Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Stochastic Analysis

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Franco Fagnola, Zheng Li

Published Jun 3, 2021• 0 citations
Publisher

Abstract

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state.In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.

DOI
10.31390/josa.2.2.06

Related papers

Self-adjoint approximations of degenerate Schrodinger operator

Vsevolod Zhanovich Sakbaev, I. V. Volovich

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

arXiv:1701.02777•Jan 10, 2017•0 citations

On Quantum Dynamics on C*-Algebras

Igor Vasil'evich Volovich, Vsevolod Zhanovich Sakbaev

We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain C *-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these C *-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on C *-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the C *-subalgebras.

Proceedings of the Steklov Institute of Mathematics•May 1, 2018•18 citations

Self-adjoint Extensions in Quantum Mechanics

Dmitri M Gitman, Igor Viktorovich Tyutin, Boris Leonidovich Voronov

Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and t

Birkhäuser Boston eBooks•Jan 1, 2012•158 citations

Self-adjoint differential operators associated with self-adjoint differential expressions

Boris Leonidovich Voronov, Igor Viktorovich Tyutin, Dmitri M Gitman

Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

•Mar 21, 2006•20 citations

Self-adjoint approximations of the degenerate Schrödinger operator

Vsevolod Zhanovich Sakbaev, Igor Vasil'evich Volovich

The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C *-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C *-algebras are investigated and it is shown that pure states could evolve into mixed states.

P-Adic Numbers Ultrametric Analysis and Applications•Jan 1, 2017•7 citations

Self-adjoint differential operators assosiated with self-adjoint differential expressions

Boris Leonidovich Voronov, Dmitri M Gitman, Igor Viktorovich Tyutin

Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

arXiv:quant-ph/0603187•Mar 21, 2006•1 citation
In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state.In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.
The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of this terms are corresponded to the unitary and shift components of Wold's decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain C *-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these C *-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on C *-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the C *-subalgebras.
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C *-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C *-algebras are investigated and it is shown that pure states could evolve into mixed states.
Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.