The Threshold Effects for the Two-Particle Hamiltonians on Lattices

For a wide class of two-body energy operators h(k) on the d-dimensional lattice ℤ^d, d≥3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k≠0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.     

Special functions related to Dedekind-type DC-sums and their applications

In this paper, we construct trigonometric functions in the form of a sum T _p (h, k) which is referred to as a Dedekind-type DC-(Dahee and Changhee) sum. We establish analytic properties of this sum, find its trigonometric representations, and prove a reciprocity theorem for these sums. Furthermore, we obtain relationships between the Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums, and the sum T _p (h, k). We also give some applications related to these sums and functions.     

On the essential spectrum of the transfer operator for expanding markov maps

The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inC ^k is a disk whose radius is related to the free energy of the Liapunov exponent.     

Neighborhood preferences in random matching problems

We consider a class of random matching problems where the distance between two points has a probability law which, for a small distance l, goes like l^r. In the framework of the cavity method, in the limit of an infinite number of points, we derive equations for p_k, the probability for some given point to be matched to its kth nearest neighbor in the optimal configuration. These equations are solved in two limiting cases: r = 0 -- where we recover p _k = 1/2^k, as numerically conjectured by Houdayer et al. and recently rigorously proved by Aldous -- and r→ + ∞. For 0 < r < + ∞, we are not able to solve the equations analytically, but we compute the leading behavior of p_k for large k. Received 14 February 2001     

Non-Commutative Martingale Inequalities

We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L ^p -martingale via its integrand, and then extend the Ito-Clifford integral theory in L ², developed by Barnett, Streater and Wilde, to L ^p for all 1<p<∞. We include an appendix on the non-commutative analogue of the classical Fefferman duality between $H ¹ and BMO. Received: 20 March 1997 / Accepted: 21 March 1997     

Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing

In this paper, we characterise the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence, restricted in a vertical plane, under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) coexists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate k^?3 scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent potential energy (PE) also approximately follows a k^?3 power-law and its flux is directed to small scales. For moderate stratification, there is no VSHF and the KE of the turbulent flow exhibits Bolgiano–Obukhov scaling that transitions from a shallow k^?11/5 form at large scales, to a steeper approximate k^?3 scaling at small scales. The entire range of scales shows a strong forward enstrophy flux, and interestingly, large (small) scales show an inverse (forward) KE flux. The PE flux in this regime is directed to small scales, and the PE spectrum is characterised by an approximate k^?1.64 scaling. Finally, for weak stratification, KE is transferred upscale and its spectrum closely follows a k^?2.5 scaling, while PE exhibits a forward transfer and its spectrum shows an approximate k^?1.6 power-law. For all stratification strengths, the total energy always flows from large to small scales and almost all the spectral indicies are well explained by accounting for the scale-dependent nature of the corresponding flux.     

Two-Tape Finite Automata with Quantum and Classical States

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and two-way two-tape deterministic finite automata (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called two-way two-tape finite automata with quantum and classical states (2TQCFA). First, we give efficient 2TFA algorithms for identifying languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as L_square={aⁿbⁿ² | n ? N}L_{\mathit{square}}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}. Third, we show that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by (k+1)-tape deterministic finite automata ((k+1)TFA). Finally, we introduce k-tape automata with quantum and classical states (kTQCFA) and prove that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by kTQCFA.     

Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities

We prove that the vertices of a curve γ⊂R ⁿ are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k+2)-vertex theorem for convex curves in the Euclidean space R ^2k . We obtain a very practical formula to calculate the vertices of a curve in R ⁿ . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R ^2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L:C ^∞(S ¹)→C ^∞(S ¹)).     

The Quantized Enveloping Algebra ? q (sl(n))¶at the Roots of Unity

In this paper we classify the irreducible, subregular representations of the quantum group at a primitive, -root of unity ɛ, for with p prime and k∈N. We show that every such a representation is induced from an irreducible -module and prove the De Concini, Kac, Procesi conjecture about the dimension of the -modules. Received: 30 December 1997 / Accepted: 15 November 1999     

On Noncommutative Multi-Solitons

 We find the moduli space of multi-solitons in noncommutative scalar field theories at large θ, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/θ is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the θ=∞ solitons. In two spatial dimensions, the parameter space for k solitons is a K?hler de-singularization of the symmetric product (ℝ²) ^k /S _k . We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: ℝ²/ℤ _k , cylinder, and T ² . However, we show that tori of area less than or equal to 2πθ do not admit stable solitons. In four dimensions the moduli space provides an explicit K?hler resolution of (ℝ⁴) ^k /S _k . In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in ℂ ^d , which for d>2 (and k>3) is not smooth and can have multiple branches. Received: 29 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 Communicated by R.H. Dijkgraaf     

Comparing mean field and Euclidean matching problems

Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional systems. Our focus here is on minimum matching problems, because they are computationally tractable while both frustrated and disordered. We first study a mean field model taking the link lengths between points to be independent random variables. For this model we find perfect agreement with the results of a replica calculation, and give a conjecture. Then we study the case where the points to be matched are placed at random in a d-dimensional Euclidean space. Using the mean field model as an approximation to the Euclidean case, we show numerically that the mean field predictions are very accurate even at low dimension, and that the error due to the approximation is O(1/d ² ). Furthermore, it is possible to improve upon this approximation by including the effects of Euclidean correlations among k link lengths. Using k=3 (3-link correlations such as the triangle inequality), the resulting errors in the energy density are already less than at . However, we argue that the dimensional dependence of the Euclidean model's energy density is non-perturbative, i.e., it is beyond all orders in k of the expansion in k-link correlations. Received: 1st December 1997 / Revised: 6 May 1998 / Accepted: 30 June 1998     

Ballistic behavior in a 1D weakly self-avoiding walk with decaying energy penalty

We consider a weakly self-avoiding walk in one dimension in which the penalty for visiting a site twice decays as exp[–|t–s| ^–p ] wheret ands are the times at which the common site is visited andp is a parameter. We prove that ifp<1 and is sufficiently large, then the walk behaves ballistically, i.e., the distance to the end of the walk grows linearly with the number of steps in the walk. We also give a heuristic argument that ifp>3/2, then the walk should have diffusive behavior. The proof and the heuristic argument make use of a real-space renormalization group transformation.     

No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p _k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN ¹⁻ k is partitioned intoN ² congruent subsquares, and each of these is retained with probabilityp _k. independent of all others. We show that if Π_k p _k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).     

Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects – such as Berry phase and Maslov index – occurring at order (^h/_2p)\hbar in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase – which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands – acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the “k-dependent Hamiltonian” H(k), where k is the Bloch wavevector.     

Smoothing Property for Schrödinger Equations¶with Potential Superquadratic at Infinity

We prove a smoothing property for one dimensional time dependent Schr?dinger equations with potentials which satisfy at infinity, k≥ 2. As an application, we show that the initial value problem for certain nonlinear Schr?dinger equations with such potentials is L ₂ well-posed. We also prove a sharp asymptotic estimate of the L ^p -norm of the normalized eigenfunctions of H=−Δ+V for large energy. Dedicated to Jean-Michel Combes on the occasion of his Sixtieth Birthday Received: 10 October 2000 / Accepted: 29 March 2001     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M×N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point pM such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme ``String Theory'of the Deutsche Forschungsgemeinschaft.     

Spectral Analysis of the Disordered Stochastic¶1-D Ising Model

We consider the generator of the Glauber dynamics for a 1-D Ising model with random bounded potential at any temperature. We prove that for any realization of the potential the spectrum of the generator is the union of separate branches (so-called k-particles branches, k= 0,1,2,…), and with probability one it is a nonrandom set. We find the location of the spectrum and prove the localization for the one-particle branch of the spectrum. As a consequence we find a lower bound for the spectral gap for any realization of the random potential. Received: 22 September 1998 / Accepted: 12 February 1999     

Diagonal and Off-Diagonal Recursion Relation for the General Potential V(r) = Arp

A recursion relation is derived for the potential V(r) = Ar ^p. Generally, this connects off-diagonal matrix elements of r ^k–2, r ^k+p, r ^k, and r ^k+2. The diagonal case is obtained by setting m = n in this relation. The relation is derived by elementary methods and without recourse to specific properties of the eigenstates. Finally, this relation is studied for the familiar potentials p = –1, 1, 2.     

Linear Stability in an Ideal Incompressible Fluid

 We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 RID="*" ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. RID="**" ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084. Acknowledgements. The authors thank Susan Friedlander for useful discussions. Communicated by P. Constantin     

Continuity of percolation probability on hyperbolic graphs

LetT _k be a forwarding tree of degreek where each vertex other than the origin hask children and one parent and the origin hask children but no parent (k2). DefineG to be the graph obtained by adding toT _k nearest neighbor bonds connecting the vertices which are in the same generation.G is regarded as a discretization of the hyperbolic planeH ² in the same sense thatZ ^d is a discretization ofR ^d . Independent percolation onG has been proved to have multiple phase transitions. We prove that the percolation probabilityO(p) is continuous on [0,1] as a function ofp.