11,231 research outputs found
Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidean space preliminaries
We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze's tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set
Effects of Newtonian gravitational self-interaction in harmonically trapped quantum systems
The Schr\"odinger-Newton equation has gained attention in the recent past as
a nonlinear modification of the Schr\"odinger equation due to a gravitational
self-interaction. Such a modification is expected from a fundamentally
semi-classical theory of gravity, and can therefore be considered a test case
for the necessity of the quantisation of the gravitational field. Here we
provide a thorough study of the effects of the Schr\"odinger-Newton equation
for a micron-sized sphere trapped in a harmonic oscillator potential. We
discuss both the effect on the energy eigenstates and the dynamical behaviour
of squeezed states, covering the experimentally relevant parameter regimes.Comment: 22 pages, 14 figure
The algebraic structure behind the derivative nonlinear Schroedinger equation
The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr\"
odinger equation (DNLSE) amongst others interesting and important nonlinear
integrable equations. In this paper, a general higher grading affine algebraic
construction of integrable hierarchies is proposed and the KN hierarchy is
established in terms of a Kac-Moody algebra and principal
gradation. In this form, our spectral problem is linear in the spectral
parameter. The positive and negative flows are derived, showing that some
interesting physical models arise from the same algebraic structure. For
instance, the DNLSE is obtained as the second positive, while the Mikhailov
model as the first negative flows, respectively. The equivalence between the
latter and the massive Thirring model is explicitly demonstrated also. The
algebraic dressing method is employed to construct soliton solutions in a
systematic manner for all members of the hierarchy. Finally, the equivalence of
the spectral problem introduced in this paper with the usual one, which is
quadratic in the spectral parameter, is achieved by setting a particular
automorphism of the affine algebra, which maps the homogeneous into principal
gradation.Comment: references adde
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