Abstract: Preface 1. Prologue 2. Introduction to elementary quantum mechanics and stability of the first kind 3. Many-particle systems and stability of the second kind 4. Lieb-Thirring and related inequalities 5. Electrostatic inequalities 6. An estimation of the indirect part of the Coulomb energy 7. Stability of non-relativistic matter 8. Stability of relativistic matter 9. Magnetic fields and the Pauli operator 10. The Dirac operator and the Brown-Ravenhall model 11. Quantized electromagnetic fields and stability of matter 12. The ionization problem, and the dependence of the energy on N and M separately 13. Gravitational stability of white dwarfs and neutron stars 14. The thermodynamic limit for Coulomb systems References Index.

Abstract: Two routes for deriving the exact two-component Hamiltonians are compared. In the first case, as already known, we start directly from the matrix representation of the Dirac operator in a restricted kinetically balanced basis and make a single block diagonalization. In the second case, not considered before, we start instead from the Foldy–Wouthuysen operator and make proper use of resolutions of the identity. The expressions are surprisingly different. It turns out that a mistake was made in the former formulation when going from the Dirac to the Schrodinger picture. The two formulations become equivalent after the mistake is corrected.

Abstract: We provide a framework for calculating holographic Green's functions from general bilinear actions and fields obeying coupled differential equations in the bulk. The matrix-valued spectral function is shown to be independent of the radial bulk coordinate. Applying this framework we improve the analysis of fluctuations in the D3/D7 system at finite baryon density, where the longitudinal perturbations of the world-volume gauge field couple to the scalar fluctuations of the brane embedding. We compute the spectral function and show how its properties are related to the quasinormal mode spectrum. We study the crossover from the hydrodynamic diffusive to the reactive regime and the movement of quasinormal modes as functions of temperature and density. We also compute their dispersion relations and find that they asymptote to the lightcone for large momenta.

Abstract: The flat, homogeneous, and isotropic universe with a massless scalar field is a paradigmatic model in loop quantum cosmology. In spite of the prominent role that the model has played in the development of this branch of physics, there still remain some aspects of its quantization which deserve a more detailed discussion. These aspects include the kinematical resolution of the cosmological singularity, the precise relation between the solutions of the densitized and nondensitized versions of the quantum Hamiltonian constraint, the possibility of identifying superselection sectors which are as simple as possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit associated with the theory in those sectors. We propose an alternative operator to represent the Hamiltonian constraint which is specially suitable to deal with all these issues in a detailed and satisfactory way. In particular, with our constraint operator, the singularity decouples in the kinematical Hilbert space and can be removed already at this level. Thanks to this fact, we can densitize the quantum Hamiltonian constraint in a well-controlled manner. Additionally, together with the physical observables, this constraint superselects simple sectors for the universe volume, with a discrete support contained in a single semiaxis of the real line and for which the basic functions that encode the information about the geometry possess optimal physical properties. Namely, they provide a no-boundary description around the cosmological singularity and admit a well-defined WDW limit in terms of standing waves. Both properties explain the presence of a generic quantum bounce replacing the classical singularity at a fundamental level, in contrast with previous studies where the bounce was proved in concrete regimes---focusing on states with a marked semiclassical behavior---or for a simplified model.

Abstract: An exact correspondence is pointed out between conformal field theories in D-dimensions and dual resonance models in Ddimensions, where Dmay differ from D Dual resonance models, pioneered by Veneziano, were forerunners of string theory The analogs of scattering amplitudes in dual reso- nance models are called Mellin amplitudes; they depend on complex variables sij which substitute for the Mandelstam variables on which scattering ampli- tudes depend The Mellin amplitudes satisfy exact duality - ie meromorphy in sij with simple poles in single variables, and crossing symmetry - and an ap- propriate form of factorization which is implied by operator product expansions (OPE) Duality is a D-independent property The position of the leading poles in s12 is given by the dimensions of fields in the OPE, but there are also satel- lites and the precise correspondence between fields in the OPE and the residues of these poles depends on D Dimensional reduction and dimensional induction DD ∓ 1 are discussed Dimensional reduction leads to the appearance of Anti de Sitter space

Abstract: We study the supersymmetry enhancement of ABJM theory Starting from a = 2 supersymmetric Chern-Simons matter theory with gauge group U(2) × U(2) which is a truncated version of the ABJM theory, we find by using the monopole operator that there is additional = 2 supersymmetry related to the gauge group We show this additional supersymmetry can combine with = 6 supersymmetry of the original ABJM theory to an enhanced = 8 SUSY with gauge group U(2) × U(2) in the case k = 1,2 We also discuss the supersymmetry enhancement of the ABJM theory with U(N) × U(N) gauge group and find a condition which should be satisfied by the monopole operator

Abstract: An exact correspondence is pointed out between conformal field theories in D dimensions and dual resonance models in D' dimensions, where D' may differ from D. Dual resonance models, pioneered by Veneziano, were forerunners of string theory. The analog of scattering amplitudes are called Mellin amplitudes; they depend on complex variables which substitute for the Mandelstam variables on which scattering amplitudes depend. The Mellin amplitudes satisfy exact duality - i.e. meromorphy with simple poles in single variables, and crossing symmetry - and an appropriate form of factorization which is implied by operator product expansions (OPE). Duality is a D-independent property. The positions of the leading poles are given by the dimensions of fields in the OPE; their residues depend on D and determine satellites. Dimensional reduction and induction D goes to D-1 and D+1 are discussed. Dimensional reduction leads to the appearence of Anti de Sitter space.

Abstract: We study the supersymmetry enhancement of ABJM theory. Starting from a ${\cal N}=2$ supersymmetric Chern-Simons matter theory with gauge group U(2)$\times$U(2) which is a truncated version of the ABJM theory, we find by using the monopole operator that there is additional ${\cal N}=2$ supersymmetry related to the gauge group. We show this additional supersymmetry can combine with ${\cal N}=6$ supersymmetry of the original ABJM theory to an enhanced ${\cal N}=8$ SUSY with gauge group U(2)$\times$U(2) in the case $k=1,2$. We also discuss the supersymmetry enhancement of the ABJM theory with U($N$)$\times$U($N$) gauge group and find a condition which should be satisfied by the monopole operator.

Abstract: This paper deals with the generalized principal eigenvalue of the parabolic operator $${\mathcal{L}\phi = \partial_{t}\phi - abla \cdot(A(t, x) abla\phi) + q(t, x) \cdot abla\phi - \mu(t, x)\phi}$$ , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.

Abstract: Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economics, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. A new application of multi-linear models to economics and social dynamics is discussed.

Abstract: We consider leading strong coupling corrections to the energy of the lightest massive string modes in AdS_5 x S^5, which should be dual to members of the Konishi operator multiplet in N=4 SYM theory. This determines the general structure of the strong-coupling expansion of the anomalous dimension of the Konishi operator. We use 1-loop results for several semiclassical string states to extract information about the leading coefficients in this expansion. Our prediction is Delta= 2 lambda^{1/4} + b_0 + b_1 lambda^{-1/4} + b_3 lambda^{-3/4} +..., where b_0 and b_1 are rational while b_3 is transcendental containing zeta(3). Explicitly, we argue that b_0= Delta_0 - 4 (where Delta_0 is the canonical dimension of the corresponding gauge-theory operator in the Konishi multiplet) and b_1=1. Our conclusions are sensitive to few assumptions, implied by a correspondence with flat-space expressions, on how to translate semiclassical quantization results into predictions for the exact quantum string spectrum.

Abstract: In this paper, we consider the problem of computing resonances in open systems. We first characterize resonances in terms of (improper) eigen-functions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in the coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large. The analysis of the second step is classical. Finally, we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.

Abstract: We derive a quantization formula of Bohr-Sommerfeld type for computing quasinormal frequencies for scalar perturbations in an AdS black hole in the limit of large scalar mass or spatial momentum. We then apply the formula to find poles in retarded Green functions of boundary CFTs on IR and IR× S. We find that when the boundary theory is perturbed by an operator of dimension ∆ ≫ 1, the relaxation time back to equilibrium is given at zero momentum by 1 ∆πT ≪ 1 πT . Turning on a large spatial momentum can significantly increase it. For a generic scalar operator in a CFT on IR, there exists a sequence of poles near the lightcone whose imaginary part scales with momentum as p d−2 d+2 in the large momentum limit. For a CFT on a sphere S we show that the theory possesses a large number of long-lived quasiparticles whose imaginary part is exponentially small in momentum.

Abstract: The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N to infinity limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N to infinity limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value K_c. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K>K_c.

Abstract: We consider probabilistic theories in which the most elementary system, a two-dimensional system, contains one bit of information. The bit is assumed to be contained in any complete set of mutually complementary measurements. The requirement of invariance of the information under a continuous change of the set of mutually complementary measurements uniquely singles out a measure of information, which is quadratic in probabilities. The assumption which gives the same scaling of the number of degrees of freedom with the dimension as in quantum theory follows essentially from the assumption that all physical states of a higher dimensional system are those and only those from which one can post-select physical states of two-dimensional systems. The requirement that no more than one bit of information (as quantified by the quadratic measure) is contained in all possible post-selected two-dimensional systems is equivalent to the positivity of density operator in quantum theory.

Abstract: The numerical properties of a deterministic Boltzmann equation solver based on a spherical harmonics expansion of the distribution function are analyzed and improved. A fully coupled discretization scheme of the Boltzmann and Poisson equations is proposed, where stable equations are obtained based on the H-transformation. It is explicitly shown that the resultant Jacobian matrix for the zeroth order component has property M for a first order expansion, which improves the stability even of higher order expansions. The detailed dependence of the free-streaming operator and the scattering operator on the electrostatic potential is exactly considered in the Newton-Raphson scheme. Therefore, convergence enhancement is achieved compared with previous Gummel-type approaches. This scheme is readily applicable to small-signal and noise analysis. As numerical examples, simulation results are shown for a silicon n+nn+ structure including a magnetic field, an SOI NMOSFET and a SiGe HBT.

Abstract: The paper is devoted to integro-differential equations arising in population dynamics The integral term describes the nonlocal consumption of resources We study the Fredholm property of the corresponding linear operators and use it to prove the existence of travelling waves when the support of the integral is sufficiently small In this case, the integro-differential operator is close to the differential operator and we can use the implicit function theorem We carry out numerical simulations in order to study the case where the support of the integral is not small We observe various regimes of wave propagation Some of them, in particular periodic waves do not exist for the usual reaction-diffusion equation

Abstract: We investigate the relation between non-unitarity of the leptonic mixing matrix and leptogenesis. We discuss how all parameters of the canonical type-I seesaw mechanism can, in principle, be reconstructed from the neutrino mass matrix and the deviation of the effective low-energy leptonic mixing matrix from unitary. When the mass M' of the lightest right-handed neutrino is much lighter than the masses of the others, we show that its decay asymmetries within flavour-dependent leptogenesis can be expressed in terms of two contributions, one depending on the unique dimension five (d=5) operator generating neutrino masses and one depending on the dimension six (d=6) operator associated with non-unitarity. In low-energy seesaw scenarios where small lepton number violation explains the smallness of neutrino masses, the lepton number conserving d=6 operator contribution generically dominates over the d=5 operator contribution which results in a strong enhancement of the flavour-dependent decay asymmetries without any resonance effects. To calculate the produced final baryon asymmetry, the flavour equilibration effects directly related to non-unitarity have to be taken into account. In a simple realization of this non-unitarity driven leptogenesis, the lower bound on M' is found to be about 10^8 GeV at the onset of the strong washout regime, more than one order of magnitude below the bound in "standard" thermal leptogenesis.

Abstract: We give conditions for the parabolic evolution operator to be an- alytic with respect to a coecient operator. We also show that the solution of a homogeneous parabolic evolution equation is analytic with respect to the coecient operator and to the initial data. We apply our results to example that can not be studied by the standard methods.

Abstract: Squared eigenfunctions are quadratic combinations of Jost functions and adjoint Jost functions which satisfy the linearized equation of an integrable equation. In this article, squared eigenfunctions are derived for the Sasa-Satsuma equation whose spectral operator is a $3\times 3$ system, while its linearized operator is a $2\times 2$ system. It is shown that these squared eigenfunctions are sums of two terms, where each term is a product of a Jost function and an adjoint Jost function. The procedure of this derivation consists of two steps: first is to calculate the variations of the potentials via variations of the scattering data by the Riemann-Hilbert method. The second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations. While this procedure has been used before on other integrable equations, it is shown here, for the first time, that for a general integrable equation, the functions appearing in these variation relations are precisely the squared eigenfunctions and adjoint squared eigenfunctions satisfying respectively the linearized equation and the adjoint linearized equation of the integrable system. This proof clarifies this procedure and provides a unified explanation for previous results of squared eigenfunctions on individual integrable equations. This procedure uses primarily the spectral operator of the Lax pair. Thus two equations in the same integrable hierarchy will share the same squared eigenfunctions (except for a time-dependent factor). In the Appendix, the squared eigenfunctions are presented for the Manakov equations whose spectral operator is closely related to that of the Sasa-Satsuma equation.

Abstract: We analyze the extension of the well known relation between Brownian motion and the Schrodinger equation to the family of the Levy processes. We consider a Levy–Schrodinger equation where the usual kinetic energy operator–the Laplacian–is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy–Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Levy process is stable we recover as a particular case the fractional Schrodinger equation. A few examples are finally given and we find that there are physically relevant models–such as a form of the relativistic Schrodinger equation–that are in the domain of the non stable Levy–Schrodinger equations.

Abstract: We consider the Schrodinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.

Abstract: We state and prove some new weighted Hardy type inequalities with an integral operator A(k) defined by A(k)f(x) := 1/K(x) integral(Omega 2) k(x,y)f(y)d mu(2) (y), where k : Omega(1) x Omega(2) --> ...

Abstract: The paper studies a natural $n$-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto zero value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.

Abstract: By revisiting previous definitions, we show that one can define an energy current operator that satisfies the continuity equation for a general Hamiltonian in one dimension. This expression is useful for studying electronic, phononic and photonic energy flow in linear systems and in hybrid structures. The definition allows us to deduce the necessary conditions that result in current conservation for general-statistics systems. The discrete form of the Fourier's law of heat conduction naturally emerges in the present definition.