Anti-isospectral Transformations, Orthogonal Polynomials and Quasi-Exactly Solvable Problems

We consider the double sinh-Gordon potential which is a quasi-exactly solvable problem and show that in this case one has two sets of Bender-Dunne orthogonal polynomials . We study in some detail the various properties of these polynomials and the corresponding quotient polynomials. In particular, we show that the weight functions for these polynomials are not always positive. We also study the orthogonal polynomials of the double sine-Gordon potential which is related to the double sinh-Gordon case by an anti-isospectral transformation. Finally we discover a new quasi-exactly solvable problem by making use of the anti-isospectral transformation.Comment: Revtex, 19 pages, No figur

Scheduling Real-time Divisible Loads in Cluster Computing Environment

The significance of cluster computing in solving massively parallel workloads is tremendous. Divisible Load Theory has proven to be very successful in optimizing the usage of the system resources by partitioning the arbitrarily divisible loads adequately among the cluster nodes. Arbitrarily divisible loads have significant real-world applications in high energy and particle physics. In this thesis, various algorithms for a cluster computing environment are studied including the ones dealing with divisible load theory confirming DLT based algorithms performing better in most cases. The loads that are considered in this thesis are hard real-time tasks with associated deadlines. Specifically, a comparison is made between clusters with one where the head node doesn't participate in processing of the work-loads with the other where the head node does participate in processing of the work-loads. A new mathematical formula is derived for the task execution time corresponding to the new scenario of head node possessing front-end processing capability. The existing algorithms corresponding to Real-Time Divisible Load Theory are then implemented using this new formula to examine the scheduling performance in this new scenario compared to the conventional scenario where the head node lacks front-end processing capability

Exact Thermodynamics of the Double sinh-Gordon Theory in 1+1-Dimensions

We study the classical thermodynamics of a 1+1-dimensional double-well sinh-Gordon theory. Remarkably, the Schrodinger-like equation resulting from the transfer integral method is quasi-exactly solvable at several temperatures. This allows exact calculation of the partition function and some correlation functions above and below the short-range order (``kink'') transition, in striking agreement with high resolution Langevin simulations. Interesting connections with the Landau-Ginzburg and double sine-Gordon models are also established.Comment: 4 pages, 3 figures (embedded using epsf), uses RevTeX plus macro (included). Minor revision to match journal version, Phys. Rev. Lett. (in press

Statistical Mechanics of Double sinh-Gordon Kinks

We study the classical thermodynamics of the double sinh-Gordon (DSHG) theory in 1+1 dimensions. This model theory has a double well potential, thus allowing for the existence of kinks and antikinks. Though it is nonintegrable, the DSHG model is remarkably amenable to analysis. Below we obtain exact single kink and kink lattice solutions as well as the asymptotic kink-antikink interaction. In the continuum limit, finding the classical partition function is equivalent to solving for the ground state of a Schrodinger-like equation obtained via the transfer integral method. For the DSHG model, this equation turns out to be quasi-exactly solvable. We exploit this property to obtain exact energy eigenvalues and wavefunctions for several temperatures both above and below the symmetry breaking transition temperature. The availability of exact results provides an excellent testing ground for large scale Langevin simulations. The probability distribution function (PDF) calculated from Langevin dynamics is found to be in striking agreement with the exact PDF obtained from the ground state wavefunction. This validation points to the utility of a PDF-based computation of thermodynamics utilizing Langevin methods. In addition to the PDF, field-field and field fluctuation correlation functions were computed and also found to be in excellent agreement with the exact results.Comment: 10 pages, 5 figures (embedded using epsfig), uses RevTeX plus macro (included). To appear in Physica

When does the Rayleigh-Schrodinger perturbation series for the energy eigenvalue have nothing to do with the exact value?

On the basis of some exactly solvable models we suggest that the Rayleigh-Schrodinger perturbation series for the energy eigenvalue may have nothing to do with the corresponding exact value in case the vacuum expectation value of the system changes discontinuously, i.e. the system undergoes first-order phase transition

Anharmonic oscillator model for first order structural phase transition

Exact solutions for the motion of a classical anharmonic oscillator in the potential V(φ)=Bφ2-|A|φ4+Cφ6 are obtained in (1 + 1) dimensions. Instanton-like solutions in (imaginary time) which takes the particle from one maximum of the potential to the other are obtained in addition to the usual oscillatory solutions. The energy dependence of the frequencies of oscillation is discussed in detail. This can be used as a model for the first order structural phase transition in the mean field approximation. The high and low temperature behaviour of the static susceptibility is obtained. Finally, a qualitative explanation is offered for the observed central peak in ferroelectrics like SrTiO2

Classical φ<SUP>6</SUP>-field theory in (1+1) dimensions. 2. Proof of the existence of domain walls above the transition point

The existence of a domain wall-like contribution to the free energy above the first order phase transition point is demonstrated for a system described by the φ6-field theory in (1+1) dimensions