Abstract: Motivated by attempts to extend AdS/CFT duality to non-BPS states we consider classical closed string solutions with several angular momenta in different directions of AdS_5 and S^5. We find a novel solution describing a circular closed string located at a fixed value of AdS_5 radius while rotating simultaneously in two planes in AdS_5 with equal spins S. This solution is a direct generalization of a two-spin flat-space solution where the string rotates in two orthogonal planes while always lying on a 3-sphere. Similar solution exists for a string rotating in S^5: it is parametrized by the angular momentum J of the center of mass and the two equal SO(6) angular momenta J_2=J_3=J' in the two rotation planes. The remarkably simple case is of J=0 where the energy depends on J' as E=[(2J')^2 + {\l}]^{1/2} with {\l}being the string tension or `t Hooft coupling. We discuss interpolation of the E(J') relation to weak coupling by identifying the N=4 SYM theory operator that should be dual to the corresponding semiclassical string state and utilizing existing results for its perturbative anomalous dimension. This opens up a possibility of studying AdS/CFT duality in this new non-BPS sector. We also investigate stability of these classical solutions under small perturbations and comment on several generalizations.

Abstract: A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.

Abstract: A parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parametrization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under local unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parametrization.

Abstract: We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.

Abstract: A quantum mechanical theory for chemical reaction rates is presented which is modeled after the [semiclassical (SC)] instanton approximation. It incorporates the desirable aspects of the instanton picture, which involves only properties of the (SC approximation to the) Boltzmann operator, but corrects its quantitative deficiencies by replacing the SC approximation for the Boltzmann operator by the quantum Boltzmann operator, exp(−βĤ). Since a calculation of the quantum Boltzmann operator is feasible for quite complex molecular systems (by Monte Carlo path integral methods), having an accurate rate theory that involves only the Boltzmann operator could be quite useful. The application of this quantum instanton approximation to several one- and two-dimensional model problems illustrates its potential; e.g., it is able to describe thermal rate constants accurately (∼10–20% error) from high to low temperatures deep in the tunneling regime, and applies equally well to asymmetric and symmetric potentials.

Abstract: The D.O.R.T. method (French acronym for Decomposition of the Time Reversal Operator) is an active remote sensing technique using arrays of antennas for the detection and localization of scatterers [Prada et al., J. Acoust. Soc. Am. 99, 2067–2076 (1996)]. The analogy between the time reversal operator and the covariance matrix used for classical sources separation in passive remote sensing [Bienvenu et al., IEEE Trans. ASSP 31, 1235–1247 (1983)] is established. Then, an experiment of subwavelength detection and localization of point-like scatterers with a linear array of transducers is presented. Using classical estimators in reception like Maximum-Likelihood and Multiple Signal Characterization (MUSIC), two point-like scatterers separated by λ/3 and placed at 100λ from the array of transducers are resolved. In these experiments, the role of multiple scattering and the existence of additional eigenvectors associated with dipolar and monopolar radiation of each scatterer is discussed.

Abstract: We consider a semiclassical multiwrapped circular string pulsating on S_5, whose center of mass has angular momentum J on an S_3 subspace. Using the AdS/CFT correspondence we argue that the one-loop anomalous dimension of the dual operator is a simple rational function of J/L, where J is the R-charge and L is the bare dimension of the operator. We then reproduce this result directly from a super Yang-Mills computation, where we make use of the integrability of the one-loop system to set up an integral equation that we solve. We then verify the results of Frolov and Tseytlin for circular rotating strings with R-charge assignment (J',J',J). In this case we solve for an integral equation found in the O(-1) matrix model when J' J. The latter region starts at J'=L/2 and continues down, but an apparent critical point is reached at J'=4J. We argue that the critical point is just an artifact of the Bethe ansatz and that the conserved charges of the underlying integrable model are analytic for all J' and that the results from the O(-1) model continue onto the results of the O(+1) model.

Abstract: This article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichlet-wave equations in the exterior of a smooth, star shaped obstacle. We can also apply our methods to systems of quasilinear wave equations having different wave speeds. This extends our work [11] for the semilinear case. Previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non-obstacle case. In [9], John and Klainerman proved almost global existence on Minkowski space for quadratic, quasilinear equations using the Lorentz invariance of the wave operator in addition to the symmetries listed above. Subsequently, in [14], Klainerman and Sideris obtained the same result for a class of quadratic, divergence-form nonlinearities without relying on Lorentz invariance. This line of thought was refined and applied to prove global-in-time results for null-form equa tions related to the theory of elasticity in Sideris [22], [23], and for multiple-speed systems of null-form quasilinear equations in Sideris and Tu [24], and Yokoyama [29]. The main difference between our approach and the earlier ones is that we ex ploit the 0(|x|-1) decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0(t~l) decay. Here, of course, x = (x\,X2,x$) is the spatial component, and t the time component, of a space time vector (t, x) G M+ x E3. Establishing 0(|x|_1) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation. A weighted L2 space-time estimate for inhomogeneous wave equations (Proposition 3.1 below, from [11]) is important in making the spatial decay useful for the long-time existence argument. For semilinear systems, one can show almost global existence from small data using only this spatial decay [11]. For quasilinear systems, however, we also have to show that both first and second derivatives of u decay like 1/t. Fortunately, we can do this using a variant of some L1 ?> L?? estimates of John, H?rmander,

Abstract: The class of Sturm-Liouville systems is defined. It appears to be a subclass of Riesz-spectral systems, since it is shown that the negative of a Sturm-Liouville operator is a Riesz-spectral operator on L /sup 2/(a,b) and the infinitesimal generator of a C/sub 0/-semigroup of bounded linear operators.

Abstract: We present density functional response theory generalized to triplet excitations. A method based on an exponential parametrization of the spin-dependent density operator is derived for the evaluation of linear and quadratic response functions for spin-dependent perturbations. The developed methodology is applicable to commonly available functionals, also hybrid functionals including exchange–correlation functionals at the general gradient-approximation level and fractional exact Hartree–Fock exchange. Illustrative calculations are presented for singlet–triplet transition moments and phosphorescence lifetimes, providing numerical data on these quantities for the first time using time-dependent density functional theory.

Abstract: We study the effective interactions induced by loops of extra-dimensional gravitons and show the special role of a specific dimension-6 four-fermion operator, product of two flavour-universal axial currents. By introducing an ultraviolet cut-off, we compare the present constraints on low-scale quantum gravity from various processes involving real-graviton emission and virtual-graviton exchange. The LEP2 limits on dimension-6 four-fermion interactions give one of the strongest constraint on the theory, in particular excluding the case of strongly-interacting gravity at the weak scale.

Abstract: The purpose of the ``bootstrap program'' for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, this program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.

Abstract: In this report we study how a time-varying system with a time-periodic integral kernel (impulse response), g(t,\tau)=g(t+T,\tau+T), can be expanded into a sum of essentially time-invariant systems. This allows us to define a linear frequency response operator for periodic systems, called the Harmonic Transfer Function (HTF). The HTF is a direct analog of the transfer function for time-invariant systems, but it captures the frequency coupling of a time-periodic system. It can, for example, be used to compute the induced L_2-norm of periodic systems. The report also includes analysis of convergence of truncated HTFs, which is essential for practical computations as the HTF is an infinite-dimensional operator.

Abstract: We introduce a Gaussian quantum operator representation, using the most general possible multimode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.

Abstract: This paper focuses on the geometric phase of entangled states of bipartite systems under bilocal unitary evolution. We investigate the relation between the geometric phase of the system and those of the subsystems. It is shown that (1) the geometric phase of cyclic entangled states with nondegenerate eigenvalues can always be decomposed into a sum of weighted nonmodular pure state phases pertaining to the separable components of the Schmidt decomposition, although the same cannot be said in the noncyclic case, and (2) the geometric phase of the mixed state of one subsystem is generally different from that of the entangled state even if the other subsystem is kept fixed, but the two phases are the same when the evolution operator satisfies conditions where each component in the Schmidt decomposition is parallel transported.

Abstract: We study the Schrodinger operator −Δ − αδ(x − Γ) in L2(3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth and admits a global normal parametrization with a uniformly elliptic metric. We show that if Γ is asymptotically planar in a suitable sense and α > 0 is sufficiently large, this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a 'two-dimensional' comparison operator determined by the geometry of the surface Γ.

Abstract: The problem of the Laplacian transfer across an irregular resistive interface (a membrane or an electrode) is investigated with use of the Brownian self-transport operator. This operator describes the transfer probability between two points of a surface, through Brownian motion in the medium neighbouring the surface. This operator governs the flux across a semi-permeable membrane as diffusing particles repetitively hit the surface until they are finally absorbed. In this paper, we first give a theoretical study of the properties of this operator for a planar membrane. It is found that the net effect of a decrease of the surface permeability is to induce a broadening of the region where a particle, first hitting the surface on one point, is finally absorbed. This result constitutes the first analytical justification of the Land Surveyor Approximation, a formerly developed method used to compute the overall impedance of a semi-permeable membrane. In a second step, we study numerically the properties of the Brownian self-transport operator for selected irregular shapes.

Abstract: The stringy picture behind the integrable spin chains governing the evolution equations in Yang-Mills theory is discussed. It is shown that one-loop dilatation operator in N=4 theory can be expressed in terms of two-point functions on 2d worldsheet. Using the relation between Neumann integrable system and the spin chains it is argued that the transition to the finite gauge theory coupling implies the discretization of the worldsheet. We conjecture that string bit model for the discretized worldsheet corresponds to the representation of the integrable spin chains in terms of the separated variables.

Abstract: The spectrum of the damped wave operator for a bounded domain in R 2 is shown to be related to the asymptotic average of the damping function by the geodesic flow. This allows the calculation of an asymptotic expression for the distribution of the imaginary parts of the eigenvalues for a radially symmetric geometry. Numerical simulations confirm the theoretical model. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the rays of geometrical optics, and the geometry of the damping region. The MATLAB code used in this paper is provided.

Abstract: A systematic method is developed to obtain increasingly accurate semiclassical initial value representation (IVR) approximations to the exact quantum propagator. The main result is a series of correction terms of increasing order in a "correction operator", which describes the difference between the exact evolution equation and the equation obeyed by the semiclassical propagator. Each term in the series involves only phase space integrals of classical trajectories and is therefore, in principle, amenable to numerical computation. The properties of the "correction operator" are studied for three different representations of the semiclassical propagator. For initial times, we find that the propagator suggested recently by Baranger et al. is superior to a thawed Gaussian propagator or the Herman-Kluk propagator.

Abstract: We construct atomic spaces S ⊂ L2(R, T 1 1 ) that are appropriate for the representation and processing of discrete tensor field data. We give conditions for these spaces to be well defined, atomic subspaces of the Wiener amalgam space W (C,L2(R, T 1 1 )) which is locally continuous and globally L2. We show that the sampling or discretization operator R from S to l2(Z, T 1 1 ) is a bounded linear operator. We introduce the dilated spaces S∆ = D∆ S parametrized by the coarseness ∆, and show that the discretization operator is also bounded with a bounded inverse for any ∆ ∈ Zn. This allows us to represent discrete tensor field data in terms of continuous tensor fields in S∆, and to obtain continous representations with fast filtering algorithms.

Abstract: In this survey article, we would like to show that the theory of reproducing kernels is fundamental, is beautiful and is applicable widely in mathematics. At the same time, we shall present some operator versions of our fundamental theory in the general theory of reproducing kernels, as original results.

Abstract: SUMMARY We carry out high-frequency analyses of Claerbout's double-square-root equation and its (numerical) solution procedures in heterogeneous media. We show that the double-square-root equation generates the adjoint of the single-scattering modelling operator upon substituting the leading term of the generalized Bremmer series for the background Green function. This adjoint operator yields the process of ‘wave-equation’ imaging. We finally decompose the wave-equation imaging process into common image point gathers in accordance with the characteristic strips in the wavefront set of the data.

Abstract: Using the harmonic analysis associated with the spherical mean operator ℛ, we define and study the Weyl transforms Wσ associated with ℛ where σ is a symbol in Sm, m ∈ ℝ, and we give criteria in terms of σ to obtain the boundedness and compactness of the transform Wσ.

Abstract: A microscopic theory of superconductivity is considered in the framework of the Hubbard p-d model for the CuO2 plane. The Dyson equation is derived in the nonintersecting diagram approximation using the projection technique for the matrix Green function of the Hubbard operator. The solution of the equation for the superconducting gap shows that interband transitions for Hubbard subbands lead to antiferromagnetic exchange pairing as in the t-J model, while intraband transitions additionally lead to spin-fluctuation pairing of the d-wave type. The calculated dependences of the superconducting transition temperature on the hole concentration and of the gap on the wave vector are in qualitative agreement with experiments.