Discrete Spectra of Semirelativistic Hamiltonians

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form H = \beta \sqrt{m^2 + p^2} + V(r) (w.l.o.g. in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation. Every Hamiltonian in this class of operators consists of the relativistic kinetic energy \beta \sqrt{m^2 + p^2} (where \beta > 0 allows for the possibility of more than one particles of mass m) and a spherically symmetric attractive potential V(r), r = |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of a numerical approximation procedure. Our main emphasis, however, is on the derivation of rigorous semi-analytical expressions for both upper and lower bounds to the energy levels of such operators. We compare the bounds obtained within different approaches and present relationships existing between the bounds.Comment: 21 pages, 3 figure

Instantaneous Bethe-Salpeter equation: improved analytical solution

Studying the Bethe-Salpeter formalism for interactions instantaneous in the rest frame of the bound states described, we show that, for bound-state constituents of arbitrary masses, the mass of the ground state of a given spin may be calculated almost entirely analytically with high accuracy, without the (numerical) diagonalization of the matrix representation obtained by expansion of the solutions over a suitable set of basis states.Comment: 7 page

Energy bounds for the spinless Salpeter equation: harmonic oscillator

We study the eigenvalues E_{n\ell} of the Salpeter Hamiltonian H = \beta\sqrt(m^2 + p^2) + vr^2, v>0, \beta > 0, in three dimensions. By using geometrical arguments we show that, for suitable values of P, here provided, the simple semi-classical formula E = min_{r > 0} {v(P/r)^2 + \beta\sqrt(m^2 + r^2)} provides both upper and lower energy bounds for all the eigenvalues of the problem.Comment: 8 pages, 1 figur

Relativistic Harmonic Oscillator

We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in compact form, i. e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.Comment: 14 pages, extended discussion of result

Instantaneous Bethe-Salpeter Equation: Analytic Approach for Nonvanishing Masses of the Bound-State Constituents

The instantaneous Bethe-Salpeter equation, derived from the general Bethe-Salpeter formalism by assuming that the involved interaction kernel is instantaneous, represents the most promising framework for the description of hadrons as bound states of quarks from first quantum-field-theoretic principles, that is, quantum chromodynamics. Here, by extending a previous analysis confined to the case of bound-state constituents with vanishing masses, we demonstrate that the instantaneous Bethe-Salpeter equation for bound-state constituents with (definitely) nonvanishing masses may be converted into an eigenvalue problem for an explicitly - more precisely, algebraically - known matrix, at least, for a rather wide class of interactions between these bound-state constituents. The advantages of the explicit knowledge of this matrix representation are self-evident.Comment: 12 Pages, LaTeX, 1 figur

Relativistic N-Boson Systems Bound by Oscillator Pair Potentials

We study the lowest energy E of a relativistic system of N identical bosons bound by harmonic-oscillator pair potentials in three spatial dimensions. In natural units the system has the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i - r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N (m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A sharper lower bound is given by the function P = P(mu), where mu = m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m --> infinity.Comment: v2: A scale analysis of P is now included; this leads to revised energy bounds, which coalesce in the large-m limi

Reflexive and Intentional Saccadic Eye Movements in Migraineurs

Background: Migraine has been postulated to lead to structural and functional changes of different cortical and subcortical areas, including the frontal lobe, the brainstem, and cerebellum. The (sub-)clinical impact of these changes is a matter of debate. The spectrum of possible clinical differences include domains such as cognition but also coordination. The present study investigated the oculomotor performance of patients with migraine with and without aura compared to control subjects without migraine in reflexive saccades, but also in intentional saccades, which involve cerebellar as well as cortical networks. Methods: In 18 patients with migraine with aura and 21 patients with migraine without aura saccadic eye movements were recorded in two reflexive (gap, overlap) and two intentional (anti, memory) paradigms and compared to 25 controls without migraine. Results: The main finding of the study was an increase of saccade latency in patients with and without aura compared to the control group solely in the anti-task. No deficits were found in the execution of reflexive saccades. Conclusions: Our results suggest a specific deficit in the generation of correct anti-saccades, such as vector inversion. Such processes are considered to need cortical networks to be executed correctly. The parietal cortex has been suggested to be involved in vector inversion processes but is not commonly described to be altered in migraine patients. It could be discussed that the cerebellum, which is recently thought to be involved in the pathophysiology of migraine, might be involved in distinct processes such as spatial re-mapping through known interconnections with parietal and frontal cortical areas

Stability in the instantaneous Bethe-Salpeter formalism: harmonic-oscillator reduced Salpeter equation

A popular three-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states in quantum field theory is the Salpeter equation, derived by assuming both instantaneous interactions and free propagation of all bound-state constituents. Numerical (variational) studies of the Salpeter equation with confining interaction, however, observed specific instabilities of the solutions, likely related to the Klein paradox and rendering (part of the) bound states unstable. An analytic investigation of this problem by a comprehensive spectral analysis is feasible for the reduced Salpeter equation with only harmonic-oscillator confining interactions. There we are able to prove rigorously that the bound-state solutions correspond to real discrete energy spectra bounded from below and are thus free of any instabilities.Comment: 23 pages, 3 figures, extended conclusions, version to appear in Phys. Rev.

Quality of Variational Trial States

Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H usually can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of the approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to precisely that Hamiltonian for which the eigenvalue problem defines the well-known spinless Salpeter equation. This bound-state wave equation may be regarded as (the most straightforward) relativistic generalization of the usual nonrelativistic Schroedinger formalism, and is frequently used to describe, e.g., spin-averaged mass spectra of bound states of quarks.Comment: LaTeX, 7 pages, version to appear in Physical Review

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