16 research outputs found

### Composition of Two Potentials

Given two potentials V0 and V1 together with a certain nodeless solution
{\phi}0 of V0, we form a composition of these two potentials. If V1 is exactly
solvable, the composition is exactly solvable, too. By combining various
solvable potentials in one-dimensional quantum mechanics, a huge variety of
solvable compositions can be made.Comment: 10 page

### 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

### 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

### 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 $\ell \geq 0$

### The Absence of Positive Energy Bound States for a Class of Nonlocal Potentials

We generalize in this paper a theorem of Titchmarsh for the positivity of
Fourier sine integrals. We apply then the theorem to derive simple conditions
for the absence of positive energy bound states (bound states embedded in the
continuum) for the radial Schr\"odinger equation with nonlocal potentials which
are superposition of a local potential and separable potentials.Comment: 23 page

### Universality of low-energy scattering in (2+1) dimensions

We prove that, in (2+1) dimensions, the S-wave phase shift, $\delta_0(k)$, k
being the c.m. momentum, vanishes as either $\delta_0 \to {c\over \ln (k/m)} or
\delta_0 \to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=\pi/2$.
This result is established first in the framework of the Schr\"odinger equation
for a large class of potentials, second for a massive field theory from proved
analyticity and unitarity, and, finally, we look at perturbation theory in
$\phi_3^4$ and study its relation to our non-perturbative result. The
remarkable fact here is that in n-th order the perturbative amplitude diverges
like $(\ln k)^n$ as $k\to 0$, while the full amplitude vanishes as $(\ln
k)^{-1}$. We show how these two facts can be reconciled.Comment: 23 pages, Late

### Generalization of Bochner's theorem for functions of the positive type

International audienceWe generalize Bochner's theorem for functions of the positive typeâtheorem 1âto more general integral transforms using the Jost solution of the radial SchrĂ¶dinger equation. The generalized theorem is theorem 2. We then use Bochner's theorem to obtain an integral representation for the phase shift, shown in theorem 4. In a forthcoming paper, this theorem will be used in inverse scattering theory. The proofs are simple, and make use of well-known theorems of real analysis and Fourier transforms of L1, L1â©L2, ... functions