Perturbations of Self-Adjoint Operators (original) (raw)
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The self-adjointness of Hermitian Hamiltonians
Foundations of Physics, 1993
For several examples of Hermitian operators, the issues involved in their possible self-adjoint extension are shown to conform with recognizable properties in the solutions to the associated classical equations of motion. This result confirms the assertion made in an earlier paper (Ref 1) that there are sufficient classical "symptoms" to diagnose any quantum "illness."
On H 4-perturbations of self-adjoint operators
Abstract. Supersingular H,4-perturbations of positive self-adjoint operators are stud- ied. It is proven that such singular perturbations can be described by non self-adjoint operators with real spectrum. The resolvent of the perturbed operator is calculated using generalization of Krein’s formula for self-adjoint extensions.
On non-self-adjoint operators for observables in Quantum Mechanics and Quantum Electrodynamics
International Journal of Modern Physics a Particles and Fields Gravitation Cosmology, 2010
The aim of this paper is to show the possible significance, and usefulness, of various nonself-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
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The aim of this paper is to show the possible significance, and usefulness, of various nonself-adjoint operators for suitable Observables in nonrelativistic and relativistic quantum mechanics, and in quantum electrodynamics. More specifically, this work deals with: (i) the maximal Hermitian (but not self-adjoint) time operator in nonrelativistic quantum mechanics and in quantum electrodynamics; (ii) the problem of the four-position and four-momentum operators, each one with its Hermitian and anti-Hermitian parts, for relativistic spin-zero particles. Afterwards, other physically important applications of non-self-adjoint (and even non-Hermitian) operators are discussed: in particular, (iii) we reanalyze in detail the interesting possibility of associating quasi-Hermitian Hamiltonians with (decaying) unstable states in nuclear physics. Finally, we briefly mention the cases of quantum dissipation, as well as of the nuclear optical potential.
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Annales Henri Poincaré, 2014
Given a unitary representation of a Lie group G on a Hilbert space H, we develop the theory of G-invariant self-adjoint extensions of symmetric operators using both von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the G-invariant unbounded operator. We also prove a G-invariant version of the rep-resentation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of G-invariant self-adjoint exten-sions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group G acts. These extensions are labeled by admissible unitaries U acting on the L 2 -space at the boundary and having spectral gap at −1. It is shown that if the unitary representation V of the symmetry group G is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by U is G-invariant if U and V commute at the boundary. Various significant examples are discussed at the end.
International Journal of Geometric Methods in Modern Physics, 2015
This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.
Spectral Properties of Non-self-adjoint Operators in the Semi-classical Regime
Journal of Differential Equations, 2001
We give a spectral description of the semi-classical Schrödinger operator with a piecewise linear, complex valued potential. Moreover, using these results, we show how an arbitrarily small bounded perturbation of a nonself-adjoint operator can completely change the spectrum of the operator.
$ mathcal{H} $ -n -perturbations of Self-adjoint Operators and Krein's Resolvent Formula
Integral Equations and Operator Theory, 2003
Supersingular H −n rank one perturbations of arbitrary positive self-adjoint operator A acting in the Hilbert space H are investigated. The operator corresponding to the formal expression A α = A + α ϕ, • ϕ, α ∈ R, ϕ ∈ H −n (A), is determined as a regular operator with pure real spectrum acting in a certain extended Hilbert space H ⊃ H. The resolvent of the operator so defined is given by a certain generalization of Krein's resolvent formula. It is proven that spectral properties of the operator are described by generalized Nevanlinna functions. The results of [23] are extended to the case of arbitrary integer n ≥ 4.