Lieb–Thirring Bounds for Complex Jacobi Matrices

We obtain various versions of classical Lieb–Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan–Mirski Lemma and seems to be fairly general.      

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schr?dinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation. Received: 26 February 2001 / Accepted: 28 May 2001     

Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u _t +H(x,u _x ) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u ^ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u ^ɛ(x,t) can be (stochastically) represented as , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity. Received: 15 February 1999 / Accepted: 14 December 1999     

Sum Rules and the Szegő Condition for Orthogonal Polynomials on the Real Line

We study the Case sum rules, especially C ₀ , for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohats theorem to cases with an infinite point spectrum and a proof that if lim n(a _n –1)= and lim nb _n = exist and 2<||, then the Szeg condition fails. Supported in part by NSF grant DMS-9707661.     

Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation

We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studied by Szegő and Baxter. We also derive a new hierarchy of coupled nonlinear difference equations satisfied by these algebro-geometric solutions.Supported in part by the US National Science Foundation under Grants No. DMS-0200219 and DMS-0405526.The research of the second and third author was supported in part by the Research Council of Norway     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

Eigenvalue Bounds for Perturbations of Schrödinger Operators and Jacobi Matrices With Regular Ground States

We prove general comparison theorems for eigenvalues of perturbed Schrödinger operators that allow proof of Lieb–Thirring bounds for suitable non-free Schrödinger operators and Jacobi matrices.     

A “Transversal” Fundamental Theorem for semi-dispersing billiards

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A Transversal Fundamental Theorem has recently been suggested by the present authors to proveglobal ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems ofN3 elastic hard balls conjectured by the celebratedBoltzmann-Sinai ergodic hypothesis. (In fact, the suggested Transversal Fundamental Theorem has been successfully applied by the authors in the casesN=3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtainlocal ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersing-billiards. Transversality means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.Dedicated to Joel L. Lebowitz on the occasion of his 60th birthdayResearch partially supported by the Hungarian National Foundation for Scientific Research, grant No. 819/1     

A No—Go Theorem for Nonlinear Canonical Quantization

We want to point out the following strengthening of the classical theorem of Groenewold and van Hove:There exists no mapping Op from polynomial observables f(p,q) on the phase space R^2n into linear operators on L^2 (R^n) which would map Poisson brackets into commutators,the position and momentum observables p and q into the usual (Schrodinger) position and momentum operators,and would obey the von Neumann rule Op (cf^k)=cDp(f)^k for k=1,2,3 and c∈R.The point is that neither linearity,nor continuity etc.of Op are assumed.     

A Perron–Frobenius Type of Theorem for Quantum Operations

We define a special class of quantum operations we call Markovian and show that it has the same spectral properties as a corresponding Markov chain. We then consider a convex combination of a quantum operation and a Markovian quantum operation and show that under a norm condition its spectrum has the same properties as in the conclusion of the Perron–Frobenius theorem if its Markovian part does. Moreover, under a compatibility condition of the two operations, we show that its limiting distribution is the same as the corresponding Markov chain. We apply our general results to partially decoherent quantum random walks with decoherence strength \(0 \le p \le 1\). We obtain a quantum ergodic theorem for partially decoherent processes. We show that for \(0 < p \le 1\), the limiting distribution of a partially decoherent quantum random walk is the same as the limiting distribution for the classical random walk.     

A Riemann–Roch Theorem¶for One-Dimensional Complex Groupoids

We consider a smooth groupoid of the form Σ⋊Γ, where Σ is a Riemann surface and Γ a discrete pseudogroup acting on Σ by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C ₀(Σ)⋊Γ generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L ^∞(Σ)⋊Γ. Received: 1 February 2000 / Accepted: 3 December 2000     

A Mermin–Wagner Theorem for Gibbs States on Lorentzian Triangulations

We establish a Mermin–Wagner type theorem for Gibbs states on infinite random Lorentzian triangulations (LT) arising in models of quantum gravity. Such a triangulation is naturally related to the distribution P of a critical Galton–Watson tree, conditional upon non-extinction. At the vertices of the triangles we place classical spins taking values in a torus M of dimension d, with a given group action of a torus G of dimension d′≤d. In the main body of the paper we assume that the spins interact via a two-body nearest-neighbor potential U(x,y) invariant under the action of G. We analyze quenched Gibbs measures generated by U and prove that, for P-almost all Lorentzian triangulations, every such Gibbs measure is G-invariant, which means the absence of spontaneous continuous symmetry-breaking.     

A Bijection Theorem for Domino Tilings with Diagonal Impurities

We consider the dimer problem on a planar non-bipartite graph G, where there are two types of dimers one of which we regard as impurities. Computer simulations reveal a reminiscence of the Cheerios effect, that is, impurities are attracted to the boundary, which is the motivation to study this particular graph. Our main theorem is a variant of the Temperley bijection: a bijection between the set of dimer coverings and the set of spanning forests with certain conditions. We further discuss some implications of this theorem: (1) the local move connectedness yielding an ergodic Markov chain on the set of all possible dimer coverings, and (2) a rough bound for the number of dimer coverings and that for the probability of finding an impurity at a given edge, which is an extension of a result in (Nakano and Sadahiro in ).     

A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation

This article presents a numerical approximation of the initial-boundary nonlinear coupled viscous Burgers’ equation based on spectral methods. A Jacobi-Gauss-Lobatto collocation (J-GL-C) scheme in combination with the implicit Runge-Kutta-Nyström (IRKN) scheme are employed to obtain highly accurate approximations to the mentioned problem. This J-GL-C method, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled viscous Burgers’ equation to a system of nonlinear ordinary differential equation which is far easier to solve. The given examples show, by selecting relatively few J-GL-C points, the accuracy of the approximations and the utility of the approach over other analytical or numerical methods. The illustrative examples demonstrate the accuracy, efficiency, and versatility of the proposed algorithm.     

A Hamilton–Jacobi type equation for computing minimum potential energy paths

A new method for computing minimum-energy reaction paths is presented. Unlike existing approaches (e.g. intrinsic reaction coordinate methods), our approach works for any reactant configuration: the structure of the transition state, reactive intermediates and product will be determined by the algorithm, and so need not be known beforehand. The method we have developed is based on solving a Hamilton–Jacobi type equation. Specifically, we introduce a speed function so that the ‘first arrival times’ from the Hamilton–Jacobi equation correspond to least-potentials. Then, adopting a back-tracing method, we can use the first arrival times to determine the minimum-energy path between any classically allowed molecular conformation and the initial (reactant) conformation. The method is illustrated by applying it to six different systems: (1) a model system with four different minima in the potential energy surface, (2) a model Muller–Brown potential, (3) the isomerization reaction of malonaldehyde using a fitting potential energy surface, (4) a model Minyaev–Quapp potential representative of con- and dis-rotations of two BH₂ groups in the BH₂–CH₂–BH₂ molecule, (5) the F?+?H₂→FH?+?H reaction and (6) the H?+?FH?→?HF?+?H reaction. Our results demonstrate that the proposed method represents a robust alternative to existing techniques for finding chemical reaction paths.     

A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations

In this paper we propose a new local discontinuous Galerkin method to directly solve Hamilton–Jacobi equations. The scheme is a natural extension of the monotone scheme. For the linear case with constant coefficients, the method is equivalent to the discontinuous Galerkin method for conservation laws. Thus, stability and error analysis are obtained under the framework of conservation laws. For both convex and noneconvex Hamiltonian, optimal (k + 1)th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations. The scheme is numerically tested on a variety of one and two dimensional problems. The method works well to capture sharp corners (discontinuous derivatives) and have the solution converges to the viscosity solution.     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

A Discrete Density Matrix Theory for Atoms¶in Strong Magnetic Fields

This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to ∞ with the magnetic field B satisfying B>> Z ^4/3 all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy. Received: 20 October 2000 / Accepted: 3 November 2000     

A Class of Counterexamples to Jeans' Theorem for the Vlasov–Einstein System

The Vlasov–Poisson and Vlasov–Einstein systems model the motion of a self gravitating system such as a galaxy. The Vlasov–Poisson system is nonrelativistic. Jeans' theorem states that every spherically symmetric solution of the Vlasov–Poisson system that is independent of time may be expressed as a function of the two invariants, energy and angular momentum. This paper shows this is not the case for the Vlasov–Einstein system. Received: 2 November 1998 / Accepted: 24 December 1998     

A Functional Central Limit Theorem for the Becker–Döring Model

We investigate the fluctuations of the stochastic Becker–Döring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the vector of the number of polymers of a given size. It is shown that the stochastic process associated to fluctuations is converging to the strong solution of an infinite dimensional stochastic differential equation (SDE) in a Hilbert space. We also prove that, at equilibrium, the solution of this SDE is a Gaussian process. The proofs are based on a specific representation of the evolution equations, the introduction of a convenient Hilbert space and several technical estimates to control the fluctuations, especially of the first coordinate which interacts with all components of the infinite dimensional vector representing the state of the process.