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arXiv:math-ph/0609055 · Journal of Physics A Mathematical and Theoretical

Spin-dependent point potentials in one and three dimensions

Claudio Cacciapuoti, Raffaele Carlone, Rodolfo Figari

Published Dec 12, 2006• 26 citations
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Abstract

We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.

DOI
10.1088/1751-8113/40/2/004

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We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimensions 1 and 3. We characterize all the Hamiltonians obtained as point perturbations of assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition, we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin-dependent point potentials that could be of interest as multi-component solvable models.
We develop a perturbative framework for time-dependent point interactions in quantum systems with purely discrete unperturbed spectrum. In two and three dimensions the point interaction is described through the renormalized resolvent obtained from heat-kernel regularization, while in one dimension the diagonal Green function is finite and no renormalization is required. Time dependence is introduced either through the interaction parameter or through the motion of the support point. In both cases we expand the non-autonomous Hamiltonian in the spectral basis of the corresponding static point-interaction problem and derive the first- and second-order pole shifts, projection corrections, and transition amplitudes. We also describe how finite degeneracies are treated by separating the coupled point-interaction direction from the unaffected orthogonal subspace in each degenerate eigenspace. The method is illustrated by explicit examples: point interactions with time-dependent coupling, harmonic oscillators perturbed by moving point interactions, and a particle on a sphere with a moving interaction center. We also discuss the one-dimensional harmonic oscillator with a moving delta potential, showing that in the non-renormalized case the present formulation reduces to the standard time-dependent perturbation theory. Finally, an exactly solvable delta center moving with constant velocity on a circle is used as a discrete-spectrum example for the local moving-center expansion and its velocity-dependent dynamical corrections.
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.
Some physical restrictions and their influences on the boundary conditions for a particle in a one-dimensional box are pointed out. We discuss the invariance of the corresponding Hamiltonian under time-reversal T ̂ and its relation with the de-generacy of the energy eigenvalues. We also discuss the effect on the boundary conditions of requiring the invariance of the Hamiltonian under parity P ̂ and si-multaneous space and time reflection P ̂ T ̂. A condition, which depends only on the boundary values of an eigenfunction of the Hamiltonian and its derivative, deter-mines if the corresponding eigenvalue satisfies the inequality E> V (x) = const, V (x) being the potential inside the box. Once these results have been presented, we choose various representative analytically solvable examples of boundary conditions (Hamiltonians) and supersymmetrize the corresponding problems. We find real and complex valued supersymmetric partner potentials among all these systems.