251 research outputs found

### 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 $\ell \geq 0$

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