Perturbations of Self-Adjoint Operators (original) (raw)

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.

On Non-Self-Adjoint Operators for Observables in Quantum Mechanics and Quantum Field Theory

International Journal of Modern Physics A, 2010

The time-dependent Schrödinger equation: the need for the Hamiltonian to be self-adjoint

Brazilian Journal of Physics, 2008

We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems-the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone´s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the timedependent Schrödinger equation.

Singular perturbations of selfadjoint operators

Communications on Pure and Applied Mathematics, 1969

Singular¯nite rank perturbations of an unbounded selfadjoint operator A 0 in a Hilbert space H 0 are de¯ned formally as A (®) = A 0 + G®G ¤ , where G is an injective linear mapping from H = C d to the scale space H ¡k (A 0), k 2 N, of generalized elements associated with the selfadjoint operator A 0 , and where ® is a selfadjoint operator in H. The cases k = 1 and k = 2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k = 2n > 1 has been considered recently by various authors from a mathematical point of view. In this paper singular¯nite rank perturbations A (®) in the general setting ran G ½ H ¡k (A 0), k 2 N, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application singular perturbations of the Dirac operator are considered.

Some Problems of Self-Adjoint Extension In the Schrodinger Equation

Arxiv preprint arXiv: …, 2009

Abstract. The Self-Adjoint Extension in the Schrodinger equation for potentials behaved as an attractive inverse square at the origin is critically reviewed. Original results are also presented. It is shown that the additional solutions must be retained for definite interval of ...

Functional Analysis and Its Applications

An approach to the similarity problem is presented, which is based on the notion of a w-perturbation of the Volterra operator and uses the theory of Muckenhoupt weights.