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Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions

Boris Leonidovich Voronov, Dmitri M Gitman, Igor Viktorovich Tyutin

Russian Physics JournalFeb 1, 200813 citations

J-self-adjoint operators with \mathcal{C} -symmetries: an extension theory approach

Sergio A Albeverio, U Günther, Sergiusz Kużel

A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ( J -self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨ n , n ⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J -self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ( J -self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨ n , n ⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J -self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.

arXiv:0811.0365Journal of Physics A Mathematical and TheoreticalFeb 13, 200936 citations

Mathematical aspects of quantum mechanics with non-self-adjoint operators

David Krejčiřı́k

Jun 6, 20182 citations

Reduction by symmetries in singular quantum-mechanical problems: General scheme and application to Aharonov-Bohm model

A. G. Smirnov

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respects a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.We develop a general technique for finding self-adjoint extensions of a symmetric operator that respects a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.

arXiv:1411.5351Journal of Mathematical PhysicsDec 1, 20153 citations

Chapter 5 Algebras of Quantum Observables

Vasily Evgenevich Tarasov

Monograph series on nonlinear science and complexityJan 1, 20080 citations

Applications of Self-Adjoint Extensions in Quantum Physics

Pavel Exner, Seba, Petr

Lecture notes in physicsJan 1, 198944 citations

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