Auxiliary field method and analytical solutions of the Schr\"{o}dinger equation with exponential potentials

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schr\"{o}dinger equation with exponential potentials of the form $-\alpha r^\lambda \exp(-\beta r)$ can also be analytically solved by using the auxiliary field method. Formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn on the Yukawa potential and the pure exponential one

Duality relations in the auxiliary field method

The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$

The quantum N-body problem and the auxiliary field method

Approximate analytical energy formulas for N-body relativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. This method has already been proved to be a powerful technique in the case of two-body problems. A general procedure is given and applied to various Hamiltonians of interest, in atomic and hadronic physics in particular. A test of formulas is performed for baryons described as a three-quark system.Comment: References adde

Extensions of the auxiliary field method to solve Schr\"{o}dinger equations

It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schr\"{o}dinger equation. This technique can generate the spectrum associated with an arbitrary potential $V(r)$ starting from the analytically known spectrum of a particular potential $P(r)$. In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of $P(r)$. The method is extended in order to find accurate analytical energy formulae for radial potentials of the form $a P(r)+V(r)$, and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed

Some equivalences between the auxiliary field method and the envelope theory

The auxiliary field method has been recently proposed as an efficient technique to compute analytical approximate solutions of eigenequations in quantum mechanics. We show that the auxiliary field method is completely equivalent to the envelope theory, which is another well-known procedure to analytically solve eigenequations, although relying on different principles \textit{a priori}. This equivalence leads to a deeper understanding of both frameworks.Comment: v2 to appear in J. Math. Phy

The few-body problem in terms of correlated gaussians

In their textbook, Suzuki and Varga [Y. Suzuki and K. Varga, {\em Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems} (Springer, Berlin, 1998)] present the stochastic variational method in a very exhaustive way. In this framework, the so-called correlated gaussian bases are often employed. General formulae for the matrix elements of various operators can be found in the textbook. However the Fourier transform of correlated gaussians and their application to the management of a relativistic kinetic energy operator are missing and cannot be found in the literature. In this paper we present these interesting formulae. We give also a derivation for new formulations concerning central potentials; the corresponding formulae are more efficient numerically than those presented in the textbook.Comment: 10 page

Semirelativistic potential model for low-lying three-gluon glueballs

The three-gluon glueball states are studied with the generalization of a semirelativistic potential model giving good results for two-gluon glueballs. The Hamiltonian depends only on 3 parameters fixed on two-gluon glueball spectra: the strong coupling constant, the string tension, and a gluon size which removes singularities in the potential. The Casimir scaling determines the structure of the confinement. Low-lying $J^{PC}$ states are computed and compared with recent lattice calculations. A good agreement is found for $1^{--}$ and $3^{--}$ states, but our model predicts a $2^{--}$ state much higher in energy than the lattice result. The $0^{-+}$ mass is also computed.Comment: 2 figure

Semirelativistic Hamiltonians and the auxiliary field method

Approximate analytical closed energy formulas for semirelativistic Hamiltonians of the form $\sigma\sqrt{\bm p^{2}+m^2}+V(r)$ are obtained within the framework of the auxiliary field method. This method, which is equivalent to the envelope theory, has been recently proposed as a powerful tool to get approximate analytical solutions of the Schr\"odinger equation. Various shapes for the potential $V(r)$ are investigated: power-law, funnel, square root, and Yukawa. A comparison with the exact results is discussed in detail

Equation of motion of an interstellar Bussard ramjet with radiation and mass losses

An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in a fusion engine to produce thrust. In recent papers, it was shown that the relativistic equation of motion of an ideal ramjet and of a ramjet with radiation loss are analytical. When a mass loss appears, the limit speed of the ramjet is more strongly reduced. But, the parametric equations, in terms of the ramjet's speed, for the position of the ramjet in the inertial frame of the interstellar medium, the time in this frame, and the proper time indicated by the clocks on board the spaceship, can still be obtained in an analytical form. The non-relativistic motion and the motion near the limit speed are studied.Comment: 4 figure