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Given a discrete set $Λ$ in the plane (we will also say a {\em lattice}) and a real number $m\ge 0$, the renormalized energy introduced in \cite{ss1} heuristically describes the interaction energy of unit charges placed at the points of $Λ$ with a uniform negative background of density $m\in\R$. It is defined in several steps, following mostly \cite{ss1}.
A full characterization of $(p,q)$-deformed Fibonacci and Lucas polynomials is given. These polynomials obey non-conventional three-term recursion relations. Their generating functions and Fourier integral transforms are explicitly computed and discussed. Relevant results known in the literature are examined as particular cases.
In this paper, we study the direct and inverse scattering theory at fixed energy for massless charged Dirac fields evolving in the exterior region of a Kerr-Newman-de Sitter black hole. In the first part, we establish the existence and asymptotic completeness of time-dependent wave operators associated to our Dirac fields. This leads to the definition of the time-dependent scattering operator that encodes the far-field behavior (with respect to a stationary observer) in the asymptotic regions of the black hole: the event and cosmological horizons. We also use the miraculous property (quoting Chandrasekhar) - that the Dirac equation can be separated into radial and angular ordinary differential equations - to make the link between the time-dependent scattering operator and its stationary counterpart. This leads to a nice expression of the scattering matrix at fixed energy in terms of stationary solutions of the system of separated equations. In a second part, we use this expression of the scattering matrix to study the uniqueness property in the associated inverse scattering problem at fixed energy. Using essentially the particular form of the angular equation (that can be solved explicitely by Frobenius method) and the Complex Angular Momentum technique on the radial equation, we are finally able to determine uniquely the metric of the black hole from the knowledge of the scattering matrix at a fixed energy.
This thesis is devoted to the study of black hole solutions in ungauged four-dimensional extended Supergravity. We characterize the most general spherically symmetric and static black-hole solution of ungauged Supergravity, and use the result to study the hidden conformal symmetries of Supergravity black holes, obtaining the full Virasoro algebra of the dual conformal field theory. We obtain also all the supersymmetric black-hole metrics of all extended Supergravities using the properties of the groups of Type E7. We introduce the H-F.G.K. formalism, which simplifies the construction of non-supersymmetric black-hole solutions in N=2 Supergravity, and apply it to a class of theories corresponding to Type-IIA String Theory compactified on a Calabi-Yau threefold. As a result we obtain the so-called "quantum" black holes, which only exist when certain quantum corrections (perturbative or non-perturbative, depending on the solution) are included in the prepotential. For the case of non-perturbative black holes, we elaborate on the potential consequences of the appearance on the solution of multi-valued functions in relation to the no-hair theorem for four-dimensional black holes. This thesis also contains the relevant mathematical background to formulate extended ungauged Supergravity in four-dimensions, that is, Special Kähler Geometry and homogeneous spaces, as well as a brief introduction to the mathematical structure of four-dimensional extended ungauged Supergravity.
The connection between real-time quantum field theory (RTQFT) [see, e.g., A.\ Kamenev and A.\ Levchenko, Advances in Physics {58} (2009) 197] and phase-space techniques [E.\ Wolf and L.\ Mandel, {\em Optical Coherence and Quantum Optics} (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [Phys.\ Rev.\ {177} (1969) 1882]. Each rotation is a phase-space mapping, generalising the corresponding ordering from free to interacting fields. In particular, response transformation [L.P.\ and S.S., Ann.\ Phys. (N.Y.) {323} (2008) 1989] extends the normal ordering of free-field operators to the time-normal ordering of Heisenberg\ operators. Structural properties of the response transformation, such as its association with the nonlinear quantum response problem and the related causality properties, hold for all generalised Keldysh rotations. Furthermore, we argue that response transformation is especially suited for RTQFT formulation of spatial, in particular, relativistic, problems, because it extends cancellation of zero-point fluctuations, characteristic of the normal ordering, to interacting fields. As an example, we consider quantised electromagnetic\ field in the Dirac sea. In the time-normally-ordered representation, dynamics of the field looks essentially classical (fields radiated by currents), without any contribution from zero-point fluctuations. For comparison, we calculate zero-point fluctuations of the interacting electromagnetic\ field under orderings other than time-normal. The resulting expression is physically inconsistent: it does not obey the Lorentz condition, nor Maxwell's equations.
In this paper we study the zero dissipation limit of the one-dimensional full compressible Navier-Stokes(CNS) equations with temperature-dependent viscosity and heat-conduction coefficient. It is proved that given a rarefaction wave with one-side vacuum state to the full compressible Euler equations, we can construct a sequence of solutions to the full CNS equations which converge to the above rarefaction wave with vacuum as the viscosity and the heat conduction coefficient tend to zero. Moreover, the uniform convergence rate is obtained. The main difficulty in our proof lies in the degeneracies of the density, the temperature and the temperature-dependent viscosities at the vacuum region in the zero dissipation limit.
We study properties of the localized solitons to the sine-Gordon equation excited on the attractive impurity by a moving kink. The cases of one- and two-dimensional spatially extended impurities are considered. For the case of one-dimensional impurity the possibility of excitation of the first even and odd high-amplitude impurity modes by the moving kink is demonstrated. For the case of two-dimensional impurity we show the possibility of excitation of the nonlinear high-amplitude waves of new type called here breathing pulson and breathing 2D soliton. We suggest different analytical expressions to model these nonlinear excitations. The dependencies of the oscillation frequency and the amplitude of the excited impurity modes on the impurity parameters are reported.
In this paper we introduce (I,J) similar method for incompressible two and three dimensional Euler equations and Navier-Stokes equations, obtain a series of explicit (I,J) similar solutions to the incompressible two dimensional Euler equations, they include all of twin wave solutions, some new singularity solutions, and some global smooth solutions with finite energy. We also discover that twin wave solution and affine solution to two dimensional incompressible Euler equations are respectively plane wave and constant vector. Finally, we supply some explicit piecewise smooth solutions to incompressible three dimensional Euler and an example to incompressible three dimensional Navier-Stokes equations which indicates that viscosity limit of a solution to Navier-Stokes equations does not need to be a solution to Euler equations.
The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the correct Schrödinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.
We use a recently found generalization of the Cauchy-Binet theorem to give a new proof of the Chebotarev-Shamis forest theorem telling that det(1+L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. More generally, we show that det(1+k L) is the number of rooted edge-k-colored spanning forests in G. If a forest with an even number of edges is called even, then det(1-L) is the difference between even and odd rooted spanning forests in G.
To study circularly symmetric field configurations in the SU(2) Hitchin system an SO(2) symmetry, [J_3, ϕ]=0 and [J_3, A_{\pm}]=\pm A_{\pm}, is imposed on the Higgs scalar ϕand the gauge fields A_{\pm} of the system, respectively, where J_3 is a sum of the third components of the orbital angular momenta and the generators of the SU(2). The circular symmetry and the equation \bar{D}ϕ=0 yield onstant, generally nonzero, vacuum expectation values for {\rm Tr}(ϕ^{2}). The equation 4F_{z\bar{z}}=[ϕ, ϕ^{*}] yields a system of differential equations which govern the circularly symmetric field configurations and an exact solution to these equations in a pure gauge form with nontrivial Higgs scalar is obtained.
In [CGN12], we proved that the renormalized critical Ising magnetization fields $Φ^a:= a^{15/8} \sum_{x\in a\, \Z^2} σ_x \, δ_x$ converge as $a\to 0$ to a random distribution that we denoted by $Φ^\infty$. The purpose of this paper is to establish some fundamental properties satisfied by this $Φ^\infty$ and the near-critical fields $Φ^{\infty,h}$. More precisely, we obtain the following results. \bi [(i)] If $A\subset \C$ is a smooth bounded domain and if $m=m_A := <{Φ^\infty, 1_A}$ denotes the limiting rescaled magnetization in $A$, then there is a constant $c=c_A>0$ such that {equation*} \log \Pb{m > x} \underset{x\to \infty}{\sim} -c \; x^{16}\,.{equation*} In particular, this provides an alternative proof that the field $Φ^\infty$ is non-Gaussian (another proof of this fact would use the $n$-point correlation functions established in \cite{CHI} which do not satisfy Wick's formula). [(ii)] The random variable $m=m_A$ has a smooth {\it density} and one has more precisely the following bound on its Fourier transform: $|\Eb{e^{i\,t m}} |\le e^{- \tilde{c}\, |t|^{16/15}}$. [(iii)] There exists a one-parameter family $Φ^{\infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field. \ei
This is a short review of the heritage of Klein's Erlangen program in modern physics.
We introduce variational methods for finding approximate eigenfunctions and eigenvalues of quantum Hamiltonians by constructing a set of orthogonal wave functions which approximately solve the eigenvalue equation.
A categorification of the Heisenberg algebra is constructed in by Khovanov using graphical calculus, and left with a conjecture on the isomorphism between the Heisenberg algebra and Grothendieck ring of the constructed category. We give a proof of Khovanov's conjectured statement in this paper from a categorification of some deformed Heisenberg algebra.
In this paper, based on the white noise analysis of square integrable pure-jump Levy process given by [1], we define the formal derivative of fractional Levy process defined by the square integrable pure-jump Levy process as the fractional Levy noises by considering fractional Levy process as the generalized functional of Levy process, and then we define the Skorohod integral with respect to the fractional Levy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional Levy noises and investigate the existence and uniqueness of their solutions; In addition, we propose a class of stochastic differential equations driven by fractional Levy noises and prove that under the Lipschtz and linear conditions there exists unique stochastic distribution-valued solution.
We develop a version of dipolar conformal field theory based on the central charge modification of the Gaussian free field with the Dirichlet boundary condition and prove that correlators of certain family of fields in this theory are martingale-observables for dipolar SLE. We prove the restriction property of dipolar SLE(8/3) and Friedrich-Werner's formula in the dipolar case.
Differential invariants for the maximal Lie invariance group of the Korteweg-de Vries equation are computed using the moving frame method and compared with existing results. Closed forms of differential invariants of any order are presented for two sets of normalization conditions. Minimal bases of differential invariants associated with the chosen normalization conditions are given.