On Integrability and Pseudo-Hermitian Systems with Spin-Coupling Point Interactions

We study the pseudo-Hermitian systems with general spin-coupling point interactions and give a systematic description of the corresponding boundary conditions for PT-symmetric systems. The corresponding integrability for both bosonic and fermionic many-body systems with PT-symmetric contact interactions is investigated.Comment: 7 page

p-Adic Schr\"{o}dinger-Type Operator with Point Interactions

A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $\delta_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of $p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed for $\alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for $\alpha>1$. In the latter case, the spectral analysis of $\eta$-self-adjoint operator realizations of $D^{\alpha}+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.Comment: 13 page

Many Body Problems with "Spin"-Related Contact Interactions

We study quantum mechanical systems with "spin"-related contact interactions in one dimension. The boundary conditions describing the contact interactions are dependent on the spin states of the particles. In particular we investigate the integrability of $N$-body systems with $\delta$-interactions and point spin couplings. Bethe ansatz solutions, bound states and scattering matrices are explicitly given. The cases of generalized separated boundary condition and some Hamiltonian operators corresponding to special spin related boundary conditions are also discussed.Comment: 13 pages, Late

Four-Parameter Point-Interaction in 1-D Quantum Systems

We construct a four-parameter point-interaction for a non-relativistic particle moving on a line as the limit of a short range interaction with range tending toward zero. For particular choices of the parameters, we can obtain a delta-interaction or the so-called delta'-interaction. The Hamiltonian corresponding to the four-parameter point-interaction is shown to correspond to the four-parameter self-adjoint Hamiltonian of the free particle moving on the line with the origin excluded.Comment: 6 pages, Plain Tex file. BU-HEP-92-

Distribution theory for Schr\"odinger's integral equation

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schr\"odinger's equation. This paper, in contrast, investigates the integral form of Schr\"odinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schr\"odinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schr\"odinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's result to hypersurfaces. Second, we derive a new closed-form solution to Schr\"odinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schr\"odinger's differential equation. Third, we derive boundary conditions for `super-singular' potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution, and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schr\"odinger's integral equation is viable tool for studying singular interactions in quantum mechanics.Comment: 23 page