Generalization of the Calogero-Cohn Bound on the Number of Bound States

It is shown that for the Calogero-Cohn type upper bounds on the number of bound states of a negative spherically symmetric potential $V(r)$, in each angular momentum state, that is, bounds containing only the integral $\int^\infty_0 |V(r)|^{1/2}dr$, the condition $V'(r) \geq 0$ is not necessary, and can be replaced by the less stringent condition $(d/dr)[r^{1-2p}(-V)^{1-p}] \leq 0, 1/2 \leq p < 1$, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on $p$ and $\ell$, and tend to the standard value for $p = 1/2$.Comment: 1 page. Correctly formatted version (replaces previous version

Potentials for which the Radial Schr\"odinger Equation can be solved

In a previous paper$^1$, submitted to Journal of Physics A -- we presented an infinite class of potentials for which the radial Schr\"odinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schr\"odinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. $^1$ We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity

Bound states in two spatial dimensions in the non-central case

We derive a bound on the total number of negative energy bound states in a potential in two spatial dimensions by using an adaptation of the Schwinger method to derive the Birman-Schwinger bound in three dimensions. Specifically, counting the number of bound states in a potential gV for g=1 is replaced by counting the number of g_i's for which zero energy bound states exist, and then the kernel of the integral equation for the zero-energy wave functon is symmetrized. One of the keys of the solution is the replacement of an inhomogeneous integral equation by a homogeneous integral equation.Comment: Work supported in part by the U.S. Department of Energy under Grant No. DE-FG02-84-ER4015

Quantum hamiltonians and prime numbers

A short review of Schroedinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose-Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approachComment: 10 pages, 2 figures, accepted as a Brief Review at MPL

Necessary and sufficient conditions for existence of bound states in a central potential

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.Comment: 7 page

New Classes of Potentials for which the Radial Schrodinger Equation can be solved at Zero Energy

Given two spherically symmetric and short range potentials $V_0$ and V_1 for which the radial Schrodinger equation can be solved explicitely at zero energy, we show how to construct a new potential $V$ for which the radial equation can again be solved explicitely at zero energy. The new potential and its corresponding wave function are given explicitely in terms of V_0 and V_1, and their corresponding wave functions \phi_0 and \phi_1. V_0 must be such that it sustains no bound states (either repulsive, or attractive but weak). However, V_1 can sustain any (finite) number of bound states. The new potential V has the same number of bound states, by construction, but the corresponding (negative) energies are, of course, different. Once this is achieved, one can start then from V_0 and V, and construct a new potential \bar{V} for which the radial equation is again solvable explicitely. And the process can be repeated indefinitely. We exhibit first the construction, and the proof of its validity, for regular short range potentials, i.e. those for which rV_0(r) and rV_1(r) are L^1 at the origin. It is then seen that the construction extends automatically to potentials which are singular at r= 0. It can also be extended to V_0 long range (Coulomb, etc.). We give finally several explicit examples.Comment: 26 pages, 3 figure

Simple function form for n+208Pb total cross section between 5 and 600 MeV

The total cross section for neutron scattering from 208Pb with energies between 5 and 600 MeV has been analyzed extending a previously defined simple function of three parameters to reveal a Ramsauer-like effect throughout the whole energy range. This effect can be parametrized in a simple way so that it may be anticipated that the complete function prescription will apply for total cross sections from other nuclei.Comment: 9 pages, 4 firgure

Asymptotic properties of the solutions of a differential equation appearing in QCD

We establish the asymptotic behaviour of the ratio $h^\prime(0)/h(0)$ for $\lambda\rightarrow\infty$, where $h(r)$ is a solution, vanishing at infinity, of the differential equation $h^{\prime\prime}(r) = i\lambda \omega (r) h(r)$ on the domain $0 \leq r <\infty$ and $\omega (r) = (1-\sqrt{r} K_1(\sqrt{r}))/r$. Some results are valid for more general $\omega$'s.Comment: 6 pages, late

Geometric spectral inversion for singular potentials

The function E = F(v) expresses the dependence of a discrete eigenvalue E of the Schroedinger Hamiltonian H = -\Delta + vf(r) on the coupling parameter v. We use envelope theory to generate a functional sequence \{f^{[k]}(r)\} to reconstruct f(r) from F(v) starting from a seed potential f^{[0]}(r). In the power-law or log cases the inversion can be effected analytically and is complete in just two steps. In other cases convergence is observed numerically. To provide concrete illustrations of the inversion method it is first applied to the Hulth\'en potential, and it is then used to invert spectral data generated by singular potentials with shapes of the form f(r) = -a/r + b\sgn(q)r^q and f(r) = -a/r + b\ln(r), a, b > 0. For the class of attractive central potentials with shapes f(r) = g(r)/r, with g(0)< 0 and g'(r)\ge 0, we prove that the ground-state energy curve F(v) determines f(r) uniquely.Comment: 13 pages, 5 figure

Some Remarks on Effective Range Formula in Potential Scattering

In this paper, we present different proofs of very recent results on the necessary as well as sufficient conditions on the decrease of the potential at infinity for the validity of effective range formulas in 3-D in low energy potential scattering (Andr\'e Martin, private communication, to appear. See Theorem 1 below). Our proofs are based on compact formulas for the phase-shifts. The sufficiency conditions are well-known since long. But the necessity of the same conditions for potentials keeping a constant sign at large distances are new. All these conditions are established here for dimension 3 and for all angular momenta $\ell \geq 0$