44 research outputs found

### New Solvable and Quasi Exactly Solvable Periodic Potentials

Using the formalism of supersymmetric quantum mechanics, we obtain a large
number of new analytically solvable one-dimensional periodic potentials and
study their properties. More specifically, the supersymmetric partners of the
Lame potentials ma(a+1)sn^2(x,m) are computed for integer values a=1,2,3,....
For all cases (except a=1), we show that the partner potential is distinctly
different from the original Lame potential, even though they both have the same
energy band structure. We also derive and discuss the energy band edges of the
associated Lame potentials pm sn^2(x,m)+qm cn^2(x,m)/ dn^2(x,m), which
constitute a much richer class of periodic problems. Computation of their
supersymmetric partners yields many additional new solvable and quasi exactly
solvable periodic potentials.Comment: 24 pages and 10 figure

### Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

Motivated by the concept of shape invariance in supersymmetric quantum
mechanics, we obtain potentials whose spectrum consists of two shifted sets of
equally spaced energy levels. These potentials are similar to the
Calogero-Sutherland model except the singular term $\alpha x^{-2}$ always falls
in the transition region $-1/4 < \alpha < 3/4$ and there is a delta-function
singularity at x=0.Comment: Latex, 12 pages, Figures available from Authors, To appear in Physics
Letters A. Please send requests for figures to [email protected] or
[email protected]

### Some Exact Results for Mid-Band and Zero Band-Gap States of Associated Lame Potentials

Applying certain known theorems about one-dimensional periodic potentials, we
show that the energy spectrum of the associated Lam\'{e} potentials
$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$ consists of
a finite number of bound bands followed by a continuum band when both $a$ and
$b$ take integer values. Further, if $a$ and $b$ are unequal integers, we show
that there must exist some zero band-gap states, i.e. doubly degenerate states
with the same number of nodes. More generally, in case $a$ and $b$ are not
integers, but either $a + b$ or $a - b$ is an integer ($a \ne b$), we again
show that several of the band-gaps vanish due to degeneracy of states with the
same number of nodes. Finally, when either $a$ or $b$ is an integer and the
other takes a half-integral value, we obtain exact analytic solutions for
several mid-band states.Comment: 18 pages, 2 figure

### Methods for Generating Quasi-Exactly Solvable Potentials

We describe three different methods for generating quasi-exactly solvable
potentials, for which a finite number of eigenstates are analytically known.
The three methods are respectively based on (i) a polynomial ansatz for wave
functions; (ii) point canonical transformations; (iii) supersymmetric quantum
mechanics. The methods are rather general and give considerably richer results
than those available in the current literature.Comment: 12 pages, LaTe

### Non-Central Potentials and Spherical Harmonics Using Supersymmetry and Shape Invariance

It is shown that the operator methods of supersymmetric quantum mechanics and
the concept of shape invariance can profitably be used to derive properties of
spherical harmonics in a simple way. The same operator techniques can also be
applied to several problems with non-central vector and scalar potentials. As
examples, we analyze the bound state spectra of an electron in a Coulomb plus
an Aharonov-Bohm field and/or in the magnetic field of a Dirac monopole.Comment: Latex, 12 pages. To appear in American Journal of Physic