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Modern Physics Letters B

BOUND STATES IN CONTINUUM GENERATED BY POINT INTERACTION AND SUPERSYMMETRIC QUANTUM MECHANICS

S. Lj. S. KOČINAC, V. Milanović

Published Sep 24, 2012• 10 citations
Publisher

Abstract

Four-parameter family of point interactions represent all possible self-adjoint extensions of kinetic energy operator. We demonstrate a method for generating a bound state in the continuum of point interactions which relies on supersymmetric quantum mechanics (SUSYQM). Both zero and nonzero transparency cases are considered.
DOI
10.1142/s0217984912501771

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There is a four-parameter family of point interactions in one-dimensional quantum mechanics. They represent all possible self-adjoint extensions of the kinetic energy operator. If time-reversal invariance is imposed, the number of parameters is reduced to three. One of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known. We present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so-called interaction. We emphasize that there is little resemblance between the interaction and what its name suggests.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.