A general class of quantum fields without cut-offs in two space-time dimensions

We consider a selfinteracting boson field in two space-time dimensions, with interaction densities of the form:V((x)): where (x) is a scalar boson field, andV() is a real positive function of exponential type. We define the space cut-off interaction by and prove thatH _r =H ₀+V _r , whereH ₀ is the free energy, is essentially self adjoint. This permits us to take away the space cut-off and we obtain a quantum field free of cut-offs.At leave from Mathematical Institute, Oslo University.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.     

Minimal Terminal ℚ-Factorial Models of Drinfeld Coarse Moduli Schemes

In this article we formally prove that the coarse moduli scheme M^r(1) of rational Drinfeld modules of rank r is an affine -factorial equivariant compactification of M^r(1) and we prove the uniqueness of such a compactification. Finally, the coarse moduli surface M³(1) is described in detail.     

Localization for some continuous random Schrödinger operators

We study the spectrum of random Schrödinger operators acting onL ²(R ^d ) of the following type . The are i.i.d. random variables. Under weak assumptions onV, we prove exponential localization forH at the lower edge of its spectrum. In order to do this, we give a new proof of the Wegner estimate that works without sign assumptions onV.

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Résumé Dans ce travail, nous étudions le spectre d'opérateurs de Schrödinger aléatoires agissant surL ²(R ^d ) du type suivant . Les sont des variables aléatoires i.i.d. Sous de faibles hypothèses surV, nous démontrons que le bord inférieur du spectre deH n'est composé que de spectre purement ponctuel et, que les fonctions propres associées sont exponentiellement décroissantes. Pour ce faire nous donnons une nouvelle preuve de l'estimée de Wegner valable sans hypothèses de signe surV.

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U.R.A. 760 C.N.R.S.     

Crystal Structure of Level Zero Extremal Weight Modules

We consider the crystal structure of the level zero extremal weight modules V() using the crystal base of the quantum affine algebra constructed in Duke Math. J. 99 (1999), 455–487. This approach yields an explicit form for extremal weight vectors in the U ^– part of each connected component of the crystal, which are given as Schur functions in the imaginary root vectors. We show the map induces a correspondence between the global crystal base of V() and elements .     

Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q( ) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q( ), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q( ) which containsV with multiplicity one and is such that all other irreducible U_q( )-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q( ) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q( )-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of , there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q( ) is of typeD ₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA _n,every affinization is isomorphic to a tensor product of representations of U_q( ) which are irreducible under U_q( ) (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.     

Scattering theory and dispersion relations for a class of long-range oscillating potentials

If a spherically symmetric potential is such that , and if an additional regularity condition is imposed^r[a sufficient one being thatrV(r) isL ¹], the partial wave amplitudes are meromorphic in a strip of width in the complex momentum plane, and the full scattering amplitude is analytic inside an ellipse at fixed energy and satisfies fixed momentum transfer dispersion relations for |t|<².Such a class of potentials includes not only exponentially decreasing potentials but also long-range oscillating potentials such as (1 +r ²)^–2 sin (exp r). In fact the results can partly be extended to a still broader class of potentials with increasing amplitude at infinity. It is argued that these results might lead to a revision of conventional ideas on what is the potential between physical hadrons.Appendices may be of interest to special functions addicts.Dedicated to Nick KhuriLaboratoire associé au C.N.R.S.     

On the Superdiffusive Behavior of Passive Tracer with a Gaussian Drift

In the present article we consider a motion of a passive tracer particle, whose trajectory satisfies the Itô stochastic differential equation d x(t) = V(t, x(t)) dt + d w(t), where w(·) is a Brownian motion, V is a stationary Gaussian random field with incompressible realizations independent of w(·) and >0. We prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field. The result is shown both for steady (time independent) and time dependent and Markovian velocity fields. In addition, we provide explicit upper and lower bounds for the Hurst exponent of the trajectory. All previous rigorous results concerned explicitely solvable shear flows cases.     

Intersections of random walks in four dimensions. II

Letf(n) be the probability that the paths of two simple random walks of lengthn starting at the origin in ⁴ have no intersection. It has previously been shown thatf(n)c(logn)^–1/2. Here it is proved that for allr>1/2, .Research Supported by NSF grant MCS-8301037     

New bounds on the number of bound states for Schrödinger operators

We consider the Schrödinger operatorH = – +V(|x|) onR ³. Letn denote the number of bound states with angular momentum (not counting the 2 + 1 degeneracy). We prove the following bounds onn . LetV 0 and d/dr r ^1-2p (-V)^{1 –p} 0 for somep [1/2, 1) then

Boson fields with bounded interaction densities

We consider interaction densities of the formV((x)), where (x) is a scalar boson field andV() is a bounded real continuous function. We define the cut-off interaction by , where _E(x) is the momentum cut-off field. We prove that the scattering operator S_r(V) corresponding to the cut-off interaction exists, and we study the behavior of the scattering operator as well as the Heisenberg picture fields, as the cut-off is removed.This research partially sponsored by the Air Force Office of Scientific Research under Contract AF 49(638)1545.At leave from Mathematical Institute, Oslo University.     

Analytic solution of the general anharmonic-oscillator problem

The regular and irregular solution of the differential three-dimensional Schrödinger equation with the confining potential , is presented. In full analogy with the recentq=1 conjecture of Singhet al. [1], the corresponding Green's function is constructed in terms of the extended continued fraction. For the simplest example withV(r)=r ²+² r ⁴, its convergence is proved and the resulting energy spectrum is numerically tested.     

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Navier-Stokes equations and area of interfaces

We present new a priori estimates for the vorticity of solutions of the three dimensional Navier-Stokes equations. These estimates imply that theL ¹ norm of the vorticity is a priori bounded in time and that the time average of the 4/(3+) power of theL ^4/(3+) spatial norm of the gradient of the vorticity is a priori bounded. Using these bounds we construct global Leray weak solutions of the Navier-Stokes equations which satisfy these inequalities. In particular it follows that vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. Next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional Hausdorff measure of level sets of the vorticity magnitude. We obtain a priori bounds on an average such measure, >. When expressed in terms of the Reynolds number and the Kolmogorov dissipation length , these bounds are

Four-quark states and nucleon-antinucleon annihilation within the quark model with QCD vacuum-induced interaction

The spectrum of , J^p=0⁺, 2⁺ mesons is discussed. We have shown that due to instanton-induced forces the physical states are strong mixtures of theSU _f (3) group basis states. The cross-sections for annihilation of the system into mesons are obtained. Thea ₀(980) meson is considered as meson consisting of 9 _f and 36 _f plets. The branchings are also predicted for the cross-sections for production of thea ₀(980) and tensor mesons in annihilation.     

Feynman path integrals and the trace formula for the Schrödinger operators

We study Schrödinger operators of the form on ^d , whereA ² is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If _n are the eigenvalues ofH we show that the theta function is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of (t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.Partially supported by the USP-Mathematisierung, University of Bielefeld (Forschungsprojekt Unendlich dimensionale Analysis)     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

Principle of limiting absorption for N-body Schrödinger operators — A remark on the commutator method

We consider a N-body Schrödinger operator H=H ₀+V. The interaction V is given by a sum of pair potentials V _jk(y)(=V _jk ^s +V _jk ^l ), y R³. We assume that: V _jk ^s =O(|y|^-(1+p)), p>0, as |y| for the short-range part V _jk ^s ; for the long-range part V _jk ^l . Under this assumption, we prove the principle of limiting absorption for H. The obtained result is essentially as good as those obtained in the two-body case. The proof is done by a slight modification of the remarkable commutator method due to Mourre.     

A bound on the number of bound states in the Schrödinger equation

A boundS _l is given for the number of bound statesn _i in thelth partial wave corresponding to a spherically symmetric potential in non-relativistic quantum mechanics. This bound is given by whereV _a(l, r) is the attractive part of the effective potentialV(r)+l(l+1)/r ². Extensive comparative study ofS _i and the Bargmann inequality is made.     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Boson fields under a general class of local relativistic invariant interactions

We consider a boson field (x) under an interaction of the form V((x))dx, whereV() is a bounded continuous real function of a real variable . IfV() has a uniformly continuous and bounded first derivative, we prove that the Heisenberg picture field exists as weak limits of the Heisenberg picture fields corresponding to the cut-off interaction.