Abstract: Motivated by recent proposals in hep-th/0202021 and hep-th/0204051 we develop semiclassical quantization of superstring in $AdS_5 x S^5$. We start with a classical solution describing string rotating in $AdS_5$ and boosted along large circle of $S^5$. The energy of the classical solution $E$ is a function of the spin $S$ and the momentum $J$ (R-charge) which interpolates between the limiting cases S=0 and J=0 considered previously. We derive the corresponding quadratic fluctuation action for bosonic and fermionic fields from the GS string action and compute the string 1-loop (large $\lambda= {R^4\over \a'^2}$) correction to the classical energy spectrum in the $(S,J)$ sector. We find that the 1-loop correction to the ground-state energy does not cancel for non-zero $S$. For large $S$ it scales as $\ln S$, i.e. as the classical term, with no higher powers of $\ln S$ appearing. This supports the conjecture made in hep-th/0204051 that the classical $E-S = a \ln S$ scaling can be interpolated to weak coupling to reproduce the corresponding operator anomalous dimension behaviour in gauge theory.

Abstract: The discrete-time recursion system $\u_{n+1}=Q[\u_n]$ with $\u_n(x)$ a vector of population distributions of species and $Q$ an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on $Q$ are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

Abstract: We present density-functional theory for linear and nonlinear response functions using an explicit exponential parametrization of the density operator. The response functions are derived using two alternative variation principles, namely, the Ehrenfest principle and the quasienergy principle, giving different but numerically equivalent formulas. We present, for the first time, calculations of dynamical hyperpolarizabilities for hybrid functionals including exchange-correlation functionals at the general gradient-approximation level and fractional exact Hartree–Fock exchange. Sample calculations are presented of the first hyperpolarizability of the para-nitroaniline molecule and of a porphyrin derived push–pull molecule, showing good agreement with available experimental data.

Abstract: We consider the scattering of time-harmonic plane waves by an inhomogeneous medium. The far field patterns u? of the scattered waves depend on the index of refraction 1 + q, the frequency, and directions and of observation and incidence, respectively. The inverse problem which is studied in this paper is to determine the support ? of q from the knowledge of u? (, ) for all , where the frequency is fixed (and known). Our new approach is based on the far field operator F which is the integral operator with kernel u? (, ). It depends on the data only and is therefore known (at least approximately). The MUSIC algorithm in signal processing uses the discrete version of F, i.e. the matrix F = (u? ( i, j)) N?N, and determines the locations of the point scatterers. The key idea in both cases is to factorize F and F in the forms where the operator S and the matrix S are 'more explicit' than F and F, respectively, and T, T are suitable isomorphisms. In a first theoretical result we show that the ranges of S and F# coincide, where F# is some suitable combination of the real and imaginary parts of F. In the finite dimensional case a simple argument from matrix theory yields that the ranges of S and F coincide. Since F# is known from the data we can decide for every function on the unit sphere whether it belongs to the range of S or not. We apply this test to the far field patterns of point sources and arrive at an explicit test whether a point z belongs to ? or not. We will demonstrate that this method also leads to a fast visualization of the obstacle.

Abstract: When solving non-stationary problems of mathematical physics, a particular attention is paid to schemes with weighted factors. Assume that we solve the Cauchy problem for the first-order evolution equation \(\frac{{du}}{{dt}} + Au = f(t),0 < t < T,u(0) = {u_0}\) where f (t),u 0 are given, whilst u(t) is the unknown function with values in a finite-dimensional Hilbert space H.

Abstract: We show how allowing non-local terms in the field equations of symmetric tensors uncovers a neat geometry that naturally generalizes the Maxwell and Einstein cases. The end results can be related to multiple traces of the generalized Riemann curvatures R_{alpha_1 ... alpha_s; beta_1 > ... beta_s} introduced by de Wit and Freedman, divided by suitable powers of the D'Alembertian operator \Box. The conventional local equations can be recovered by a partial gauge fixing involving the trace of the gauge parameters Lambda_{alpha_1 ... alpha_{s-1}}, absent in the Fronsdal formulation. The same geometry underlies the fermionic equations, that, for all spins s+(1/2), can be linked via the operator (not hskip 1pt pr)/(\Box) to those of the spin-s bosons.

Abstract: This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

Abstract: In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.

Abstract: As an extension of the intertwining operator idea, an algebraic method which provides a link between supersymmetric quantum mechanics and quantum (super)integrability is introduced. By realization of the method in two dimensions, two infinite families of superintegrable and isospectral stationary potentials are generated. The method makes it possible to perform Darboux transformations in such a way that, in addition to the isospectral property, they acquire the superintegrability preserving property. Symmetry generators are second and fourth order in derivatives and all potentials are isospectral with one of the Smorodinsky–Winternitz potentials. Explicit expressions of the potentials, their dynamical symmetry generators, and the algebra they obey as well as their degenerate spectra and corresponding normalizable states are presented.

Abstract: Using the data of eigenvalues and eigenvectors of Neumann matrices in the 3-string vertex, we prove analytically that the ghost kinetic operator of vacuum string field theory obtained by Hata and Kawano is equal to the ghost operator inserted at the open string midpoint. We also comment on the values of determinants appearing in the norm of sliver state.

Abstract: By taking into account that the equations of motion of classical mechanics can be expressed in terms of differential operators in phase space, we develop a simple method for obtaining exact solutions for several models in one and more dimensions. We also propose a simple procedure for the systematic construction of exactly solvable models in one dimension.

Abstract: We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrodinger equation with harmonic potential.

Abstract: We present an iterative path integral algorithm for computing multitime correlation functions of a quantum system coupled to a dissipative bath of harmonic oscillators. By splitting the Boltzmann operator into two parts and reordering the propagators in the expression for canonical correlation functions, we are able to transform the evolution time contour into a symmetric one so that a forward propagation and a backward one are specified. Because the memory induced by the bath through the Feynman–Vernon influence functional decays rapidly in the complex time plane, long-time correlations are negligible. Taking advantage of this fact, we show that the correlation function can be obtained via an iterative procedure. The method is used to calculate three-time correlation functions of a dissipative two-level system.

Abstract: In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian $H={1\over2}p^2+{1\over2}x^2 +i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H={1\over2}p^2+{1\over2}x^2-\epsilon x^4$ using nonperturbative methods.

Abstract: We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions. We show that the operator effecting the change from closed to open, or from open to closed, is a boundary primary field of weight -1/8, belonging to a c=-2 logarithmic conformal field theory.

Abstract: We perform a finite energy sum rule analysis of the flavor ud two-point V-A current correlator, Delta Pi (Q^2). The analysis, which is performed using both the ALEPH and OPAL databases for the V-A spectral function, Delta rho, allows us to extract the dimension six V-A OPE coefficient, a_6, which is related to the matrix element of the electroweak penguin operator, Q_8, by chiral symmetry. The result for a_6 leads directly to the improved (chiral limit) determination epsilon'/epsilon = (- 15.0 +- 2.7) 10^{-4}. Determination of higher dimension OPE contributions also allows us to perform an independent test using a low-scale constrained dispersive analysis, which provides a highly nontrivial consistency check of the results.

Abstract: We demonstrate that the Q matrix introduced in Baxter's 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for Q(v) which both contains the Bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues.

Abstract: A semiclassical methodology for evaluating the Boltzmann operator entering semiclassical approximations for finite temperature correlation functions is described. Specifically, Miller’s imaginary time semiclassical approach is applied to the Herman–Kluk coherent state initial value representation (IVR) for the time evolution operator in order to obtain a coherent state IVR for the Boltzmann operator. The phase-space representation gives rise to exponentially decaying factors for the coordinates and momenta of the real time trajectories employed in the dynamical part of the calculation. A Monte Carlo procedure is developed for evaluating dynamical observables, in which the absolute value of the entire exponential part of the integrand serves as the sampling function. Numerical tests presented show that the methodology is accurate as well as stable over the temperature range relevant to chemical applications.

Abstract: We investigate a reformulation of the dynamics of interacting fermion systems in terms of a stochastic extension of time-dependent Hartree-Fock equations. From a path-integral representation of the evolution operator, we show that the exact N-body state can be interpreted as a coherent average over Slater determinants evolving in a random mean-field. The imaginary time propagation is also presented and gives a similar scheme which converges to the exact ground state. In addition, the growth of statistical errors is examined to show the stability of this stochastic formulation.

Abstract: In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator representations used previously. We derive generalized T-Q relations in which 3 of these parameters shift. After a suitable restriction of parameters, we give an explicit expression for the Q-operator of the 6-vertex model and show the connection with Baxter's expression for the central block of his corresponding operator.

Abstract: We first summarize some well-known, however instructive facts from the theory of autonomous abstract Cauchy problems for a closed operator (A,D(A)) on some Banach space X (compare [5], Chapter II.6).

Abstract: Recently, it was shown that the time-reversal operator for a single, small spherical scatterer could have up to four distinguishable eigenstates [Chambers and Gautesen, J. Acoust. Soc. Am. 109, 2616–2624 (2001)]. In this paper, that analysis is generalized for scatterers of arbitrary shape and larger size. It is shown that the time-reversal operator may have an indefinitely large number of distinguishable eigenstates, with the exact number depending on the nature of the scatterer and the geometry of the time-reversal mirror. In addition, the case of a multiple number of well-separated scatterers is investigated, with the result that the total spectrum is the direct combination of the eigenstates associated with each scatterer. As an example, the singular value spectrum of the time-reversal operator for a linear array is calculated explicitly for bubbles and hard rubber spheres of finite size. Both resonance peaks and apparent crossing points can be observed in the spectrum as the size of the scatterer increases.

Abstract: We refer to Krupka's variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator.

Abstract: When A E B(X) and B E B(Y) are given we denote by M C an operator acting on the Banach space X ○+ Y of the form M C = ( A 0 C B), where C ∈ B(Y,X). In this note we examine the relation of Weyl's theorem for A ○+ B and M C through local spectral theory.

Abstract: We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (\lambda-A)^{-1} exists in a sector of the complex plane and decays like 1/|\lambda| as |\lambda| tends to infinity. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem \dot{u}-\Delta u=f, u(0)=0.