On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

Fractal drums and then-dimensional modified Weyl-Berry conjecture

In this paper, we study the spectrum of the Dirichlet Laplacian in a bounded (or, more generally, of finite volume) open set R ⁿ (n1) with fractal boundary of interior Minkowski dimension (n–1,n]. By means of the technique of tessellation of domains, we give the exact second term of the asymptotic expansion of the counting functionN() (i.e. the number of positive eigenvalues less than ) as +, which is of the form ^/2 times a negative, bounded and left-continuous function of . This explains the reason why the modified Weyl-Berry conjecture does not hold generally forn2. In addition, we also obtain explicit upper and lower bounds on the second term ofN().     

Dynamical Localization II with an Application to the Almost Mathieu Operator

Several recent works have established dynamical localization for Schrödinger operators, starting from control on the localization length of their eigenfunctions, in terms of their centers of localization. We provide an alternative way to obtain dynamical localization, without resorting to such a strong condition on the exponential decay of the eigenfunctions. Furthermore, we illustrate our purpose with the almost Mathieu operator, H _{, ,} =–+ cos(2(+x)), 15 and with good Diophantine properties. More precisely, for almost all , for all q>0, and for all functions ²( ) of compact support, we show that The proof applies equally well to discrete and continuous random Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes, supplied in the random case by the multi-scale analysis.     

Определение дефекто в решетки третъего ро да и величины когерентн ых областей решетки у порошкооБразного в олъфрама

, . , , -, , , .     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

Anderson localization for the almost Mathieu equation: II. Point spectrum for λ>2

We prove that for any >2 and a.e. , the pure point spectrum of the almost Mathieu operator (H()) _n = _n-1 + _n+1 + cos(2( +n)) _n contains the essential closure of the spectrum. Corresponding eigenfunctions decay exponentially. The singular continuous component, if it exists, is concentrated on a set of zero measure which is nowhere dense in .     

Influence of saturable absorber on critical phenomenon analogies in single mode laser driven by external coherent field

The influence of the existence of absorbing atoms in a laser cavity on the critical exponents of abrupt transitions between steady states in the system of single mode onephoton laser driven by an external coherent field is discussed. An isolated threshold when ,and a kind of threshold when, where is the pump parameter of the absorbing atoms, is obtained. The asymptotic forms of the families of potential functions for the system near these threshold are also found. The well-known scaling hypothesis in the general homogeneous function form is shown to be a characteristic of these asymptotic families. Four threshold exponents, , , and and four threshold amplitudes,B, D, andA, near each of these thresholds, are obtained. The threshold exponents near each threshold obey the scaling laws of critical phenomena, while the corresponding amplitudes obey the definite relations between exponents and amplitudes.     

Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.     

A\overline {Poincar\acute{e}} gauge theory of gravity

We develop a gauge theory of gravity on the basis of the principal fiber bundle over the four-dimensional space-timeM with the covering groupP¯ ₀ of the proper orthochronous Poincaré group. The field components are constructed with the connection coefficients , and with a Higgs-type field. A Lorentz metricg is introduced with , which are then identified with the components of duals of the Vierbein fields. Associated with there is a spinor structure onM. For Lagrangian densityL, which is a function of , ,, matter field , and oftheir first derivatives, we give the conditions imposed by the requirement of the gauge invariance. The Lagrangian densityL is restricted to be of the formL =L ^tot (, T ^klm ,R ^klmn , _k , ), in whichT ^klm ,R ^klmn are the field strengths of , , respectively. Identities and conservation laws following from the gauge invariance are given. Particularly noteworthy is the fact that the energy momentum conservation law follows from theinternal translational invariance. The field equation of is automatically satisfied, if those of and of are both satisfied. The possible existence of matter fields with intrinsic energy momentum is pointed out. When is a field with vanishing intrinsic energy momentum, the present theory practically agrees with the conventional Poincaré gauge theory of gravity, except for the seemingly trivial terms in the expression of the spin-angular momentum density. A condition leading to a Riemann-Cartan space-time is given. The field holds a key position in the formulation.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Strong scattering of radio waves by a two-layer medium with rough boundaries and inhomogeneous permittivity of the layer

An analytical expression for the two-frequency correlation function of reflected radiation is derived in the framework of the Kirchhoff approximation, assuming that the mean square roughness heights _1,2 of the upper (₁) and lower (₂) boundaries are large compared to the wavelength and taking account of large-scale permittivity fluctuations in the layer. The condition under which p cannot be small when _i ^{2 2} is specified. In particular, it is shown that if the scattering is only at the upper boundary of the layer (when ₁ 0, ₂ = = 0), then this condition is , where m, n=0, 1,.... The potential of the layered medium sounding methods based on the relations obtained is estimated.Institute of Physics, State University, Rostov-on-Don. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 7, pp. 619–630, July, 1995.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

Random media and eigenvalues of the Laplacian

Let be a fixed number >1. We remove [m ]-balls of centersw ₁, ..., with the same radius /m from a bounded domain inR ³. We consider the asymptotic behaviour of thek ^th eigenvalue of the Laplacian in under the Dirichlet condition as a random variable on a probability space , whenm.     

On the Schrödinger equation and the eigenvalue problem

If _k is thek ^th eigenvalue for the Dirichlet boundary problem on a bounded domain in ⁿ , H. Weyl's asymptotic formula asserts that , hence . We prove that for any domain and for all . A simple proof for the upper bound of the number of eigenvalues less than or equal to - for the operator –V(x) defined on ⁿ (n3) in terms of is also provided.Research partially supported by a Sloan Fellowship and NSF Grant No. 81-07911-A1     

Screening Effect Due to Heavy Lower Tails in One-Dimensional Parabolic Anderson Model

We consider the large-time behavior of the solution to the parabolic Anderson problem _tu=u+u with initial data u(0, ·)=1 and non-positive finite i.i.d. potentials . Unlike in dimensions d2, the almost-sure decay rate of u(t, 0) as t is not determined solely by the upper tails of (0); too heavy lower tails of (0) accelerate the decay. The interpretation is that sites x with large negative (x) hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to d=1. The result answers an open question from our previous study [BK00] of this model in general dimension.     

Effective stimulated emission and excited-state absorption cross-section spectra of para-phenylene-ethynylene polymers

The effective stimulated emission cross-section spectra, em,eff(), the stimulated emission cross-section spectra, em(), and the excited-state absorption cross-section spectra, ex()=em()–em,eff(), of the conjugated polymers poly(2,5-dioctadecyloxy-paraphenylene-ethynylene-co-2,5-thienyl) (OPT) and poly(2,5-dioctadecyloxy-paraphenylene-ethynylene-co-2,5-pyridinyl) (OPP) in tetrahydrofuran are determined in the fluorescence region. em,eff() determined by a fluorescence amplification technique is found to be small compared to em() indicating that ex() is nearly as large as em(). The laser excitation to the wide inhomogeneously broadened S1-absorption band leads to the formation of spectrally structured inhomogeneously broadened lattice relaxed exciton conformations extending on average over two monomer units which are thought to be responsible for upper-to-lower-state stimulated emission and lower-to-upper-state (excited-state) absorption.     

En">Characters and Composition Factor Multiplicities for the Lie Superalgebra $${\mathfrak{g}}{\mathfrak{l}}$$ ( m / n )

The multiplicities a of simple modules L in the composition series of Kac modules V lambda for the Lie superalgebra (m/n ) were described by Serganova, leading to her solution of the character problem for (m/n ). In Serganova's algorithm all with nonzero a are determined for a given this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero a for any given weight . In particular, we claim that for an r-fold atypical weight there are 2r distinct weights such that a = 1, and a = 0 for all other weights . Some related properties on the multiplicities a are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.     

Projectively Equivariant Symbol Calculus

The spaces of linear differential operators acting on -densities on and the space of functions on which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect (ⁿ) of vector fields of ⁿ. However, these modules are isomorphic as sl(n + 1,)-modules where is the Lie algebra of infinitesimal projective transformations. In addition, such an -equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the -equivariant symbol map to study the of kth-order linear differential operators acting on -densities, for an arbitrary manifold M and classify the quotient-modules .     

Layering transition in SOS model with external magnetic field

For the SOS model defined by the Hamiltonian , where _x , _x ,{1,2,...},h>0,x ^d ,d2 it is shown that in the low-temperature region an infinite sequence of first-order phase transitions takes place whenh»0 and the temperature is fixed.     

TM-polarized nonlinear waves guided by asymmetric dielectric layered structures

We found the field structure, exact dispersion relations and power flow ofp-polarized nonlinear guided and surface waves travelling along a three-component layered structure consisting of a film of thicknessd with dielectric constant _b bounded at the negativez-side by a linear medium with dielectric constant _a and at the positivez-side by a nonlinear uniaxial substrate characterized by the diagonal dielectric tensor ₁₁ = ₂₂ = + (|E ₁|² + |E ₂|²), ₃₃ = , <0 (self-defocusing medium),E ₁ andE ₂ being the components of the electric field in thex andy-direction, respectively. It is shown that for sufficiently smalld/ (: wavelength) the nonlinear wave may exist only at power flows exceeding some certain minimum values. For sufficiently larged/ to some values of the power flow there correspond two distinct values of the propagation constant. In this case with increasing of the power flow the number of waveguide modes is decreasing and for higher-order modes the film-waveguide exhibits an optical-power limiter from the above behaviour.