arXiv subjects

Rhine Samajdar

Publications and source records attributed to Rhine Samajdar.

59 records · Page 4Linked to original sources

arXiv2018

Exact eigenfunction amplitude distributions of integrable quantum billiards

The exact probability distributions of the amplitudes of eigenfunctions, $Ψ(x, y)$, of several integrable planar billiards are analytically calculated and shown to possess singularities at $Ψ= 0$; the nature of this singularity is shape-dependent. In particular, we prove that the distribution function for a rectangular quantum billiard is proportional to the complete elliptic integral, ${\bf K}(1 - Ψ^2)$, and demonstrate its universality, modulo a weak dependence on quantum numbers. On the other hand, we study the low-lying states of nonseparable, integrable triangular billiards and find the distributions thereof to be described by the Meijer G-function or certain hypergeometric functions. Our analysis captures a marked departure from the Gaussian distributions for chaotic billiards in its survey of the fluctuations of the eigenfunctions about $Ψ= 0$.

math-ph↗

arXiv2017

Nodal portraits of quantum billiards: Domains, lines, and statistics

We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches to counting the nodal domains---using trace formulae, graph theory, and difference equations---are also illustrated with examples. The nodal patterns addressed pertain to waves on vibrating plates and membranes, acoustic and electromagnetic modes, wavefunctions of a "particle in a box'" as well as to percolating clusters, and domains in ferromagnets, thus underlining the diversity---and far-reaching implications---of the problem.

quant-ph↗

arXiv2016

A difference-equation formalism for the nodal domains of separable billiards

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1-12 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

quant-ph↗

arXiv2014

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, $ν$, of the eigenfunctions are considered. The billiards for which the time-independent Schrödinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, $ν$ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of $m\mod kn$, given a particular $k$, for a set of quantum numbers, $m, n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.

nlin.SI↗

arXiv2014

Nodal domains of the equilateral triangle billiard

We characterise the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains. The number of nodal domains has a quadratic form in terms of the quantum numbers, with a non-trivial number-theoretic factor. The patterns of the eigenfunctions follow a group-theoretic connection in a way that makes them predictable as one goes from one state to another. Extensive numerical investigations bring out the distribution functions of the mode number and signed areas. The statistics of the boundary intersections is also treated analytically. Finally, the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula. We believe that the results belong generically to non-separable systems, thus extending the previous works which are concentrated on separable and chaotic systems.

quant-ph↗