Generalized Kochen-Specker theorem

A generalized Kochen-Specker theorem is proved. It is shown that there exist sets of n projection operators, representing n yes-no questions about a quantum system, such that none of the 2 possible answers is compatible with sum rules imposed by quantum mechanics. Namely, if a subset of commuting projection operators sums up to a matrix having only even or only odd eigenvalues, the number of yes answers ought to he even or odd, respectively. This requirement may lead to contradictions. An example is provided, involving nine projection operators in a 4-dimensional space.Dedicated to Professor Max Jammer on the occasion of his 80th birthday.I am grateful to N. D. Mermin for patiently explaining to me that ref. 11 was a Kochen-Specker argument, not one about locality, as I had wrongly thought. This work was supported by the Gerard Swope Fund, and the Fund for Encouragement of Research.     

On a theorem of Deift and Hempel

We provide an alternative proof of the main result of Deift and Hempel [1] on the existence of eigenvalues ofv-dimensional Schrödinger operatorsH =H ₀+W in spectral gaps ofH ₀.Research partially supported by USNSF under Grant DMS-8416049On leave of absence from the Institute for Theoretical Physics, University of Graz, A-8010 Graz, Austria; Max Kade Foundation Fellow     

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the 0 limit.Research partially supported by U.S.N.S.F. under Grant MCS-78-01885.     

Yang-Baxter equation and representation theory: I

The problem of constructing the GL(N,) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered. In caseN=2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2,) are obtained and their eigenvalues are calculated. Some results for the caseN>2 are also presented.     

A limit theorem for stochastic acceleration

We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)= (t). The equation of motion is (t)=F(x(t),v(t), ), wherex(·) andv(·) take values in ^d ,d3, and ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as 0,v (t)v(t/²) converges weakly to a diffusion Markov processv(t), and ² x (t) converges weakly to , wherex=lim ² x (0).     

The central limit theorem and the problem of equivalence of ensembles

In this paper we show that the local limit theorem is a consequence of the integral central limit theorem in the case of a Gibbs random field _t ,tZ corresponding to a finite range potential.We apply this theorem to show that the equivalence between Gibbs and canonical ensemble is a consequence of the integral central limit theorem and of very weak conditions on decrease of correlations.Research supported by a C.N.R.-Ak. Nauk. U.R.S.S. fellowship     

Renormalizing rectangles and other topics in random matrix theory

We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM–N zero eigenvalues (forM>N). These zero eigenvalues are kinematical in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the rectangularityr=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical -function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r–1| is held fixed asN, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.     

Antilocality and a Reeh-Schlieder theorem on manifolds

It is shown that the operatorA ^1/2, whereA is any positive self-adjoint extension of a positive operator of the form -Laplace-Beltrami operator +potential on ann-dimensional Riemannian manifold, is strongly antilocal. Using this result, a Reeh-Schlieder theorem for the canonical vacuum of the Klein-Gordon field propagating in ultrastatic spacetimes is derived. In a further application, we gain weaker versions of the Reeh-Schlieder theorem for more general situations.Supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Spectral Analysis and Zeta Determinant on the Deformed Spheres

We consider a class of singular Riemannian manifolds, the deformed spheres , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian , we study the associated zeta functions . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular and . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. Partially supported by FAPESP: 2005/04363-4     

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τ_ess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τ_ess(H) the following asymptotics is found

Ensemble average of an arbitrary number of pairs of different eigenvalues using Grassmann integration

An identity satisfied by the eigenvalues of a real-symmetric matrix and an integral representation of a determinant using Grassmann variables are used to show that the ensemble average ofS different pairs of eigenvalues of a GOE is given by (–1) ^S 2^{–S –1/2}(S+1/2).     

Converse of the Eilers-Horst theorem

We generalize the theorem of Eilers and Horst, showing that any finite as well as any-finite measure on a quantum logic of all closed subspaces of a Hilbert spaceH of dimension 2 is a Gleason one iff the dimension ofH is a nonmeasurable cardinal.     

PT-Symmetry, Canonical Decomposition, Perturbation Theory

Let H be any PT-symmetric Schrödinger operator of the type H=- ² +x ² +igW(x), where W is a real polynomial, odd under reflection of all coordinates, gR, acting on L ² ( R ^d ). The proof is outlined of the following statements: PH is self-adjoint and its eigenvalues coincide with the eigenvalues of (H*H). Moreover the eigenvalues of (H*H), known as the singular values of H, can be computed via perturbation theory by Borel summability.     

Theu=0 structure theorem

The previous theorem of the author on the analytic structure of the bubble diagram functions that occur in unitary equations (and are kernels of products of connected scattering operatorsS _m,n ^c or (S ^–1) _m,n ^c , and related quantities), is extended to a class of situations, called here in generalu=0 points, that were not covered by this earlier result.This new theorem, which is proved on the basis of a refined macrocausality condition, resolves one of the remaining crucial problems in the derivation of discontinuity formulae and related results inS-matrix theory: all points are in factu=0 points for some of the bubble diagram functions, such as ((S ^–1) _3,3 ^c S _3,3 ^c ), that are encountered even in the simplest cases. In all previous approaches, ad hoc technical assumptions with no a priori physical basis were required for these terms.The origin of theu=0 problem is the absence of information, in general, on a product of distributions that are boundary values of analytic functions from opposite directions, and more generally on the essential support, or singular spectrum, of a product of distributions whose essential supports contain opposite directions. On the other hand, the recent results obtained by Kashiwara-Kawai-Stapp in the framework of hyperfunction theory apply mainly to phase-space factors, whose bubbles are constants times conservation -functions rather than actual scattering operators. The present work has basically required the development of new physical and mathematical ideas and methods. In particular, a new general result on the essential support of a product of bounded operators is presented inu=0 situations, under a general regularity property on individual terms. The latter follows in the application from the refined macrocausality condition, in the same time as information on the essential support ofS-matrix kernels.     

The structure of projection-valued states: A generalization of Wigner's theorem

A projection-valued state is defined to be a completely orthoadditive map from the projections on one Hilbert space into the projections on another Hilbert space, which preserves the unit. Any such mapping is shown to have the formP U ₁(P 1₁)U ₁ ^–1 U ₂(P 1₂)U ₂ ^–1 , whereU ₁ is unitary andU ₂ is antiunitary, generalizing Wigner's theorem on symmetry transformations. A physical interpretation is given and the relation to quantum logic is discussed.The contents of this paper are a portion of the author's dissertation at the University of Massachusetts at Amherst.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

Discrete Structure of Some Schlesinger Systems on the Riemann Sphere

In the present work we investigate the group structure of the Schlesinger transformations for isomonodromic deformations of the Fuchsian differential equations. We perform these transformations as isomorphisms between the moduli spaces of the logarithmic sl(N)-connections with fixed eigenvalues of the residues at singular points. We give a geometrical interpretation of the Schlesinger transformations and perform our calculations using the techniques of the modifications of bundles with connections, or, the Hecke correspondences for the loop group SL(N)C(z).     

Goldstone theorem for bose systems

In three or more dimensions (3) it is proved that if the correlations decay faster than |x|^-(-20) then gauge symmetry breaking is excluded. In one and two dimensions (=1 or 2) the gauge symmetry is always preserved.     

The Spectrum of Heavy Tailed Random Matrices

Let X _N be an N → N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X _N , once renormalized by , converges almost surely and in expectation to the so-called semicircular distribution as N goes to infinity. In this paper we study the same question when P is in the domain of attraction of an α-stable law. We prove that if we renormalize the eigenvalues by a constant a _N of order , the corresponding spectral distribution converges in expectation towards a law which only depends on α. We characterize and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero. This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.