Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Z^n which are discrete analogs of the Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of so-called pseudodifference operators (i.e., pseudodifferential operators on the group Z^n) with analytic symbols and on the limit operators method. We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schr\"{o}dinger operators on Z^n, Dirac operators on Z^3, and square root Klein-Gordon operators on Z^n

Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory

We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at large x and we give a canonical representation of their essential spectrum in terms of spectra of limits at infinity of translations of H. The configuration space is an arbitrary abelian locally compact not compact group.Comment: 63 pages. This is the published version with several correction

Algorithmic Bayesian Persuasion

Persuasion, defined as the act of exploiting an informational advantage in order to effect the decisions of others, is ubiquitous. Indeed, persuasive communication has been estimated to account for almost a third of all economic activity in the US. This paper examines persuasion through a computational lens, focusing on what is perhaps the most basic and fundamental model in this space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow. Here there are two players, a sender and a receiver. The receiver must take one of a number of actions with a-priori unknown payoff, and the sender has access to additional information regarding the payoffs. The sender can commit to revealing a noisy signal regarding the realization of the payoffs of various actions, and would like to do so as to maximize her own payoff assuming a perfectly rational receiver. We examine the sender's optimization task in three of the most natural input models for this problem, and essentially pin down its computational complexity in each. When the payoff distributions of the different actions are i.i.d. and given explicitly, we exhibit a polynomial-time (exact) algorithm, and a "simple" $(1-1/e)$-approximation algorithm. Our optimal scheme for the i.i.d. setting involves an analogy to auction theory, and makes use of Border's characterization of the space of reduced-forms for single-item auctions. When action payoffs are independent but non-identical with marginal distributions given explicitly, we show that it is #P-hard to compute the optimal expected sender utility. Finally, we consider a general (possibly correlated) joint distribution of action payoffs presented by a black box sampling oracle, and exhibit a fully polynomial-time approximation scheme (FPTAS) with a bi-criteria guarantee. We show that this result is the best possible in the black-box model for information-theoretic reasons

Essential spectra of difference operators on \sZ^n-periodic graphs

Let (\cX, \rho) be a discrete metric space. We suppose that the group \sZ^n acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a \sZ^n-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on $l^p(X)$ where $X$ is a \sZ^n-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on \sZ^n and their limit operators. In case $X$ is the set of vertices of a combinatorial graph, the graph structure defines a Schr\"{o}dinger operator on $l^p(X)$ in a natural way. We illustrate our approach by determining the essential spectra of Schr\"{o}dinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures

Demonstration of the difference Casimir force for samples with different charge carrier densities

A measurement of the Casimir force between a gold coated sphere and two Si plates of different carrier densities is performed using a high vacuum based atomic force microscope. The results are compared with the Lifshitz theory and good agreement is found. Our experiment demonstrates that by changing the carrier density of the semiconductor plate by several orders of magnitude it is possible to modify the Casimir interaction. This result may find applications in nanotechnology.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Let

A Simple Theory of Condensation

A simple assumption of an emergence in gas of small atomic clusters consisting of $c$ particles each, leads to a phase separation (first order transition). It reveals itself by an emergence of ``forbidden'' density range starting at a certain temperature. Defining this latter value as the critical temperature predicts existence of an interval with anomalous heat capacity behaviour $c_p\propto\Delta T^{-1/c}$. The value $c=13$ suggested in literature yields the heat capacity exponent $\alpha=0.077$.Comment: 9 pages, 1 figur

Nonlinear properties of left-handed metamaterials

We analyze nonlinear properties of microstructured materials with negative refraction, the so-called left-handed metamaterials. We consider a two-dimensional periodic structure created by arrays of wires and split-ring resonators embedded into a nonlinear dielectric, and calculate the effective nonlinear electric permittivity and magnetic permeability. We demonstrate that the hysteresis-type dependence of the magnetic permeability on the field intensity allows changing the material from left- to right-handed and back. These effects can be treated as the second-order phase transitions in the transmission properties induced by the variation of an external field.Comment: 4 pages, 3 figure