2,144 research outputs found

### A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied - in particular,
the number of nodal domains, $\nu$, of the eigenfunctions are considered. The
billiards for which the time-independent Schr\"odinger 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, $\nu$ 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.Comment: 13 pages, 5 figure

### Quantum Mechanics of a Rotating Billiard

Integrability of a square billiard is spontaneously broken as it rotates
about one of its corners. The system becomes quasi-integrable where the
invariant tori are broken with respect to a certain parameter, $\lambda =
2E/\omega^{2}$ where E is the energy of the particle inside the billiard and
$\omega$ is the angular frequency of rotation of billiard. We study the system
classically and quantum mechanically in view of obtaining a correspondence in
the two descriptions. Classical phase space in Poincar\'{e} surface of section
shows transition from regular to chaotic motion as the parameter $\lambda$ is
decreased. In the Quantum counterpart, the spectral statistics shows a
transition from Poisson to Wigner distribution as the system turns chaotic with
decrease in $\lambda$. The wavefunction statistics however show breakdown of
time-reversal symmetry as $\lambda$ decreases

### 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.Comment: 26 pages, 13 figure

### Geometric phase for neutrino propagation in magnetic field

The geometric phase for neutrinos propagating in an adiabatically varying
magnetic field in matter is calculated. It is shown that for neutrino
propagation in sufficiently large magnetic field the neutrino eigenstates
develop a significant geometric phase. The geometric phase varies from 2$\pi$
for magnetic fields $\sim$ fraction of a micro gauss to $\pi$ for fields $\sim
10^7$ gauss or more. The variation of geometric phase with magnetic field
parameters is shown and its phenomenological implications are discussed

- …