Abstract: Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ``basic'' solution corresponding to the identity operator provides a ``natural'' quantization of the Lobachevskiy plane.

Abstract: We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrodinger operator. We show that this problem is severely ill-posed. The results extend to electrical impedance tomography. They show that the logarithmic stability results of Alessandrini are optimal (up to the value of the exponent).

Abstract: The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitian conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

Abstract: We re-derive the renormalization group equation for the effective coupling of the dimension five operator which corresponds to a Majorana mass matrix for the Standard Model neutrinos. We find a result which differs somewhat from earlier calculations, leading to modifications in the evolution of leptonic mixing angles and CP phases. We also present a general method for calculating beta-functions from counterterms in MS-like renormalization schemes, which works for tensorial quantities.

Abstract: An air bag device for knee protection (M1), wherein an air bag (26) covers the substantial lower surface (9a) of a column cover (9) projected to an operator side to minimize the volume thereof without deteriorating the protection of the knee of the operator so as to reduce a time required for the air bag to complete the development and inflation, and the shape of the air bag after completing the inflation is formed generally in a plate shape capable of covering at least the lower surface of the column cover, whereby, even if the knee of the operator is close to the column cover in the state of the operator depressing a brake pedal, the developed and inflated air bag can be disposed smoothly in a small space between the knee of the operator and the lower surface of the column cover without being interferred with the knee of the operator, and thus the knee of the operator can be protected accurately by rapidly developing and inflating the air bag in the small space between the column cover and the knee of the operator.

Abstract: We study perturbative and non-perturbative properties of the Konishi multiplet in = 4 SYM theory in D = 4 dimensions. We compute two-, three- and four-point Green functions with single and multiple insertions of the lowest component of the multiplet, 1, and of the lowest component of the supercurrent multiplet, 20'. These computations require a proper definition of the renormalized operator, 1, and lead to an independent derivation of its anomalous dimension. The O(g2) value found in this way is in agreement with previous results. We also find that instanton contributions to the above correlators vanish. From our results we are able to identify some of the lowest dimensional gauge-invariant composite operators contributing to the OPE of the correlation functions we have computed. We thus confirm the existence of an operator belonging to the representation 20', which has vanishing anomalous dimension at order g2 and g4 in perturbation theory as well as at the non-perturbative level, despite the fact that it does not obey any of the known shortening conditions.

Abstract: We study perturbative and non-perturbative properties of the Konishi multiplet in N=4 SYM theory in D=4 dimensions. We compute two-, three- and four-point Green functions with single and multiple insertions of the lowest component of the multiplet, and of the lowest component of the supercurrent multiplet. These computations require a proper definition of the renormalized operator and lead to an independent derivation of its anomalous dimension. The O(g^2) value found in this way is in agreement with previous results. We also find that instanton contributions to the above correlators vanish. From our results we are able to identify some of the lowest dimensional gauge-invariant composite operators contributing to the OPE of the correlation functions we have computed. We thus confirm the existence of an operator belonging to the representation 20', which has vanishing anomalous dimension at order g^2 and g^4 in perturbation theory as well as at the non-perturbative level, despite the fact that it does not obey any of the known shortening conditions.

Abstract: We construct a classical solution of vacuum string field theory (VSFT) and study whether it represents the perturbative open string vacuum. Our solution is given as a squeezed state in the Siegel gauge, and it fixes the arbitrary coefficients in the BRST operator in VSFT. We identify the tachyon and massless vector states as fluctuation modes around the classical solution. The tachyon mass squared α'mt2 is given in a closed form using the Neumann coefficients defining the three-string vertex, and it reproduces numerically the expected value of -1 to high precision. The ratio of the potential height of the solution to the D25-brane tension is also given in terms of the Neumann coefficients. However, the behavior of the potential height in level truncation does not match our expectation, though there are subtle points in the analysis.

Abstract: Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ``basic'' solution corresponding to the identity operator provides a ``natural'' quantization of the Lobachevskiy plane.

Abstract: Space of states of PT symmetrical quantum mechanics is examined. Requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to the space with an indefinite metric. The self consistent expressions for the probability amplitude and average value of operator are suggested. Further specification of space of state vectors yield the superselection rule, redefining notion of the superposition principle. The expression for the probability current density, satisfying equation of continuity and vanishing for the bound state, is proposed.

Abstract: Chemists regularly exploit point group symmetry in their analyses of molecular structure and properties but they rarely utilize time reversal symmetry. The time reversal operator T reverses the momenta and spins of all particles in a system and distinguishes properties which are even under T, such as the electric dipole moment, from those that are odd, such as the magnetic dipole moment. We review the role of T in quantum mechanics and discuss its application to the properties of molecules in electric and magnetic fields. Among the properties considered are natural and magnetic optical activity, magneto-chiral effects, antisymmetric Raman scattering, optical NMR and ESR, chirality, and absolute enantioselection.

Abstract: We point out that the leading infrared singular terms in the effective actions of noncommutative gauge theories arising from nonplanar loop diagrams have a natural interpretation in terms of the matrix model (operator) formulation of these theories. In this formulation (for maximal spatial noncommutativity), noncommutative space arises as a configuration of an infinite number of D-particles. We show that the IR singular terms correspond to instantaneous linear potentials between these D-particles resulting from the zero point energies of fluctuations about this background. For theories with fewer fermionic than bosonic degrees of freedom, such as pure noncommutative gauge theory, the potential is attractive and renders the theory unstable. With more fermionic than bosonic degrees of freedom, the potential is repulsive and we argue that the theory is stable, though oddly behaved.

Abstract: We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equations that gives $O(1/t)$ decay of solutions of the inhomomogeneous linear wave equation based in terms of $O(1/|x|)$ estimates for the forcing term.

Abstract: The three-time correlation function that describes resonance Raman (RR) spectra is computed directly using the Herman–Kluk semiclassical propagator. The trace expression for this correlation function {C(t1,t2,t3)=Tr[e−βĤe−iĤg(t1+t2)e−iĤet3e+iĤg(t2+t3)e+iĤet1]} allows forward–backward time propagation of trajectories over closed time-circuits, leading to efficient convergence in multidimensional systems. A local harmonic approximation is used to derive an expression for the density operator in the coherent state representation (〈p1 q1|e−βĤ|p2 q2〉). This allows efficient sampling of phase space as well as simulations at arbitrary temperatures and in arbitrary coordinates. The resulting method is first analyzed for a one-dimensional problem, where the results are shown to be in excellent agreement with exact quantum calculations. The method is then applied to the problem of RR scattering of iodine in the condensed phase. The RR spectrum of an I2 molecule in a xenon fluid at 230 K is calculated and also found...

Abstract: We show that the time reversal operator for a planar time reversal mirror (TRM) can have up to four distinct eigenvalues with a small spherical acoustic scatterer. Each eigenstate represents a resonance between the TRM and an induced scattering moment of the sphere. Their amplitude distributions on the TRM are orthogonal superpositions of the radiation patterns from a monopole and up to three orthogonal dipoles. The induced monopole moment is associated with the compressibility contrast between the sphere and the medium, while the dipole moments are associated with density contrast. The number of eigenstates is related to the number of orthogonal orientations of each induced multipole. For hard spheres (glass, metals) the contribution of the monopole moment to the eigenvalues is much greater than that of the dipole moments, leading to a single dominant eigenvalue. The other eigenvalues are much smaller, making it unlikely multiple eigenvalues could have been observed in previous experiments using hard materials. However, for soft materials such as wood, plastic, or air bubbles the eigenvalues are comparable in magnitude and should be observable. The presence of multiple eigenstates breaks the one-to-one correspondence between eigenstates and distinguishable scatterers discussed previously by Prada and Fink [Wave Motion 20, 151–163 (1994)]. However, eigenfunctions from separate scatterers would have different phases for their eigenfunctions, potentially restoring the ability to distinguish separate scatterers. Since relative magnitudes of the eigenvalues for a single scatterer are governed by the ratio of the compressibility contrast to the density contrast, measurement of the eigenvalue spectrum would provide information on the composition of the scatterer.

Abstract: We present a cohomological method for obtaining the non-Abelian Seiberg-Witten map for any gauge group and to any order in theta. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator.

Abstract: Binary symmetry constraints of the N-wave interaction equations in 1+1 and 2+1 dimensions are proposed to reduce the N-wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of the constrained flows of the N-wave interaction equations. The constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the N-wave interaction equations, and thus the integrability by quadratures are shown for the N-wave interaction equations by the constrained flows.

Abstract: The method of intertwining with n-dimensional (nD) linear intertwining operator L is used to construct nD isospectral, stationary potentials. It has been proven that differential part of L is a series in Euclidean algebra generators. Integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n=2,3,4,5. The most general forms of 2D and 3D isospectral potentials are considered in detail and construction of their hierarchies is exhibited. The followed approach provides coordinate systems which make it possible to perform separation of variables and to apply the known methods of supersymmetric quantum mechanics for 1D systems. It has been shown that in choice of coordinates and L there are a number of alternatives increasing with $n$ that enlarge the set of available potentials. Some salient features of higher dimensional extension as well as some applications of the results are presented.

Abstract: The operator realizing a Dehn twist in quantum Teichmuller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c>1. The completeness condition of the eigenvectors includes the integration measure which appeared in the representation theoretic approach to quantum Liouville theory by Ponsot and Teschner. The underlying quantum group structure is also revealed.

Abstract: In this article, we propose to use van Vleck’s contact transformation method for the study of time-dependent interactions in solid state nuclear magnetic resonance (NMR) by Floquet theory [A. Schmidt and S. Vega, J. Chem. Phys. 96, 2655 (1992)]. Floquet theory has been used for studying the spin dynamics of magic angle spinning (MAS) NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. Here the above method is combined with Floquet theory to study the dynamics of a dipolar coupled spin (I=1/2) system. In order to determine the frequencies and intensities of bands in MAS spectra, we need to diagonalize the Floquet–Hamiltonian matrix. This is generally done numerically, by truncating the infinite dimensional Floquet–Hamiltonian matrix. Here we propose an effective Floquet–Hamiltonian for the system. We demonstrate the application of the above method to a homonuclear dipolar coupled spin system (I=1/2). The eigenvalues obtained from the above are compared with those obtained using numerical diagonalization. The eigenvalues of the effective Hamiltonian compare quite well with those computed using numerical diagonalization of Floquet matrices.

Abstract: The paper studies the spectral properties of the Schrodinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation $gV$ gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed sign. Assuming that the latter condition is satisfied and that $V$ is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$, this asymptotics is either of the Weyl type or is completely determined by the behaviour of $V$ at infinity.

Abstract: We analyze the time evolution of a one-dimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t→∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very non-generic η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.

Abstract: The forward−backward semiclassical treatment of ensemble averaged quantities is combined with a discretized path integral description of the Boltzmann operator describing the initial density. We present a practical Monte Carlo methodology for calculating time-dependent expectation values and time correlation functions, applicable to polyatomic systems.

Abstract: The rise in linear entropy of a subsystem in the N-atom Jaynes-Cummings model is shown to be strongly influenced by the shape of the classical orbits of the underlying classical phase space: we find a one-to-one correspondence between maxima ~minima! of the linear entropy and maxima ~minima! of the expectation value of atomic excitation Jz . Since the expectation value of this operator can be viewed as related to the orbit radius in the classical phase-space projection associated with the atomic degree of freedom, the proximity of the quantum wave packet to this atomic phase-space borderline produces a maximum rate of entanglement. The consequence of this fact for initial conditions centered at periodic orbits in regular regions is a clear periodic recoherence. For chaotic situations the same phenomenon ~proximity of the atomic phase-space borderline! is, in general, responsible for oscillations in the entanglement properties. The importance of studying in detail the decoherence pro- cess is twofold. First, it may be viewed as a key to the un- derstanding of some of the striking differences between the quantum and classical descriptions of the world such as ''the nonexistence at the classical level of the majority of states allowed by quantum mechanics'' @1#. The decoherence pro- cess is believed to be the agent that eliminates interference between two or more macroscopically separated localized states @2#. Second, given the impressive technological ad- vances in several experimental areas ~quantum optics, con- densed matter, atomic physics, etc.!, it is nowadays possible to realize a system of two interacting degrees of freedom and watch the time evolution of the corresponding entanglement process @3#. It is therefore also of importance to understand the entanglement process in simple Hamiltonian systems. Hamiltonian systems with two degrees of freedom often present a very rich dynamics, which in many cases is not yet completely understood from a general point of view. In par- ticular, if the interaction is nonlinear the system may present chaotic behavior in the classical limit. The consequences of this fact to the quantum dynamics is yet an unsettled issue. A step in this direction was taken a few years ago, as it was conjectured that ''the rate of entropy production can be used as an intrinsically quantum test of the chaotic versus regular nature of the evolution' ' @4#. The idea has been tested in some models @5,6#. More specifically, in the context of the N-atom Jaynes-Cummings model, the reduced-density linear entropy ~or idempotency defect! has been used as a measure of the entanglement of the quantum subsystems. For a given classical energy, initial conditions for the quantum states are prepared as coherent wave packets centered at regular and chaotic regions of the classical phase space. For short times, a fast increase in decoherence for chaotic initial conditions is found when compared to regular ones. Typically the linear entropy in this model rises from zero to a plateau. In the present contribution, we show that this rise is strongly influenced by the shape of the atomic projection of classical orbits. There is a clear correlation between the rate of increase of linear entropy and the increase of the expec- tation value of the atomic excitation ^Jz&(t) for short enough times. In classical terms, this can be visualized as follows: ^Jz&(t) is a measure of the instantaneous radius of the pro- jection of the trajectory in the atomic phase space. This atomic phase space is limited, the maximum radius corre- sponds to A4J5A2N, meaning therefore that the maximum rate of entanglement can be directly associated with the proximity of this borderline. If the quantum evolution is such that the initial wave packet is centered at a classically regular region of the atomic phase space, in particular on a ~noncir-