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.     

Existence of Resonances for Metric Scattering in Even Dimensions

Suppose _g is the (negative) Laplace–Beltrami operator of a Riemannian metric g on ⁿ which is Euclidean outside some compact set. It is known that the resolvent R()=(– _g –²)^–1, as the operator from L ² _comp( ⁿ ) to H ² _loc( ⁿ ), has a meromorphic extension from the lower half plane to the complex plane or the logarithmic plane when n is odd or even, respectively. Resonances are defined to be the poles of this meromorphic extension. We prove that when n is 4 or 6, there always exist infinitely many resonances provided that g is not flat. When n is greater than 6 and even, we prove the same result under the condition that the metric is conformally Euclidean or is close to the Euclidean metric.     

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     

Landau Hamiltonians with Unbounded Random Potentials

We prove the almost sure existence of a pure point spectrum for the two-dimensional Landau Hamiltonian with an unbounded Anderson-like random potential, provided that the magnetic field is sufficiently large. For these models, the probability distribution of the coupling constant is assumed to be absolutely continuous. The corresponding densityg has support equal to , and satisfies , for some > 0. This includes the case of Gaussian distributions. We show that the almost sure spectrum is , provided the magnetic field B0. We prove that for each positive integer n, there exists a field strength B _n , such that for all B>B _n , the almost sure spectrum is pure point at all energies except in intervals of width about each lower Landau level , for m < n. We also prove that for any B0, the integrated density of states is Lipschitz continuous away from the Landau energiesE _n (B). This follows from a new Wegner estimate for the finite-area magnetic Hamiltonians with random potentials.     

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.     

Binary reactions in $$\bar pp$$ collisions at intermediate energies

We discuss here the binary reactions of strange and charmed particle production in collisions at intermediate energies. In the case of baryon production with only one strange or charmed quark the cross section is determined by planar diagrams withK ^*,K ^** orD ^*,D ^**-meson poles in thet-channel. We calculated these diagrams in the frame of quark-gluon string model (QGSM) proposed earlier. We obtained also the cross-sections for reactions with baryon exchange in thet-channel with and pair in the final state. Predicted cross-sections for the reactions of production are of the order of hundred nanobarns. Using reggeon calculus we estimated cross-sections of binary reactions with two or three strange quarks in the final state: and . We discuss also the possible manifestation of color transparency effects in reactions with antiprotons on nuclei where all antiproton quarks annihilate.     

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in ((n/k+logk)n ^2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes (n ^1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in (logn) parallel time, and so is in the complexity class NC.     

New quantum groups by double-bosonisation

We obtain new family of quasitriangular Hopf algebras via the author's recent double-bosonisation construction for new quantum groups. They are versions of U _q(su _n+1) with a fermionic rather than bosonic quantum plane of roots adjoined to U _q(su _n). We give the n = 2 case in detail. We also consider the anyonic-double of an anyonic ( ) braided group and the double-bosonisation of the free braided group in n variables.     

Finitary Spacetime Sheaves of Quantum Causal Sets: Curving Quantum Causality

A locally finite, causal, and quantal substitute for a locally Minkowskian principal fiber bundle of modules of Cartan differential forms over a bounded region X of a curved C -smooth spacetime manifold M with structure group G that of orthochronous Lorentz transformations L ⁺ := SO(1,3), is presented. is usually regarded as the kinematical structure of classical Lorentzian gravity when the latter is viewed as a Yang-Mills type of gauge theory of a sl(2, {})-valued connection 1-form . The mathematical structure employed to model this replacement of is a principal finitary spacetime sheaf of quantum causal sets with structure group G _n, which is a finitary version of the continuous group G of local symmetries of General Relativity, and a finitary Lie algebra g _n-valued connection 1-form on it, which is a section of its subsheaf . is physically interpreted as the dynamical field of a locally finite quantum causality, whereas its associated curvature as some sort of finitary and causal Lorentzian quantum gravity.     

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     

On the Leibniz Homology of Poisson Algebras

We study the Leibniz homology of the Poisson algebra of polynomial functions over (²ⁿ ,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some _2n ( )-modules and obtain some explicit result in low degree.     

Ellipsoidal bag model for heavy quark system

The ellipsoidal bag model is used to describe heavy quark systems such asQ ,Q g andQ ² . Instead of two step model, these states are described by an uniform picture. The potential derived from the ellipsoidal bag forQ is almost equivalent to the Cornell potential. For aQ ² system with large quark pair separation, an improvement of 70 MeV is obtained comparing with the spherical bag.     

On the parametrisation of unitary matrices by the moduli of their elements

The parametrisation of ann×n unitary matrix by the moduli of its elements is not a well posed problem, i.e. there are continuous and discrete ambiguities which naturally appear. We show that the continuous ambiguity is (n–1)(n–3)-dimensional in the general case and in the symmetric caseS _ij=S_ij. We give also lower bounds on the number of discrete ambiguities, the number of solutions being at least in the first case and for the symmetric one, where [r] denotes the integral part ofr.     

The Chern classes of Sobolev connections

AssumeF is the curvature (field) of a connection (potential) onR ⁴ with finiteL ² norm . We show the chern number (topological quantum number) is an integer. This generalizes previous results which showed that the integrality holds forF satisfying the Yang-Mills equations. We actually prove the natural general result in all even dimensions larger than 2.     

Updated estimate of the muon magnetic momentusing revised results from <Emphasis Type="Italic">e</Emphasis><Superscript> + </Superscript><Emphasis Type="Italic">e</Emphasis><Superscript>-</Superscript>annihilation

A new evaluation of the hadronic vacuum polarization contribution to the muon magnetic moment is presented. We take into account the reanalysis of the low-energy e ⁺ e ^-annihilation cross section into hadrons by the CMD-2 Collaboration. The agreement between e ⁺ e ^-and spectral functions in the channel is found to be much improved. Nevertheless, significant discrepancies remain in the center-of-mass energy range between 0.85 and , so that we refrain from averaging the two data sets. The values found for the lowest-order hadronic vacuum polarization contributions are where the errors have been separated according to their sources: experimental, missing radiative corrections in e ⁺ e ^-data, and isospin breaking. The corresponding Standard Model predictions for the muon magnetic anomaly read where the errors account for the hadronic, light-by-light (LBL) scattering and electroweak contributions. The deviations from the measurement at BNL are found to be (1.9 ) and (0.7 ) for the e ⁺ e ^-- and -based estimates, respectively, where the second error is from the LBL contribution and the third one from the BNL measurement.Received: 7 September 2003, Published online: 30 October 2003     

Equilibrium states of gravitational systems

We formulate the equilibrium correlation functions for local observables of an assembly of non-relativistic, neutral gravitating fermions in the limit where the number of particles becomes infinite, and in a scaling where the region , to which they are confined, remains fixed. We show that these correlation functions correspond, in the limit concerned, to states on the discrete tensor product , where the are copies of the gauge invariantC*-algebra of the CAR overL ²(R ³). The equilibrium states themselves are then given by , where , is the Gibbs state on for an infinitely extended ideal Fermi gas at density , and where ₀ is the normalised density function that minimises the Thomas-Fermi functional, obtained in [2], governing the equilibrium thermodynamics of the system.     

Masses ofq 2 \bar q 2 states in a Bethe-Salpeter model

Ground-state masses ofq ^{2 –2} states (true and mock baryonium) are investigated in the framework of a Bethe-Salpeter formalism motivated from QCD. The four-particle system is described by pairwise interactions betweenqq orq pairs with a spectator approximation for the non-interacting pair. The quark-quark interactions are Coulomb plus harmonic interactions; the harmonic terms have been modified to produce linear confinement for heavier quarks, in agreement with experimental spectra. The confining interaction is proportional to the strong coupling constant _s. Apart from the quark masses, the confining interaction is characterized by three basic parameters: (i) a universal spring constant ₀; (ii) a constantC ₀/ ₀ ² , which defines the vacuum structure; (iii) a constantA ₀, which provides a smooth transition from quadratic to linear confinement as one goes from light to heavy quark systems. These three constants [ ₀ = 0.158 GeV;C ₀=0.296;A ₀=0.0283] have been shown to produce excellent fits to all quarkonia states [q ,q ,Q ] as well as baryon spectra (qqq); thus our predictions forq ^{2 2} states contain no free parameters. In this model, theL=0 ground states occur in the range 1.8–2 GeV, 2.15–2.3 GeV and 6.72–6.75 GeV foru ^{2 2},s ^{2 2} andc ^{2 2} states, respectively. We discuss the prospects for these states to be seen experimentally. In the case of thes ^{2 2} state, this is likely to have a rather narrow width, and may correspond to theX(2.22 GeV) meson observed in radiative decays of theJ/ meson. Thec ^{2 2} state might also be visible as a resonance with an appreciable width.Research supported in part by the National Science Foundation under grant NSF-PHY 86-06364Research supported in part by the U.S. Department of Energy     

Hearing the zero locus of a magnetic field

We investigate the ground state of a two-dimensional quantum particle in a magnetic field where the field vanishes nondegenerately along a closed curve. We show that the ground state concentrates on this curve ase/h tends to infinity, wheree is the charge, and that the ground state energy grows like (e/h)^2/3. These statements are true for any energy level, the level being fixed as the charge tends to infinity. If the magnitude of the gradient of the magnetic field is a constantb ₀ along its zero locus, then we get the precise asymptotics(e/h) ^2/3 (b ₀) ^2/3 E _* +O(1) for every energy level. The constantE _* .5698 is the infimum of the ground state energiesE() of the anharmonic oscillator family .     

Remarks on the Schur—Howe—Sergeev Duality

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur–Sergeev duality between q(n) and a central extension of the hyperoctahedral group H _k. We show that the zero-weight space of a q(n)-module with highest weight given by a strict partition of n is an irreducible module over the finite group parameterized by . We also discuss some consequences of this Howe duality.     

Crystal field of Dy implanted in Cu single crystals obtained by Mössbauer spectroscopy

The field dependence of the hyperfine interaction of¹⁶¹Dy impurities in Cu has been studied in external magnetic fields up to 3.21 T by means of Mössbauer spectroscopy. ¹⁶¹Dy was introduced into a single crystal of Cu by means of low temperature ion implantation. From the measurements we determine the parameters of the cubic crystalline electric field acting on the Dy nuclei:A ₄<r⁴>=–28±58 K and . The ground state is a doublet, separated well from a excited quartet by .