Supersymmetry,Shape Invariance and Exactly Solvable Noncentral Potentials

Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss in detail the example of the potential $$V(r,\theta,\phi)={\omega^2\over 4}r^2 + {\delta\over r^2}+{C\over r^2 sin^2\theta}+{D\over r^2 cos^2\theta} + {F\over r^2 sin^2\theta sin^2 \alpha\phi} +{G\over r^2 sin^2\theta cos^2\alpha\phi}$$ with 7 parameters.Other algebraically solvable examples are also given.Comment: 16 page

Relationship Between the Energy Eigenstates of Calogero-Sutherland Models With Oscillator and Coulomb-like Potentials

We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these models which plays a crucial role in such an identification. Further, we show that our analysis is in fact valid for any many-particle system in arbitrary dimensions whose potential term (apart from the oscillator or the Coulomb-like potential) is a homogeneous function of coordinates of degree -2. The explicit coordinate transformation which maps the Coulomb-like problem to the oscillator one has also been determined in some specific cases.Comment: 23 pages, RevTeX, no figure, some clarifications added, version to appear in Journal of Physics

Manipulation of wetting morphologies in topographically structured substrates

In the present work, static liquid morphologies confined to linear micron sized surface grooves were studied experimentally and analyzed theoretically. Geometries with increasing complexities, from wedges to trapezoidal grooves, were explored with the main focus on triangular grooves. In contrast to chemically structured substrates where only liquid morphologies with positive Laplace pressure are found, topographically structured substrates exhibit liquid morphologies with both positive and negative Laplace pressure. Depending upon the wettability and the exact geometry of substrates, either drop-like morphologies or elongated filaments with positive or negative Laplace pressure represent the generic equilibrium structures on the substrates. For very high contact angles, drop-like morphologies are dominant irrespective of the underlying substrate geometry. Transitions between these liquid morphologies can be triggered by varying the wettability or the geometry of substrates. In the present work, various cross sections of the grooves were explored while the wettability was controlled by various self-assembly monolayers or by means of the electrowetting effect. Upon changing the apparent contact angle of an aqueous drop by electrowetting, the transition between the drop-like and elongated filament morphologies could be triggered and thus a liquid can be transported along prefabricated grooves on demand. A clear threshold behavior for filling of the grooves was observed which corresponds to the stability boundaries of the static wetting morphologies in the respective groove geometry. The length of the liquid filament that advances into the groove depends on the exact geometry of the groove and the electrical properties of the system. An electrical model is presented to explain this behavior. Unlike liquid filaments in rectangular grooves, liquid filaments in triangular grooves become unstable when they are quenched from a filling into a non-filling regime. This instability of liquid filaments in triangular grooves was studied in detail using homogeneous filaments of glassy polymer (polystyrene) which have been prepared in a non equilibrium state by deposition from a solution. At elevated temperature, molten polystyrene restores its material contact angle with the substrate thus forming filaments with positive Laplace pressure. After dewetting, this liquid filament decays into isolated droplets with a characteristic spacing, depending upon wedge geometry, wettability and filament width. This instability is driven by the interplay of local filament width and Laplace pressure and constitutes a wide class of one-dimensional instabilities which also includes the Rayleigh - Plateau instability as a special case. The dynamics of this instability was also studied via in situ AFM experiments which allows to determine the time constant of the instability. A careful analysis of the time constant of the instability allows for the quantitative determination of the slip length in the system

Degenerate Topological Vortex solutions from a generalized Abelian Higgs Model with a Chern-Simons term

We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits topological vortices satisfying Bogomol'nyi bound for which the magnetic flux is not quantized even though the energy is quantized. Furthermore, the vortex solution in each topological sector is infinitely degenerate.Comment: 13 pages (one figure not included), Revtex, IP/BBSR/93-5