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F. F. Schoberl

Publications and source records attributed to F. F. Schoberl.

At least 19 recordsLinked to original sources
arXiv2006

Generalized Instantaneous Bethe-Salpeter Equation and Exact Quark Propagators

A three-dimensional reduction of the homogeneous Bethe-Salpeter equation retaining, in contrast to the Salpeter equation, the exact propagators (crucial for, e.g., a proper incorporation of dynamical chiral symmetry breakdown) is proposed. This formalism may be easily extended to Bethe-Salpeter equations for bound states composed of particles that are not, or not all, identical to spin-1/2 fermions as well as to three-dimensional reductions of the Bethe-Salpeter equation different from Salpeter's equation.

hep-ph↗
arXiv2005

Exact-propagator instantaneous Bethe-Salpeter equation for quark-antiquark bound states

Recently, an instantaneous approximation to the Bethe-Salpeter formalism for the analysis of bound states in quantum field theory has been proposed which retains, in contrast to the Salpeter equation, as far as possible the exact propagators of the bound-state constituents, extracted nonperturbatively from Dyson-Schwinger equations or lattice gauge theory. The implications of this improvement for the solutions of this bound-state equation, that is, the spectrum of the mass eigenvalues of its bound states and the corresponding wave functions, when considering the quark propagators arising in quantum chromodynamics are explored.

hep-ph↗
arXiv2005

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.

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arXiv2005

Instantaneous Bethe-Salpeter Equation with Exact Propagators

Consequent application of the instantaneous approximation to both the interaction and all propagators of the bound-state constituents allows us to forge, within the framework of the Bethe-Salpeter formalism for the description of bound states, an instantaneous form of the Bethe-Salpeter equation with exact (i.e., full) propagators of the bound-state constituents. This instantaneous equation generalizes the well-known Salpeter equation the derivation of which needs the additional assumption of free propagation of the bound-state constituents.

hep-th↗
arXiv2004

Facets of the spinless Salpeter equation

The spinless Salpeter equation represents the simplest and most straightforward generalization of the Schroedinger equation of standard nonrelativistic quantum theory towards the inclusion of relativistic kinematics. Moreover, it can be also regarded as a well-defined approximation to the Bethe-Salpeter formalism for descriptions of bound states in relativistic quantum field theories. The corresponding Hamiltonian is, in contrast to all Schroedinger operators, a nonlocal operator. Because of the nonlocality, constructing analytical solutions for such kind of equation of motion proves difficult. In view of this, different sophisticated techniques have been developed in order to extract rigorous analytical information about these solutions. This review introduces some of these methods and compares their significance by application to interactions relevant in physics.

hep-ph↗
arXiv2003

The charmed strange meson system

Motivated by the recent surprising discovery of two new meson states with c s-bar quark content but unexpectedly low masses and narrow total decay widths, we work out, in a nonrelativistic potential-model approach developed already some two decades ago, the predictions for the energy levels of the corresponding charm-antistrange quark bound states. In spite of the fact that this simple quark model reproduces the mass spectrum of the previously observed hadrons remarkably well, we are led to the conclusion that, without considerable modifications, both the new states do not fit into this framework.

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arXiv2002

Electric Polarizability of Mesons in Semirelativistic Quark Models

The electric polarizability of mesons, in particular, that of the charged pion, is studied in the framework of a semirelativistic description of hadrons as bound states of valence quarks in terms of a Hamiltonian composed of the relativistic kinetic energy as well as a phenomenological potential describing the strong interactions between the quarks. The quark-core contribution to the electric polarizability obtainable in quark models of this kind turns out to be in the semirelativistic approaches even smaller than in the nonrelativistic limit.

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arXiv2001

Energy bounds for the spinless Salpeter equation

We study the spectrum of the spinless-Salpeter Hamiltonian H = β\sqrt{m^2 + p^2} + V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, then upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ β\sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of P here provided. At the critical point the relative growth to the Coulomb potential h(r)=-1/r must be bounded by dV/dh < 2β/π.

hep-th↗
arXiv2001

Discrete spectra of semirelativistic Hamiltonians from envelope theory

We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = β\sqrt{m^2 + p^2} + V(r), beta > 0, where V(r) represents an attractive, spherically symmetric potential in three dimensions. In order to locate the eigenvalues of H, we extend the ``envelope theory,'' originally formulated only for nonrelativistic Schroedinger operators, to the case of Hamiltonians H involving the relativistic kinetic-energy operator. If V(r) is a convex transformation of the Coulomb potential -1/r and a concave transformation of the harmonic-oscillator potential r^2, both upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E = min_{r>0} [ β\sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of the numbers P here provided. At the critical point, the relative growth to the Coulomb potential h(r) = -1/r must be bounded by dV/dh < 2 β/π.

hep-th↗
arXiv2001

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.

hep-ph↗
arXiv2000

Instantaneous Bethe-Salpeter equation: utmost analytic approach

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.

hep-ph↗
arXiv2000

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.

hep-ph↗
arXiv2000

Accuracy of Approximate Eigenstates

Besides perturbation theory, which requires, of course, 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 some variational method, are proposed here as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the "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.

hep-ph↗
arXiv2000

Numerical Solution of the Spinless Salpeter Equation by a Semianalytical Matrix Method (a Mathematica 4.0 routine)

In quantum theory, the so-called "spinless Salpeter equation," the relativistic generalization of the nonrelativistic Schroedinger equation, is used to describe both bound states of scalar particles and the spin-averaged spectra of bound states of fermions. A numerical procedure solves the spinless Salpeter equation by approximating this eigenvalue equation by a matrix eigenvalue problem with explicitly known matrices.

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