We obtain, using the Birman-Schwinger method, upper limits on the total number of bound states and on the number of $\ell$-wave bound states of the semirelativistic spinless Salpeter equation. We also obtain a simple condition, in the ultrarelativistic case ($m=0$), for the existence of at least one $\ell$-wave bound states: $C(\ell,p/(p-1))$ $\int_0^{\infty}dr r^{p-1} |V^-(r)|^p\ge 1$, where $C(\ell,p/(p-1))$ is a known function of $\ell$ and $p>1$.
Using the auxiliary field method, a generic upper bound is obtained for the spinless Salpeter equation with two different masses. Analytical results are presented for the cases of the Coulomb and linear potentials when a mass is vanishing.
We study the relativistic version of the $d$-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension $d$. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors o…
Motivated by the observation of a recent renewal of rather strong interest in the description of bound states by (semi-) relativistic equations of motion, we revisit, for the example of the Woods-Saxon interactions, the eigenvalue problem posed by the spinless Salpeter equation and recall various elementary knowledge, considerations, and techniques that practitioners seeking solutions to this specific reduction of the Bethe-Salpeter equation may find helpful.
The probablity current for a quantum spinless relativistic particle is introduced based on the Hamiltonian dynamics approach utilizing the Salpeter equation as an alternative of the Klein-Gordon equation. The correctness of the presented formalism is illustrated by examples of exact solutions to the Salpeter equation including the new ones introduced in this work.
The spinless Salpeter equation can be regarded as the eigenvalue equation of a Hamiltonian that involves the relativistic kinetic energy and therefore is, in general, a nonlocal operator. Accordingly, it is hard to find solutions of this bound-state equation by exclusively analytic means. Nevertheless, a lot of tools enables us to constrain the resulting bound-state spectra rigorously. We illustrate some of these techniques for the example of the Hulthén potential.
By application of the 'geometric spectral inversion' technique, which we have recently generalized to accommodate also singular interaction potentials, we construct from spectral data emerging from the solution of the Minkowski-space formulation of the homogeneous Bethe-Salpeter equation describing bound states of two spinless particles a Schroedinger approach to such states in terms of nonrelativistic potential models. This spectrally equivalent modeling of bound states yields their qualitative features (masses, form factors, etc.) without having to deal with the more involved Bethe-Salpeter …
Noticing renewed or increasing interest in the possibility to describe semirelativistic bound states (of either spin-zero constituents or, upon confining oneself to spin-averaged features, constituents with nonzero spin) by means of the spinless Salpeter equation generalizing the Schrödinger equation towards incorporation of effects caused by relativistic kinematics, we revisit this problem for interactions between bound-state constituents of Yukawa shape, by recalling and applying several well-known tools enabling to constrain the resulting spectra.
We compile some easily deducible information on the discrete eigenvalue spectra of spinless Salpeter equations encompassing, besides a relativistic kinetic term, interactions which are expressible as superpositions of an attractive Coulomb potential and an either attractive or repulsive Yukawa potential and, hence, generalizations of the Hellmann potential employed in several areas of science. These insights should provide useful guidelines to all attempts of finding appropriate descriptions of bound states by (semi-) relativistic equations of motion.
We constrain the possible bound-state solutions of the spinless Salpeter equation (the most obvious semirelativistic generalization of the nonrelativistic Schrödinger equation) with an interaction between the bound-state constituents given by the kink-like potential (a central potential of hyperbolic-tangent form) by formulating a bunch of very elementary boundary conditions to be satisfied by all solutions of the eigenvalue problem posed by a bound-state equation of this type, only to learn that all results produced by a procedure very much liked by some quantum-theory practitioners prove to …
Observing renewed interest in long-standing (semi-) relativistic descriptions of bound states, we would like to make a few comments on the eigenvalue problem posed by the spinless Salpeter equation and, illustrated by the examples of the nonsingular Woods-Saxon potential and the singular Hulthén potential, recall elementary tools that practitioners looking for analytic albeit approximate solutions might find useful in their quest.
The van der Waals-London's law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring [LT], by proving an upper bound which confirms this law. Recently the van der Waals-London's law was proven under some assumptions by I.M. Sigal and the author [AS]. Following the strategy of [AS] and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, u…
