Almost periodic functions, constructively

The almost periodic functions form a natural example of a non-separable normed space. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. Here we present a simple proof of Bohr's fundamental theorem for almost periodic functions which we then generalize to almost periodic functions on general topological groups

An Algorithm for Finding the Periodic Potential of the Three-dimensional Schrodinger Operator from the Spectral Invariants

In this paper, we investigate the three-dimensional Schrodinger operator with a periodic, relative to a lattice {\Omega} of R3, potential q. A special class V of the periodic potentials is constructed, which is easily and constructively determined from the spectral invariants. First, we give an algorithm for the unique determination of the potential q in V of the three-dimensional Schrodinger operator from the spectral invariants that were determined constructively from the given Bloch eigenvalues. Then we consider the stability of the algorithm with respect to the spectral invariants and Bloch eigenvalues. Finally, we prove that there are no other periodic potentials in the set of large class of functions whose Bloch eigenvalues coincides with the Bloch eigenvalues of q in V

Quantization of generic chaotic 3D billiard with smooth boundary I: energy level statistic

Numerical calculation and analysis of extremely high-lying energy spectra, containing thousands of levels with sequential quantum number up to 62,000 per symmetry class, of a generic chaotic 3D quantum billiard is reported. The shape of the billiard is given by a simple and smooth de formation of a unit sphere which gives rise to (almost) fully chaotic classical dynamics. We present an analysis of (i) quantum length spectrum whose smooth part agrees with the 3D Weyl formula and whose oscillatory part is peaked around the periods of classical periodic orbits, (ii) nearest neighbor level spacing distribution and (iii) number variance. Although the chaotic classical dynamics quickly and uniformly explores almost entire energy shell, while the measure of the regular part of phase space is insignificantly small, we find small but significant deviations from GOE statistics which are explained in terms of localization of eigenfunctions onto lower dimensional classically invariant manifolds.Comment: 10 pages in plain Latex (6 figures in PCL format available upon request) submitted to Phys. Lett.

Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems

We present a semiclassical calculation of the generalized form factor which characterizes the fluctuations of matrix elements of the quantum operators in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some recently developed techniques for the spectral form factor of systems with hyperbolic and ergodic underlying classical dynamics and f=2 degrees of freedom, that allow us to go beyond the diagonal approximation. First we extend these techniques to systems with f>2. Then we use these results to calculate the generalized form factor. We show that the dependence on the rescaled time in units of the Heisenberg time is universal for both the spectral and the generalized form factor. Furthermore, we derive a relation between the generalized form factor and the classical time-correlation function of the Weyl symbols of the quantum operators.Comment: some typos corrected and few minor changes made; final version in PR

Interference enhanced thermoelectricity in quinoid type structures

Quantum interference (QI) effects in molecular junctions may be used to obtain large thermoelectric responses. We study the electrical conductance G and the thermoelec- tric response of a series of molecules featuring a quinoid core using density functional theory (DFT), as well as a semi-empirical interacting model Hamiltonian describing the {\pi}-system of the molecule which we treat in the GW approximation. Molecules with a quinoid type structure are shown to have two distinct destructive QI features close to the frontier orbital energies. These manifest themselves as two dips in the transmission, that remain separated, even when either electron donating or withdraw- ing side groups are added. We find that the position of the dips in the transmission and the frontier molecular levels can be chemically controlled by varying the electron donating or withdrawing character of the side groups as well as the conjugation length inside the molecule. This feature results in a very high thermoelectric power factor S^2G and figure of merit ZT, where S is the Seebeck coefficient, making quinoid type molecules potential candidates for efficient thermoelectric devices.Comment: 22 pages, 11 figure