

2011 ◽  
Vol 141 (2) ◽  
pp. 417-430 ◽  
Author(s):  
Jiangang Qi ◽  
Shaozhu Chen
Keyword(s):  
Essential Spectrum ◽  
Open Problem ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Essential Spectra ◽  
Entire Interval ◽  
Linearly Independent ◽  
Adjoint Extension ◽  
Set Up ◽  
The Right

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

Adversarial Medical and Scientific Testimony and Lay Jurors: A Proposal for Medical Malpractice Reform

1995 ◽  
Vol 21 (2-3) ◽  
pp. 281-300
Author(s):  
Jody Weisberg Menon
Keyword(s):  
General Theory ◽  
Legal System ◽  
Medical Malpractice ◽  
Common Knowledge ◽  
Common Law ◽  
Expert Witnesses ◽  
Malpractice Reform ◽  
The Right ◽  
Scholarly Attention ◽  
Scientific Testimony

Pleas for reform of the legal system are common. One area of the legal system which has drawn considerable scholarly attention is the jury system. Courts often employ juries as fact-finders in civil cases according to the Seventh Amendment of the Constitution: “In Suits at common law, where the value in controversy shall exceed twenty dollars, the right of trial by jury shall be preserved … .” The general theory behind the use of juries is that they are the most capable fact-finders and the bestsuited tribunal for arriving at the most accurate and just outcomes. This idea, however, has been under attack, particularly by those who claim that cases involving certain difficult issues or types of evidence are an inappropriate province for lay jurors who typically have no special background or experience from which to make informed, fair decisions.The legal system uses expert witnesses to assist triers of fact in understanding issues which are beyond their common knowledge or difficult to comprehend.

scholarly journals EL DERECHO A PROBAR EN EL PROCEDIMIENTO DE INVESTIGACIÓN Y SANCIÓN DEL HOSTIGAMIENTO SEXUAL: ¿UNA GARANTÍA RECORTADA?

2020 ◽  
pp. 91-102
Author(s):  
LUIS MARTÍN BRAVO SENMACHE
Keyword(s):  
Sexual Harassment ◽  
General Theory ◽  
Due Process ◽  
The Other ◽  
Legislative Reform ◽  
Due Process Of Law ◽  
The Right

Con base en la teoría general del proceso, la investigación determina que en el Procedimiento de Investigación y Sanción del Hostigamiento Sexual (PISHS)es identificable la estructura del contradictorio, por lo que su naturaleza es la de un proceso. Sin embargo, la revisión del tratamiento normativo que el PISHS ha dedicado al derecho a la prueba de la parte acusada pone en evidencia que, en la estructura de dicho proceso, el contradictorio no ha sido implementado más que parcialmente, dado que su dimensión sustancial (específicamente, el poder de influencia) no ha sido cabalmente asegurada a favor del presunto/a hostigador/a. Dos escenarios se erigen como posible solución al problema: uno a través de la vía de hecho (preferencia del principio del debido proceso) y otro mediante la reforma legislativa del art. 17.2 del reglamento. Based on the general theory of the process, the investigation determines that in the Investigation and Sanction Procedure for Sexual Harassment (PISHS) the structure of the contradictory is identifiable, so its nature is that of a process. However, the review of the normative treatment that the PISHS has dedicated to the right to proof of the accused party shows that, in the structure of said process, the contradictory has only been partially implemented, given that its substantial dimension (specifically, the power of influence) has not been fully secured in favor of the alleged harasser. Two scenariosare erected as a possible solution to the problem: one through the facto route (preference for the principle of due process of law) and the other through the legislative reform of the art. 17.2 of the reglament.

M-embedded symmetric operator spaces and the derivation problem

2019 ◽  
Vol 169 (3) ◽  
pp. 607-622
Author(s):  
JINGHAO HUANG ◽  
GALINA LEVITINA ◽  
FEDOR SUKOCHEV
Keyword(s):  
Symmetric Spaces ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Symmetric Operator Spaces ◽  
Order Continuous

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

scholarly journals Self-Adjoint Extensions with Friedrichs Lower Bound

2020 ◽  
Vol 14 (7) ◽  
Author(s):  
Matteo Gallone ◽  
Alessandro Michelangeli
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Symmetric Operator ◽  
Simple Criterion ◽  
Abstract Result ◽  
Friedrichs Extension ◽  
Lower Semi

Abstract We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

scholarly journals Congruences induced by transitive representations of inverse semigroups

1987 ◽  
Vol 29 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Mario Petrich ◽  
Stuart Rankin
Keyword(s):  
General Theory ◽  
Inverse Semigroup ◽  
Symmetric Group ◽  
Transitive Group ◽  
Inverse Semigroups ◽  
Group Representations ◽  
Symmetric Inverse Semigroup ◽  
The Right ◽  
Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

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