Exterior-Interior Duality for Discrete Graphs

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in $\mathbb{R}^2$, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Interior duality is derived for both methods.Comment: 15 pages 1 figur

On the semiclassical theory for universal transmission fluctuations in chaotic systems: the importance of unitarity

The standard semiclassical calculation of transmission correlation functions for chaotic systems is severely influenced by unitarity problems. We show that unitarity alone imposes a set of relationships between cross sections correlation functions which go beyond the diagonal approximation. When these relationships are properly used to supplement the semiclassical scheme we obtain transmission correlation functions in full agreement with the exact statistical theory and the experiment. Our approach also provides a novel prediction for the transmission correlations in the case where time reversal symmetry is present

An estimate for the average spectral measure of random band matrices

For a class of random band matrices of band width $W$, we prove regularity of the average spectral measure at scales $\epsilon \geq W^{-0.99}$, and find its asymptotics at these scales.Comment: 19 pp., revised versio

Counting nodal domains on surfaces of revolution

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional to the product of the angular and the "surface" quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the nodal sequence, whose statistical properties we study. In particular we investigate the distribution of the normalized counts $\frac{\nu_n}{n}$ for sequences of eigenfunctions with $K \le n\le K + \Delta K$ where $K,\Delta K \in \mathbb{N}$. We show that the distribution approaches a limit as $K,\Delta K\to\infty$ (the classical limit), and study the leading corrections in the semi-classical limit. With this information, we derive the central result of this work: the nodal sequence of a mirror-symmetric surface is sufficient to uniquely determine its shape (modulo scaling).Comment: 36 pages, 8 figure

Characterization of Quantum Chaos by the Autocorrelation Function of Spectral Determinants

The autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general, and for the study of the intimate link between quantum chaos and random matrix theory, in particular. For this purpose, the correlation functions of spectral determinants are evaluated for various random matrix ensembles, and are compared with the corresponding semiclassical expressions. The method is demonstrated by applying it to the spectra of the quantized Sinai billiards in two and three dimensions.Comment: LaTeX, 32 pages, 6 figure

Isospectral domains with mixed boundary conditions

We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M. Kac.Comment: 9 figures. Statement of Theorem 5.1 correcte

Can One Hear the Shape of a Graph?

We show that the spectrum of the Schrodinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: A compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.Comment: 9 pages, 1 figur

Effect of phase relaxation on quantum superpositions in complex collisions

We study the effect of phase relaxation on coherent superpositions of rotating clockwise and anticlockwise wave packets in the regime of strongly overlapping resonances of the intermediate complex. Such highly excited deformed complexes may be created in binary collisions of heavy ions, molecules and atomic clusters. It is shown that phase relaxation leads to a reduction of the interference fringes, thus mimicking the effect of decoherence. This reduction is crucial for the determination of the phase--relaxation width from the data on the excitation function oscillations in heavy--ion collisions and bimolecular chemical reactions. The difference between the effects of phase relaxation and decoherence is discussed.Comment: Extended revised version; 9 pages and 3 colour ps figure

“Free Will and Affirmation: Assessing Honderich’s Third Way”

In the third and final part of his A Theory of Determinism (TD) Ted Honderich addresses the fundamental question concerning “the consequences of determinism.” The critical question he aims to answer is what follows if determinism is true? This question is, of course, intimately bound up with the problem of free will and, in particular, with the question of whether or not the truth of determinism is compatible or incompatible with the sort of freedom required for moral responsibility. It is Honderich’s aim to provide a solution to “the problem of the consequences of determinism” and a key element of this is his articulation and defence of an alternative response to the implications of determinism that collapses the familiar Compatibilist/Incompatibilist dichotomy. Honderich offers us a third way – the response of “Affirmation” (HFY 125-6). Although his account of Affirmation has application and relevance to issues and features beyond freedom and responsibility, my primary concern in this essay will be to examine Honderich’s theory of “Affirmation” as it concerns the free will problem

The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl