Quantum Limits of Eisenstein Series and Scattering states

We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.Comment: 12 pages, Corrects a typo and its ramification from previous versio

Random Walks Along the Streets and Canals in Compact Cities: Spectral analysis, Dynamical Modularity, Information, and Statistical Mechanics

Different models of random walks on the dual graphs of compact urban structures are considered. Analysis of access times between streets helps to detect the city modularity. The statistical mechanics approach to the ensembles of lazy random walkers is developed. The complexity of city modularity can be measured by an information-like parameter which plays the role of an individual fingerprint of {\it Genius loci}. Global structural properties of a city can be characterized by the thermodynamical parameters calculated in the random walks problem.Comment: 44 pages, 22 figures, 2 table

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as $\lambda \to \infty$) of complex zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold $(M, g)$ with ergodic geodesic flow. If $\{\phi_{j_k} \}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula \frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial} {\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of integration over the complex zeros and where $\bar{\partial}$ is with respect to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and corrected some typo

On The Isoperimetric Spectrum of Graphs and Its Approximations

In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method

Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear relations amongst the gradients of the integrals of motion, coincide with points where there are k (doubly) degenerate eigenvalues of representatives L and Lbar of the two inequivalent classes of Lax matrices (corresponding to degenerate periodic or antiperiodic solutions of the associated second-order difference equation). The singularities are shown to be nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold. Sigma_k is shown to be of elliptic type, and the frequencies of transverse oscillations under Hamiltonians which fix Sigma_k are computed in terms of spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is given by the product of the holonomies (equal to +/- 1) of the even- (or odd-) indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio

Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs

Transport through generalized trees is considered. Trees contain the simple nodes and supernodes, either well-structured regular subgraphs or those with many triangles. We observe a superdiffusion for the highly connected nodes while it is Brownian for the rest of the nodes. Transport within a supernode is affected by the finite size effects vanishing as $N\to\infty.$ For the even dimensions of space, $d=2,4,6,...$, the finite size effects break down the perturbation theory at small scales and can be regularized by using the heat-kernel expansion.Comment: 21 pages, 2 figures include

Autocorrelation function of eigenstates in chaotic and mixed systems

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation length. For localized states, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low resolution only. For a version with pictures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab

Dirac and magnetic Schr\"odinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schr\"odinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with R\"ockner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet

Evolution of Intermediate Water Masses Based on Argo Float Displacements

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