arXiv subjects

S. R. Jain

Publications and source records attributed to S. R. Jain.

7 recordsLinked to original sources

arXiv2026

On The Spectral Properties of Discrete Landscape Functions

We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$ and obtain the bound $$[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leqφ(N)$$ for every odd $k$. The Chandra--Jain conjecture, asserting that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=φ(N)$ for all $N\geq3$, has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814). We then reduce higher modes to the coprime case. Exact computations for all 624 pairs with $3\leq N\leq50$ reveal only the degrees $φ(N/d)$ and $2φ(N/d)$ for non-coprime modes, where $d=\gcd(N,k)$. This motivates the Second Chandra--Jain Conjecture that this degree dichotomy holds generally. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers.

math.DS↗

arXiv2013

Pseudo-Hermitian random matrix theory

Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present applications to problems in statistical mechanics where new results have become possible. We have found it important to mention the precise directions where advances could be made if further results become available.

quant-ph↗

arXiv2003

Time-delay, energy-continuum, and systematics of particles

Finding systematics in the mass-lifetime data for all the hadrons has been an outstanding problem. In this work, we show that the product of mass and lifetime for unstable particles is very well-approximated by \hbar 2^n/n where n is an integer specific for a particle. In doing so, we have employed a relation between time-delay and resonances. The energy-continuum has been treated in a way to take advantage of Cantor's mathematical work on continuum. Thus, even though the resonances are designated by complex energy variables where ordering is not possible, in terms of stability, the index n labels these resonances; larger the n, more stable a resonance is.

quant-ph↗

arXiv2002

Eigenvalue spectrum for single particle in a spheroidal cavity: A Semiclassical approach

Following the semiclassical formalism of Strutinsky et al., we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit $η\to1.0$. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of sphere and an exact comparison with the quantum mechanical results is observed upto the second decimal place for $kR_{0}\geq{7}$. The contributions of the equatorial, the planar (in the axis of symmetry plane) and the non-planar(3-Dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with $L_z=0$ except for L=1. The remaining quantum mechanical eigenvalues are observed to arise from the 3-dimensional(3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations.

nucl-th↗

arXiv2002

Quantum Limit on Computational Time and Speed

We investigate if physical laws can impose limit on computational time and speed of a quantum computer built from elementary particles. We show that the product of the speed and the running time of a quantum computer is limited by the type of fundamental interactions present inside the system. This will help us to decide as to what type of interaction should be allowed in building quantum computers in achieving the desired speed.

quant-ph↗

arXiv1999

Off-diagonal long-range order in one-dimensional many-body problem

We prove that there is off-diagonal long-range order in the symmetrised version of the one-dimensional many-body problem presented by Jain and Khare (Phys. Lett. A262 (1999)35). This model is related to the short-range Dyson model employed to study intermediate statistics in systems like the Anderson model in three dimensions at the metal-insulator transition point and pseudointegrable billiards. To the best of our knowledge, this is the only example showing quantum phases and possibility of Bose-Einstein condensation in one-dimensional statistical mechanics.

cond-mat↗

arXiv1998

Spacing distributions for rhombus billiards

We show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the Wigner-Dyson family of random matrix theory and the Poisson distribution. Some of the distributions find explanation in a recent work of Bogomolny, Gerland and Schmit. For the irrational billiards, despite ergodicity, we get the same distributions for the examples considered - once again, distinct from the Wigner-Dyson distributions. All results are obtained numerically by a method that allows us to reach very high energies.

chao-dyn↗