17 research outputs found
New Shape Invariant Potentials in Supersymmetric Quantum Mechanics
Quantum mechanical potentials satisfying the property of shape invariance are
well known to be algebraically solvable. Using a scaling ansatz for the change
of parameters, we obtain a large class of new shape invariant potentials which
are reflectionless and possess an infinite number of bound states. They can be
viewed as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for energy eigenvalues,
eigenfunctions and transmission coefficients are given. Included in our
potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.Comment: 8pages, Te
Slowly Rotating Homogeneous Stars and the Heun Equation
The scheme developed by Hartle for describing slowly rotating bodies in 1967
was applied to the simple model of constant density by Chandrasekhar and Miller
in 1974. The pivotal equation one has to solve turns out to be one of Heun's
equations. After a brief discussion of this equation and the chances of finding
a closed form solution, a quickly converging series solution of it is
presented. A comparison with numerical solutions of the full Einstein equations
allows one to truncate the series at an order appropriate to the slow rotation
approximation. The truncated solution is then used to provide explicit
expressions for the metric.Comment: 16 pages, uses document class iopart, v2: minor correction
New Eaxactly Solvable Hamiltonians: Shape Invariance and Self-Similarity
We discuss in some detail the self-similar potentials of Shabat and
Spiridonov which are reflectionless and have an infinite number of bound
states. We demonstrate that these self-similar potentials are in fact shape
invariant potentials within the formalism of supersymmetric quantum mechanics.
In particular, using a scaling ansatz for the change of parameters, we obtain a
large class of new, reflectionless, shape invariant potentials of which the
Shabat-Spiridonov ones are a special case. These new potentials can be viewed
as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for the energy eigenvalues,
eigenfunctions and transmission coefficients for these potentials are obtained.
We show that these potentials can also be obtained numerically. Included as an
intriguing case is a shape invariant double well potential whose supersymmetric
partner potential is only a single well. Our class of exactly solvable
Hamiltonians is further enlarged by examining two new directions: (i) changes
of parameters which are different from the previously studied cases of
translation and scaling; (ii) extending the usual concept of shape invariance
in one step to a multi-step situation. These extensions can be viewed as
q-deformations of the harmonic oscillator or multi-soliton solutions
corresponding to the Rosen-Morse potential.Comment: 26 pages, plain tex, request figures by e-mai
Quantum Mechanics of Multi-Prong Potentials
We describe the bound state and scattering properties of a quantum mechanical
particle in a scalar -prong potential. Such a study is of special interest
since these situations are intermediate between one and two dimensions. The
energy levels for the special case of identical prongs exhibit an
alternating pattern of non-degeneracy and fold degeneracy. It is shown
that the techniques of supersymmetric quantum mechanics can be used to generate
new solutions. Solutions for prongs of arbitrary lengths are developed.
Discussions of tunneling in -well potentials and of scattering for piecewise
constant potentials are given. Since our treatment is for general values of
, the results can be studied in the large limit. A somewhat surprising
result is that a free particle incident on an -prong vertex undergoes
continuously increased backscattering as the number of prongs is increased.Comment: 17 pages. LATEX. On request, TOP_DRAW files or hard copies available
for 7 figure
Peculiarities of the hidden nonlinear supersymmetry of Poschl-Teller system in the light of Lame equation
A hidden nonlinear bosonized supersymmetry was revealed recently in
Poschl-Teller and finite-gap Lame systems. In spite of the intimate
relationship between the two quantum models, the hidden supersymmetry in them
displays essential differences. In particular, the kernel of the supercharges
of the Poschl-Teller system, unlike the case of Lame equation, includes
nonphysical states. By means of Lame equation, we clarify the nature of these
peculiar states, and show that they encode essential information not only on
the original hyperbolic Poschl-Teller system, but also on its singular
hyperbolic and trigonometric modifications, and reflect the intimate relation
of the model to a free particle system.Comment: 10 pages, typos corrected; to appear in J. Phys.
Negaton and Positon Solutions of the KDV Equation
We give a systematic classification and a detailed discussion of the
structure, motion and scattering of the recently discovered negaton and positon
solutions of the Korteweg-de Vries equation. There are two distinct types of
negaton solutions which we label and , where is the
order of the Wronskian used in the derivation. For negatons, the number of
singularities and zeros is finite and they show very interesting time
dependence. The general motion is in the positive direction, except for
certain negatons which exhibit one oscillation around the origin. In contrast,
there is just one type of positon solution, which we label . For
positons, one gets a finite number of singularities for odd, but an
infinite number for even values of . The general motion of positons is in
the negative direction with periodic oscillations. Negatons and positons
retain their identities in a scattering process and their phase shifts are
discussed. We obtain a simple explanation of all phase shifts by generalizing
the notions of ``mass" and ``center of mass" to singular solutions. Finally, it
is shown that negaton and positon solutions of the KdV equation can be used to
obtain corresponding new solutions of the modified KdV equation.Comment: 20 pages plus 12 figures(available from authors on request),Latex
fil
Stability of non-time-reversible phonobreathers
Non-time reversible phonobreathers are non-linear waves that can transport
energy in coupled oscillator chains by means of a phase-torsion mechanism. In
this paper, the stability properties of these structures have been considered.
It has been performed an analytical study for low-coupling solutions based upon
the so called {\em multibreather stability theorem} previously developed by
some of the authors [Physica D {\bf 180} 235]. A numerical analysis confirms
the analytical predictions and gives a detailed picture of the existence and
stability properties for arbitrary frequency and coupling.Comment: J. Phys. A.:Math. and Theor. In Press (2010
Hierarchy of QM SUSYs on a Bounded Domain
We systematically formulate a hierarchy of isospectral Hamiltonians in
one-dimensional supersymmetric quantum mechanics on an interval and on a
circle, in which two successive Hamiltonians form N=2 supersymmetry. We find
that boundary conditions compatible with supersymmetry are severely restricted.
In the case of an interval, a hierarchy of, at most, three isospectral
Hamiltonians is possible with unique boundary conditions, while in the case of
a circle an infinite tower of isospectral Hamiltonians can be constructed with
two-parameter family of boundary conditions.Comment: 15 pages, 3 figure