Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

The asymmetric contact process at its second critical value

The asymmetric contact process onZ has two distinct critical values ₁ > ₂ (at least with sufficient asymmetry). One can consider the process on {0,...,N} and analyze the time (which we call _N ) till complete vacany starting from complete occupation. Its behavior has already been resolved for all regions of except for =₂. For this value, Schinazi proved that lim _N log _N /logN=2 in probability and conjectured that _N /N ² converges in distribution. It is that result that we prove in this paper. We rely heavily on the Brownian motion behavior of the edge particle, which comes from Galves and Presutti and Kuczek.     

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraU at a ^th root of 1 has maximal dimension (= ^N ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU , the last one being aq-commutative algebraU ^(2N) . This allows us to reduce many problems concerningU to that concerningU ^(2N) .To Armand Borel on his 70^th birthdaySupported in part by the NSF grant DMS-9103792     

The effective mass of holes in CdSb from reflectivity measurements

From the measured reflectivity near the plasma edge the effective mass of free holes in Ag-doped CdSb was determined. In the studied concentration rangem _b ^* >m _a ^* >m _c ^* is valid. The dependence of the effective mass on the concentration of free holes indicates a strong non-parabolicity of the bands.The authors wish to thank Dr. B. Rheinländer for discussions and information about his results before publication. Mrs. D. Vaková carried out the measurements on UR-10 Zeiss, Mrs. V. Míková prepared the single crystals, Mr. J. Kapar oriented them by X-ray diffraction and Dr. L. Satoura determinedN and. The authors thank them for their help.     

Thermal fluctuations and ac response of weakly pinned charge density waves

The Fukuyama-Lee theory for the ac response () of weakly pinned charge density waves is extended to include thermal fluctuations. The equation of motion for the local phase includes an extrinsic damping and a distinction is made between static and dynamic parameters in it. It is split into static, thermally fluctuating and response contributions to the phase, respectively. The static problem is treated using a result from Feigel'man's theory which provides a revised value for the weak pinning constant. The impurity averaging of the response equation is performed using the simplifying statistical properties of the stochastic pinning force following Bleher's recent work. The main emphasis is on the treatment of the thermal fluctuations via a thermal field _th. The non-linear Langevin equation for _th is linearized and further simplified by an RPA type approximation which eliminates the impurity fluctuations from _th. The resulting equation is solved exactly. It is shown that the correlation function of the thermal field decays initially with a short time constant. This allows to treat the thermal fluctuations on an equal footing with the impurity fluctuations in the self-consistent Born approximation. The main contribution of the thermal fluctuations results in powers of a thermal factor exp(- _th ² /2) to the first and second order self energies of the phason Green's function. Numerical results due to these modifications are given for (,T). It is found that the absorption peak in Re () broadens and shifts to lower frequencies when the temperature is raised. The corresponding treatment for three spatial dimensional is indicated. The thermal factor is evaluated for this case and differences to Maki's result are noted. The questions of analyticity and conductivity sum rule are also dealt with.Dedicated to Professor Helmut Reik on the occasion of his 60th birthday     

Theory of electric field-induced optical second harmonic generation in semiconductors

The nonlinear optical coefficientd _ij(kl) (2, 0,, ) for electric field-induced optical second harmonic generation in semiconductors is calculated by means of a perturbation treatment. The result is given directly as a Brillouin zone integral over a resonant energy denominator. A simplified energy band structure model is used to carry out the Brillouin zone integral. The analytic closed-form expression ford _ij(kl) (2, 0,, ) thus obtained permits the calculation of the absolute value of its spectrum from available energy band parameters. The dispersion ofd ₁₁₍₁₁₎ (2, 0,, ) of Ge is numerically calculated, second harmonic photon energies being close to theE ₀ andE ₁ gaps. The results show pronounced resonant behaviours.     

Parametric relaxation of spectra

We study the change of an quasienergy spectrum upon variation of the weight of a perturbation in the Floquet operatorF=F ₀e^–iV . Employing ideas from level dynamics and random matrix-theory we show that the distribution of nearest-neighbor spacings can display effectively irreversible behavior. Small deviations from equilibrium relax in a certain collision time which scales with the numberN of levels as _collN ^–3/2.     

Simulation calculation of dielectric constants: Comparison of methods on an exactly solvable model

A mean spherical model of classical dipoles on a simple cubic lattice of sideM=2N+1 sites is considered. Exact results are obtained for finite systems using periodic boundary conditions with an external dielectric constant and using reaction field boundary conditions with a cutoff radiusR _c N and an external dielectric constant. The dielectric constant in the disordered phase is calculated using a variety of fluctuation formulas commonly implemented in Monte Carlo and molecular dynamics simulations of dipolar systems. The coupling in the system is measured by the parametery=4 ²/9kT, where ² is the fixed mean square value of the dipole moments on the lattice. The system undergoes a phase transition aty2.8, so that very high dielectric constants cannot be obtained in the disordered phase. The results show clearly the effects of system size, cutoff radius, external dielectric constant, and different measuring techniques on a dielectric constant estimate. It is concluded that with periodic boundary conditions, the rate of approach of the dielectric constant estimate to its thermodynamic limit is asN ^–2/3 and depends only weakly on. Methods of implementing reaction field boundary conditions to give rapid convergence to the thermodynamic limit are discussed.     

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

Correlation inequalities are used to show that the two component (²)² model (with HD, D, HP, P boundary conditions) has a unique vacuum if the field does not develop a non-zero expectation value. It follows by a generalized Coleman theorem that in two space-time dimensions the vacuum is unique for all values of the coupling constant. In three space-time dimensions the vacuum is unique below the critical coupling constant.For then-componentP(||²)₂+₁ model, absence of continuous symmetry breaking, as goes to zero, is proven for all states which are translation invariant, satisfy the spectral condition, and are weak* limit points of finite volume states satisfyingN _loc and higher order estimates.     

Escape times in interacting biased random walks

The dynamics ofN particles with hard core exclusion performing biased random walks is studied on a one-dimensional lattice with a reflecting wall. The bias is toward the wall and the particles are placed initially on theN sites of the lattice closest to the wall. ForN=1 the leading behavior of the first passage timeT _FP to a distant sitel is known to follow the Kramers escape time formulaT _FP ^l where is the ratio of hopping rates toward and away from the wall. ForN > 1 Monte Carlo and analytical results are presented to show that for the particle closest to the wall, the Kramers formula generalizes toT _FR ^IN. First passage times for the other particles are studied as well. A second question that is studied pertains to survival timesT _s in the presence of an absorbing barrier placed at sitel. In contrast to the first passage time, it is found thatT _s follows the leading behavior independent ofN.     

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS _r ⁿ of radiusr (n 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys. 16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS _r ⁿ . The above class includes the uniform measures onxs Rⁿ ¦x¦ r for any 0 p r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys. 8:478 (1967)] and the underlying symmetry property.     

Higher Conformal Multifractality

We derive, from conformal invariance and quantum gravity, the multifractal spectrum f() of the harmonic measure (i.e., electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions. It gives the Hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r . First examples are a random walk, i.e., a Brownian motion, a self-avoiding walk, or a critical percolation cluster. The generalized dimensions D(n) as well as the multifractal functions f() are derived, and are all identical for these three cases. The external frontiers of a Brownian motion and of a percolation cluster are thus identical to a self-avoiding walk in the scaling limit. The multifractal (MF) function f(,c) of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a Q-state Potts cluster, is given as a function of the central charge c of the associated conformal field theory. The dimensions D _EP of the external perimeter and D _H of the hull of a critical scaling curve or cluster obey the superuniversal duality equation . Finally, for a conformally invariant scaling curve which is simple, i.e., without double points, we derive higher multifractal functions, like the universal function f ₂(,) which gives the Hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve, and as r on the other. The general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated. The results apply to critical O(N) loops, Potts clusters, and to the SLE process. We present a duality between external perimeters of Potts clusters and O(N) loops at their critical point, as well as the corresponding duality in the SLE process for =16.     

Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A ₀ in a Hilbert space ₀ are defined formally as A ₍₎=A ₀+GG ^*, where G is an injective linear mapping from = ^d to the scale space _-k(A₀)k , kN, of generalized elements associated with the self-adjoint operator A ₀, and where is a self-adjoint operator in . The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A ₍₎ in the general setting ran G – _k (A ₀), kN, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

LetN, be von-Neumann-Algebras on a Hilbert space , a comon cyclic and separarting vector. Assume to be cyclic and separating also forN . Denote byJ , J _N the modular conjugations to (, ), and _N the associated modular operators. If and these data define in a canonical way a conformal quantum field theory in a cricle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentiageometrie und Quantenphysik     

Measurement of the reaction γp → K0Σat photon energies up to 2.6 GeV⋆

The reaction p K⁰⁺ was measured in the photon energy range from threshold up to 2.6 GeV with the SAPHIR detector at the electron stretcher facility, ELSA, in Bonn. Results are presented on the reaction cross-section and the polarization of the ⁺ as a function of the kaon production angle in the centre-of-mass system, cos^c.m._K, and the photon energy. The cross-section is lower and varies less with photon energy and kaon production angle than that of p K⁺⁰. The ⁺ is polarized predominantly at cos^c.m._K 0. The data presented here are more precise than previous ones obtained with SAPHIR and extend the photon energy range to higher values. They are compared to isobar model calculations.     

A solution spectrum of the nonlinear Schrödinger equation. II

It was shown in a previous communication that the nonlinear Schrödinger equation exhibits a spectrum of eigenfunctions of the form = _k,A _k (coshkx) ^–k and = _k B _k (coshkx) ^–k–1sinhkx, and the corresponding eigenvalues of the energy are related to a band structure with a characteristic energy gap as a significant feature. In the present paper, it is shown that a further spectrum exists exhibiting the general structure = _k=0 A _k(cosh kx)^–k–1/2and = _k=0 B_k(cosh kx)^–k–3/2sinhkx and yielding also a band structure. An extension of the solution spectrum to a nonlinear Klein-Gordon equation and a nonlinear Dirac equation does not imply essential difficulties, and the corresponding characteristic band structure has to be related to a mass spectrum.     

Some results for the exponential interaction in two or more dimensions

We show that for the regularized exponential interaction :e : ind space-time dimensions the Schwinger functions converge to the Schwinger functions for the free field ifd>2 for all or ifd=2 for all such that ||>₀.Partially sponsored by the I.H.E.S. through the Stiftung Volkswagenwerk     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

On the photo-electroluminescence of ZnS-Cu

The influence of weak ultra-violet irradiation on the brightness waves of electroluminescence is investigated for two types of ZnS-Cuphosphors. The observed effects (increase in brightness in the primary peak and its phase shift, the disappearance of the secondary peak) are explained on the basis of present-day conceptions on electroluminescence.

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ZnS-Cu

ZnS-Cu. ( , ) .