On the epsilon - 0 limit of the Lippmann-Schwinger-Low states

2004 ◽  
Vol 82 (12) ◽  
pp. 1085-1095
Author(s):  
V J Menon ◽  
Ritesh Kumar Dubey
Keyword(s):  
Hilbert Space ◽  
Projection Operator ◽  
State Vector ◽  
Resolvent Operator ◽  
Quantum Scattering ◽  
Scattering States ◽  
Nonlinear Relation ◽  
New Type ◽  
Transition Amplitudes ◽  
Separable Interaction

The Lippmann–Schwinger–Low (LSL) quantum scattering states involve a resolvent operator depending on an infinitesimal adiabatic parameter ε. We reexamine the LSL formalism by taking the ε → + 0 limit at the end of the analysis (rather than at the outset). It is found that the LSL state vector |ψ kL > does not coincide with the Schrödinger eigen vector in Hilbert space as a whole, and the pair |ψ nL >, |ψ kL > is mutually nonorthogonal if the energy En = Ek, n ≠ k. For this purpose we carefully use a new type of projection operator ηk, a novel nonlinear relation among transition amplitudes, and a separable interaction as illustration. PACS Nos.: 0.3.65.Nk, 0.3.80.+r

scholarly journals Resolvent operator and spectrum of new type boundary value problems

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1671-1680 ◽  
Author(s):  
Oktay Mukhtarov ◽  
Hayati Olğar ◽  
Kadriye Aydemir
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Boundary Value Problems ◽  
Resolvent Operator ◽  
Boundary Value ◽  
Lebesgue Spaces ◽  
Direct Sum ◽  
New Type ◽  
Type Boundary ◽  
The Resolvent Operator

The aim of this study is to investigate a new type boundary value problems which consist of the equation -y''(x) + (By)(x) = ?y(x) on two disjoint intervals (-1,0) and (0,1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L2(-1,0)( L2(0,1). By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

scholarly journals NEW CONCEPT OF SOLVABILITY IN QUANTUM MECHANICS

2013 ◽  
Vol 53 (5) ◽  
pp. 473-482 ◽  
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Ad Hoc ◽  
Current Concept ◽  
Quantum Models ◽  
Solvable Quantum ◽  
New Type

In quite a few recent quantum models one is allowed to make a given Hamiltonian <em>H</em> self-adjoint only after an ad hoc generalization of Hermitian conjugation, <em>H</em><sup>†</sup>→<em>H</em><sup>‡</sup>:= Θ <sup>−1</sup><em>H</em><sup>†</sup>Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠<em> I</em> the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of <em>H</em>) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.

scholarly journals Some Properties of D-Operator on Hilbert Space

2020 ◽  
pp. 3366-3371
Author(s):  
Eiman Al-janabi
Keyword(s):  
Hilbert Space ◽  
Direct Product ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Invertible Operator ◽  
D Operator ◽  
New Type ◽  
Equivalent Property ◽  
Drazin Invertible Operator

In this paper, we introduce a new type of Drazin invertible operator on Hilbert spaces, which is called D-operator. Then, some properties of the class of D-operators are studied. We prove that the D-operator preserves the scalar product, the unitary equivalent property, the product and sum of two D-operators are not D-operator in general but the direct product and tenser product is also D-operator.

scholarly journals Quantum Corrections to the Dynamics of the Gravitational System

2019 ◽  
Vol 64 (12) ◽  
pp. 1143
Author(s):  
V. V. Kuzmichev ◽  
V. E. Kuzmichev
Keyword(s):  
State Vector ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
The State ◽  
Constrained Systems ◽  
Short Introduction ◽  
Gravitational Systems ◽  
New Type ◽  
The Universe ◽  
Insight Into

A short introduction into the theory of quantum gravitational systems with a finite number of degrees of freedom is given. The theory is based on the method of quantization of constrained systems. The state vector of the system satisfies a set of wave equations which describes the time evolution of the system in the space of quantum fields. The state vector in such an approach can be normalized to unity. The theory permits a generalization to negative values of the scale factor and, being applied to cosmology, leads to the new understanding of the evolution of the universe. It gives an insight into the reasons why the regime of the expansion may change from acceleration to deceleration or vice versa, revealing a new type of quantum forces acting like dark matter and dark energy in the universe.

scholarly journals Hilbert space multidimensional modelling of continuous measurements

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190142
Author(s):  
J. R. Busemeyer ◽  
Z. Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Data Fusion ◽  
State Vector ◽  
Joint Distribution ◽  
Context Effect ◽  
Theme Issue ◽  
Multiple Contexts ◽  
Complete Set ◽  
Multidimensional Modelling

Data fusion problems arise when a researcher needs to analyse results obtained by measuring empirical variables under different measurement contexts. A context is defined by a subset of variables taken from a complete set of variables under investigation. Multiple contexts can be formed from different subsets, which produce a separate distribution of measurements associated with each context. A context effect occurs when the distributions produced by the different contexts cannot be reproduced by marginalizing over a complete joint distribution formed by all the variables. We propose a Hilbert space multidimensional theory that uses a state vector and measurement operators to account for multiple distributions produced by different contexts. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

Generalized variational-like inclusion involving relaxed monotone operators

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Syed Shakaib Irfan ◽  
Mohammad F. Khan ◽  
Ali P. Farajzadeh ◽  
Allahkaram Shafie
Keyword(s):  
Hilbert Space ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Recent Literature ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Proximal Operator ◽  
New Class

Abstract In this paper, we introduce a new class of resolvent operator, the η-proximal operator, and discuss some of its properties. We consider a new generalized variational-like inclusion problem involving relaxed monotone operators in Hilbert space and construct a new iterative algorithm for proving the existence of the solutions of our problem. Our results improve and generalize many corresponding results in the recent literature.

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