Self-adjoint Extension Method Research Articles

Proper incorporation of the self-adjoint extension method to the Green function formalism: one-dimensional -function potential case

The one-dimensional -function potential is discussed in the framework of the Green function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green function for this case is crucially dependent on the boundary conditions which are provided by the self-adjoint extension method. The most general Green function which contains four real self-adjoint extension parameters is constructed. The relation between the bare coupling constant and the self-adjoint extension parameter is also derived.

Green’s-function approach to two- and three-dimensional delta-function potentials and application to the spin-1/2 Aharonov–Bohm problem

Delta-function potentials in two- and three-dimensional quantum mechanics are analyzed by incorporating the self-adjoint extension method within the Green’s-function method. The energy-dependent Green’s functions for free particle plus delta-function potential systems are explicitly determined and similar calculations are carried out for the spin-1/2 Aharonov–Bohm system. It is found that the time-dependent propagator for the latter cannot be evaluated analytically except for a particular value of the self-adjoint extension parameter. It corresponds to the case in which the singular solution alone contributes for the partial wave defined by ‖m+α‖<1.

Schrödinger self-adjoint extension and quantum field theory

We argue that the results obtained using the quantum mechanical method of self-adjoint extension of the Schrödinger Hamiltonian can also be derived using Feynman perturbation theory in the investigation of the corresponding non-relativistic field theories. We show that this is indeed what happens in the study of an anyon system, and, in doing so, we establish a field theoretical description for “colliding anyons”, i.e. anyons whose quantum mechanical wave functions satisfy the non-conventional boundary conditions obtained with the method of self-adjoint extension. We also show that analogous results hold for a system of non-abelian Chern-Simons particles.

Self-adjoint extension approach to the spin-1/2 Aharonov-Bohm-Coulomb problem.

The spin-1/2 Aharonov-Bohm problem is examined in the Galilean limit for the case in which a Coulomb potential is included. It is found that the application of the self-adjoint extension method to this system yields singular solutions only for one-half the full range of the flux parameter, which is allowed in the limit of a vanishing Coulumb potential. Thus one has a remarkable example of a case in which the condition of normalizability is necessary but not sufficient for the occurrence of singular solutions. Expressions for the bound state energies are derived. Also the conditions for the occurrence of singular solutions are obtained when the nongauge potential is \ensuremath{\xi}/${\mathit{r}}^{\mathit{p}}$ (0\ensuremath{\le}p2).

Charged particle with magnetic moment in the Aharonov-Bohm potential

We considered a charged quantum mechanical particle with spin ${1\over 2}$ and gyromagnetic ratio $g\ne 2$ in the field af a magnetic string. Whereas the interaction of the charge with the string is the well kown Aharonov-Bohm effect and the contribution of magnetic moment associated with the spin in the case $g=2$ is known to yield an additional scattering and zero modes (one for each flux quantum), an anomaly of the magnetic moment (i.e. $g>2$) leads to bound states. We considered two methods for treating the case $g>2$. \\ The first is the method of self adjoint extension of the corresponding Hamilton operator. It yields one bound state as well as additional scattering. In the second we consider three exactly solvable models for finite flux tubes and take the limit of shrinking its radius to zero. For finite radius, there are $N+1$ bound states ($N$ is the number of flux quanta in the tube).\\ For $R\to 0$ the bound state energies tend to infinity so that this limit is not physical unless $g\to 2$ along with $R\to 0$. Thereby only for fluxes less than unity the results of the method of self adjoint extension are reproduced whereas for larger fluxes $N$ bound states exist and we conclude that this method is not applicable.\\ We discuss the physically interesting case of small but finite radius whereby the natural scale is given by the anomaly of the magnetic moment of the electron $a_e=(g-2)/2\approx 10^{-3}$.

Time substitutions in stochastic processes as a tool in path integration: Alice Young and Cécile DeWitt-Morette. Department of Physics, Center for Relativity, The University of Texas, Austin, Texas 78712

The propagator of the spinless Aharonov–Bohm–Coulomb system is derived by following the Duru–Kleinert method. We use this propagator to explore the spin-1/2 Aharonov–Bohm–Coulomb system which contains a point interaction as a Zeeman term. Incorporation of the self-adjoint extension method into the Green's function formalism properly allows us to derive the finite propagator of the spin-1/2 Aharonov–Bohm–Coulomb system. As a by-product, the relation between the self-adjoint extension parameter and the bare coupling constant is obtained. Bound-state energy spectra of both spinless and spin-1/2 Aharonov–Bohm–Coulomb systems are examined.

Some Unusual Applications of Lie Algebra Representations in Quantum Theory

Several new applications of Lie algebra representations in quantum theory are discussed. These include a method of self adjoint extension of the Dirac Hamiltonian, description of continuous spectra, dynamical groups and infinite-component wave equations, induced and indecomposable representations and infinite parameter Lie algebras.