arXiv subjects

S. Pratik Khastgir

Publications and source records attributed to S. Pratik Khastgir.

16 recordsLinked to original sources

arXiv2026

Sum of Three Squares and Particle in a Cubic Box - Degeneracies

We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica$^{\circledR}$.

math-ph↗

arXiv2018

From Classical Periodic Orbits in Integrable $π$-Rational Billiards to Quantum Energy Spectrum

In the present note, we uncover a remarkable connection between the length of periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact one-to-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions and exploiting the method of reflective tiling and folding of classical trajectories. We have further extended the method to 3-dimensional billiards for which exact analytical results are scarcely available - exploiting the geometric construction, we determine the exact energy spectra of two new tetrahedral domains which we believe are integrable. We test the veracity of our results by comparing them with numerical results.

quant-ph↗

arXiv2013

Metric deformation and boundary value problems in 3D

A novel perturbative method, proposed by Panda {\it et al.} [1] to solve the Helmholtz equation in two dimensions, is extended to three dimensions for general boundary surfaces. Although a few numerical works are available in the literature for specific domains in three dimensions such a general analytical prescription is presented for the first time. An appropriate transformation is used to get rid of the asymmetries in the domain boundary by mapping the boundary into an equivalent sphere with a deformed interior metric. The deformed metric produces new source terms in the original homogeneous equation. A deformation parameter measuring the deviation of the boundary from a spherical one is introduced as a perturbative parameter. With the help of standard Rayleigh-Schr{ö}dinger perturbative technique the transformed equation is solved and the general solution is written down in a closed form at each order of perturbation. The solutions are boundary condition free and which make them widely applicable for various situations. Once the boundary conditions are applied to these general solutions the eigenvalues and the wavefunctions are obtained order by order. The efficacy of the method has been tested by comparing the analytic values against the numerical ones for three dimensional enclosures of various shapes. The method seems to work quite well for these shapes for both, Dirichlet as well as Neumann boundary conditions. The usage of spherical harmonics to express the asymmetries in the boundary surfaces helps us to consider a wide class of domains in three dimensions and also their fast convergence guarantees the convergence of the perturbative series for the energy. Direct applications of this method can be found in the field of quantum dots, nuclear physics, acoustical and electromagnetic cavities.

math-ph↗

arXiv2009

The Effect of $w-term$ on Visibility Correlation and Power Spectrum Estimation

Visibility-visibility correlation has been proposed as a technique for the estimation of power spectrum, and used extensively for small field of view observations, where the effect of $w-term$ is usually ignored. We consider power spectrum estimation from the large field of view observations, where the $w-term$ can have a significant effect. Our investigation shows that a nonzero $w$ manifests itself as a modification of the primary aperture function of the instrument. Using a gaussian primary beam, we show that the modified aperture is an oscillating function with a gaussian envelope. We show that the two visibility correlation reproduces the power spectrum beyond a certain baseline given by the width, $U_{w}$ of the modified aperture. Further, for a given interferometer, the maximum $U_{w}$ remains independent of the frequencies of observation. This suggests that, the incorporation of large field of view in radio interferometric observation has a greater effect for larger observing wavelengths.

astro-ph.IM↗

arXiv2008

Steiner trees and spanning trees in six-pin soap films

We have studied the Steiner tree problem using six-pin soap films in detail. We extend the existing method of experimental realisation of Steiner trees in $n$-terminal problem through soap films to observe new non-minimal Steiner trees. We also produced spanning tree configurations for the first time by our method. Experimentally, by varying the pin diameter, we have achieved these new stable soap film configurations. A new algorithm is presented for creating these Steiner trees theoretically. Exact lengths of these Steiner tree configurations are calculated using a geometrical method. An exact two-parameter empirical formula is proposed for estimating the lengths of these soap film configurations in six-pin soap film problem.

cs.CG↗

arXiv2003

Affine Toda field theory from tree unitarity

Elasticity property (i.e. no-particle creation) is used in the tree level scattering of scalar particles in 1+1 dimensions to construct the affine Toda field theory(ATFT) associated with root systems of groups $a_2^{(2)}$ and $c_2^{(1)}$. A general prescription is given for constructing ATFT (associated with rank two root systems) with two self conjugate scalar fields. It is conjectured that the same method could be used to obtain the other ATFT associated with higher rank root systems.

hep-th↗

arXiv1998

S-matrices and bi-linear sum rules of conserved charges in affine Toda field theories

The exact quantum $S$-matrices and conserved charges are known for affine Toda field theories(ATFTs). In this note we report on a new type of bi-linear sum rules of conserved quantities derived from these exact $S$ matrices. They exist when there is a multiplicative identity among $S$-matrices of a particular ATFT. Our results are valid for simply laced as well as non-simply laced ATFTs. We also present a few explicit examples.

hep-th↗

arXiv1997

S-matrices of non-simply laced affine Toda theories by folding

The exact factorisable quantum S-matrices are known for simply laced as well as non-simply laced affine Toda field theories. Non-simply laced theories are obtained from the affine Toda theories based on simply laced algebras by folding the corresponding Dynkin diagrams. The same process, called classical `reduction', provides solutions of a non-simply laced theory from the classical solutions with special symmetries of the parent simply laced theory. In the present note we shall elevate the idea of folding and classical reduction to the quantum level. To support our views we have made some interesting observations for S-matrices of non-simply laced theories and give prescription for obtaining them through the folding of simply laced ones.

hep-th↗

arXiv1995

Non-canonical folding of Dynkin diagrams and reduction of affine Toda theories

The equation of motion of affine Toda field theory is a coupled equation for $r$ fields, $r$ is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than $r$ fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In this paper we present many new reductions. In other words the symmetry of affine Dynkin diagrams could be extended and it leads to non-canonical foldings. We investigate these reductions in detail and formulate general rules for possible reductions. We will show that eventually most of the theories end up in $a_{2n}^{(2)}$ that is the theory cannot have a further dimension $m$ reduction where $m<n$.

hep-th↗

arXiv1995

Instability of Solitons in imaginary coupling affine Toda Field Theory

Affine Toda field theory with a pure imaginary coupling constant is a non-hermitian theory. Therefore the solutions of the equation of motion are complex. However, in $1+1$ dimensions it has many soliton solutions with remarkable properties, such as real total energy/momentum and mass. Several authors calculated quantum mass corrections of the solitons by claiming these solitons are stable. We show that there exists a large class of classical solutions which develops singularity after a finite lapse of time. Stability claims, in earlier literature, were made ignoring these solutions. Therefore we believe that a formulation of quantum theory on a firmer basis is necessary in general and for the quantum mass corrections of solitons, in particular.

hep-th↗

arXiv1993

Singular Limits and String Solutions

It is shown that new leading ($\al'$) as well as all-order solutions of String theory can be obtained by taking appropriate singular limits of the known solutions. We give several leading order solutions for the bosonic as well as the heterotic string. We then present all-order forms of the previously known two dimensional cosmological solutions. An all-order form for the cosmological solution in three dimensions is also predicted. The physical implications of our results are discussed.

hep-th↗

arXiv1993

Topology changing processes and symmetries of string effective action

Wormhole solutions corresponding to space-time geometries $R^1\times S^1\times S^2$ and $R^1\times S^3$ are obtained from reduced string effective action and the action is written in a manifestly $O(d,d)$ invariant form. A general treatment is given for obtaining wormhole solutions of different topologies from dimensional reduction. For specific ansatz of internal metric and antisymmetric field the reduced action is shown to have a global $SL(2,C)$ symmetry. The $SL(2,C)$ and duality symmetries have been exploited to generate new configurations of internal fields which produce wormhole solutions in four space-time dimensions. The $SL(2,C)$ symmetry discussed in this paper arises due to specific form of the moduli and these transformations belong to a subgroup of $O(d,d)$ global symmetry.

hep-th↗

arXiv1992

Wormholes and symmetries of string effective action

The dimensional reduction technique is adopted to derive string effective action. Wormhole solutions corresponding to space-time geometries $R^1\times S^1\times S^2$ and $R^1\times S^3$ are presented. The duality and SL(2,R) symmetries are implemented to generate new wormhole solutions.

hep-th↗

arXiv1992

Symmetries of String Effective Action and Space-Time Geometry

Two dimensional charged black hole solution is obtained by implementing an $O(2,2)$ transformation on the three dimensional black string solution. Two different monopole backgrounds in five dimensions are related through an $O(2,2)$ transformation. It has been shown in these examples that the particular $O(2,2)$ transformation corresponds to duality transformation.

hep-th↗

arXiv1992

Four Dimensional Stringy Black Membrane

An exact conformal field theory describing a four dimensional singular string background is obtained by chiral gauging a $U(1)$ subgroup along with translations in $R$ of an $SL(2,R)\times R$ Wess-Zumino-Witten model. It is shown that the target space-time describes a four dimensional black membrane. Furthermore various duality transformed solutions are constructed. These are also shown to correspond to various forms of four dimensional black membranes.

hep-th↗

arXiv1991

String Effective Action and Two Dimensional Charged Black Hole

Graviton-dilaton background field equations in three space-time dimensions, following from the string effective action are solved when the metric has only time dependence. By taking one of the two space dimensions as compact, our solution reproduces a recently discovered charged black hole solution in two space-time dimensions. Solutions in presence of nonvanishing three dimensional background antisymmetric tensor field are also discussed.

hep-th↗