Coherent states for the Hartmann potential

We obtain the coherent states for a particle in the noncentral Hartmann potential by transforming the problem into four isotropic oscillators evolving in a parametric time. We use path integration over the holomorphic coordinates to find the quantum states for these oscillators. The decomposition of the transition amplitudes gives the coherent states and their parametric-time evolution for the particle in the Hartmann potential. We also derive the coherent states in the parabolic coordinates by considering the transition amplitudes between the coherent states and eigenstates in the configuration space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 439–452, June, 2008.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree

We study translation-invariant Gibbs measures for the ferromagnetic Potts model with q states on the Cayley tree of order k and generalize some earlier results. We consider the question of the extremality of the known translation-invariant Gibbs measures for the Potts model with three states on the Cayley tree of order k = 3.     

Inequality in population weights,majority threshold and the inversion probability in the case of three states

Two-tier voting systems are prone to majority inversions, when the outcome of an election is not backed by a majority of popular vote. We study the inversion probability in a model with two candidates, three states and uniformly distributed fractions of supporters for each candidate. We show that the inversion probability in a two-tier voting system with three states eventually decreases with a majority threshold in the states and increases with the inequality in the size distribution of the states.     

Conditions for the discovery of solution horizons

We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.This work was supported in part by NSF Grant ECS-8700836 to The University of Michigan.     

A continuous model for the theory of organizational structure

We investigate a general class of linear models of dyadic interactions with a constant discrete time delay. We prove that the changes in stability of the stationary states occur for various intervals of the parameters that determine the strength and nature of emotional interactions between the partners. The dynamics of interactions depend on both reactivity of partners to their own emotional states as well as to the partner's states. The results suggest that reactivity to the partner's states has greater impact on the dynamics of the relationship than the reactivity to one's own states. Moreover, the results underscore the importance of deliberation in maintaining the stability of the relationship. Moreover, we have found that multiple stability switches are only possible when one of the partners reacts with delay to their own emotional states. We also propose a generalization to triadic interactions.     

Evaluating the geometric measure of quantum discord

We derive analytic formulas for the geometric measure of quantum discord introduced by Dakic, Vedral, and Brukner for pure states and (2×n)-dimensional states and establish a general lower bound for arbitrary states.     

Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions

We consider a family of Cuntz–Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz–Pimsner algebra; if the self-similar group is contracting, then the Cuntz–Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.     

Semiclassical asymptotic approximations and the density of states for the two-dimensional radially symmetric Schrödinger and Dirac equations in tunnel microscopy problems

We consider the two-dimensional stationary Schrödinger and Dirac equations in the case of radial symmetry. A radially symmetric potential simulates the tip of a scanning tunneling microscope. We construct semiclassical asymptotic forms for generalized eigenfunctions and study the local density of states that corresponds to the microscope measurements. We show that in the case of the Dirac equation, the tip distorts the measured density of states for all energies.     

Spectral properties of three-electron systems in the Hubbard model

We investigate spectral properties of a three-electron system in the Hubbard model framework. We prove that the essential spectrum of the system in a quartet state consists of a single segment and the three-electron bound state is absent. We show that the essential spectrum of the system in doublet states is the union of at most three segments. We also prove that three-electron bound states exist in doublet states.     

Using the Evolution Operator Method to Describe a Particle in a Homogeneous Alternating Field

Using the evolution operator method, we construct coherent states of a nonrelativistic free particle with a variable mass M(t) and a nonrelativistic particle with a variable mass M(t) in a homogeneous alternating field. Under certain physical conditions, they can be regarded as semiclassical states of particles. We discuss the properties (in particular, the completeness relation, the minimization of the uncertainty relations, and the time evolution of the corresponding probability density) of the found coherent states in detail. We also construct exact wave functions of the oscillator type and of the plane-wave type for the considered systems and compute the quantum Wigner distribution functions for the wave functions of coherent and oscillator states. We establish the unitary equivalence of the problems of a free particle and a particle in a homogeneous alternating field.     

Quaternionic vector coherent states and the supersymmetric harmonic oscillator

The quaternionic vector coherent states are realized as coherent states of the supersymmetric harmonic oscillator with broken symmetry in analogy with the standard canonical coherent states of the ordinary harmonic oscillator. We study the nonclassical properties of the oscillator, such as the photon number distribution and signal-to-quantum-noise ratio in terms of these states and discuss the squeezing properties and the temporal stability of the coherent states. We obtain the orthogonal polynomials associated with the quaternionic vector coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 80–98, October, 2006.     

非谐振子Gazeau-Klauder与Klauder-Perelomov相干态

利用Kinani-Daoud方法构造了非谐振子势的Gazeau-Klauder(GK)相干态和Klauder-Perelomov(KP)相干态，表明两种相干态在非线性谐振子势下具有完全不同的形式，并对两种相干态的完备性以及各自构成的Hilbert空间进行了讨论. 对相干态的Mandel Q参数的研究表明：GK相干态服从亚Poisson统计分布，KP相干态服从超Poisson统计分布.     

Scale transformations in phase space and stretched states of a harmonic oscillator

We consider scale transformations (q, p) → (λq, λp) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson–Schrödinger uncertainty relations for stretched states.     

The Lifschitz singularity for the density of states on the Sierpinski gasket

Summary We prove the existence of the density of states for the Laplacian on the infinite Sierpinski gasket. Then the Lifschitz-type singularity of the density of states is established. We also investigate the long-time asymptotics of the Brownian trajectory on the Sierpinski gasket, getting bounds similar to those in the ^d-case.     

Barut–Girardello Coherent States for the Gegenbauer Oscillator

We define the Gegenbauer oscillator and introduce a family of Barut–Girardello coherent states (eigenstates of a relevant annihilation operator) for this oscillator. Gegenbauer (ultraspherical) polynomials play the same role for this oscillator as Hermite polynomials do in the case of the usual boson oscillator. We establish the validity of unity decomposition for the introduced states and evaluate their overlap. These results reproduce similar results obtained earlier by the authors for Legendre and Chebyshev polynomials. Bibliography: 38 titles.     

A lower bound for the length of the shortest carefully synchronizing words

We introduce the notion of careful synchronization for partial finite automata as a natural generalization of the synchronization notion for complete finite automata. We obtain a lower bound for the careful synchronization threshold for automata with a given number of states.     

Wigner distribution functions for a relativistic linear oscillator

We construct the Wigner representation for a relativistic model of the linear harmonic oscillator governed by a finite-difference equation. We find Wigner functions for the stationary states, the thermodynamic equilibrium states, and the coherent states. We examine their nonrelativistic limits and the high and low temperature limits for the equilibrium states. We compute the mean values of the position and momentum coordinates for these Wigner functions. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 114, No. 3. pp. 410 425, March, 1998.     

A note on the non-formation of vacuum states for compressible Navier-Stokes equations

We give a refinement of Lemma 2.2 in [D. Hoff, J.A. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys. 216 (2001) 255-276] and complete the proof of non-formation of vacuum states for one-dimensional compressible Navier-Stokes equation given there.