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arXiv2013

Convective Equations and a Generalized Cole-Hopf Transformation

Differential equations with convective terms such as the Burger's equation appear in many applications and have been the subject of intense research. In this paper we use a generalized form of Cole-Hopf transformation to relate the solutions of some of these nonlinear equations to the solutions of linear equations. In particular we consider generalized forms of Burger's equation and second order nonlinear ordinary differential equations with convective terms which can represent steady state one-dimensional convection.

math-ph↗

arXiv2013

Continuous Renormalization Group Analysis of Spectral Problems in Quantum Field Theory

The isospectral renormalization group is a powerful method to analyze the spectrum of operators in quantum field theory. It was introduced in 1995 [see \cite{BachFrohlichSigal1995}, \cite{BachFrohlichSigal1998}] and since then it has been used to prove several results for non-relativistic quantum electrodynamics. After the introduction of the method there have been many works in which extensions, simplifications or clarifications are presented (see \cite{BachChenFrohlichSigal2003}, \cite{GriesemerHasler2008}, \cite{FrohlichGriesemerSigal2009}). In this paper we present a new approach in which we construct a flow of operators parametrized by a continuous variable in the positive real axis. While this is in contrast to the discrete iteration used before, this is more in spirit of the original formulation of the renormalization group introduced in theoretical physics in 1974 \cite{KogutWilson1974}. The renormalization flow that we construct can be expressed in a simple way: it can be viewed as a single application of the Feshbach-Schur map with a clever selection of the spectral parameter. Another advantage of the method is that there exists a flow function for which the renormalization group that we present is the orbit under this flow of an initial Hamiltonian. This opens the possibility to study the problem using different techniques coming from the theory of evolution equations.

math-ph↗

arXiv2013

Nonlinear differential identities for cnoidal waves

This article presents a family of nonlinear differential identities for the spatially periodic function $u_s(x)$, which is essentially the Jacobian elliptic function $\cn^2(z;m(s))$ with one non-trivial parameter $s$. More precisely, we show that this function $u_s$ fulfills equations of the form {equation*} \big(u_s^{(α)}u_s^{(β)}\big)(x)=\sum_{n=0}^{2+α+β}b_{α,β}(n)u_s^{(n)}(x)+c_{α,β}, {equation*} for any $s>0$ and for all $α,β\in\N_0$. We give explicit expressions for the coefficients $b_{α,β}(n)$ and $c_{α,β}$ for given $s$. Moreover, we show that for any $s$ satisfying $\sinh(π/(2s))\geq 1$ the set of functions $\{1,u^{\vphantom{a}}_s,u'_s,u"_s,...\}$ constitutes a basis for $L^2(0,2π)$. By virtue of our formulas the problem of finding a periodic solution to any nonlinear wave equation reduces to a problem in the coefficients. A finite ansatz exactly solves the KdV equation (giving the well-known cnoidal wave solution) and the Kawahara equation. An infinite ansatz is expected to be especially efficient if the equation to be solved can be considered a perturbation of the KdV equation.

math-ph↗

arXiv2013

Finite field dependent BRST transformations and its applications to gauge field theories

The Becchi-Rouet-Stora and Tyutin (BRST) transformation plays a crucial role in the quantization of gauge theories. The BRST transformation is also very important tool in characterizing the various renormalizable field theoretic models. The generalization of the usual BRST transformation, by making the infinitesimal global parameter finite and field dependent, is commonly known as the finite field dependent BRST (FFBRST) transformation. In this thesis, we have extended the FFBRST transformation in an auxiliary field formulation and have developed both on-shell and off-shell FF-anti-BRST transformations. The different aspects of such transformation are studied in Batalin-Vilkovisky (BV) formulation. FFBRST transformation has further been used to study the celebrated Gribov problem and to analyze the constrained dynamics in gauge theories. A new finite field dependent symmetry (combination of FFBRST and FF-anti-BRST) transformation has been invented. The FFBRST transformation is shown useful in connection of first-class constrained theory to that of second-class also. Further, we have applied the Batalin-Fradkin-Vilkovisky (BFV) technique to quantize a field theoretic model in the Hamiltonian framework. The Hodge de Rham theorem for differential geometry has also been studied in such context.

hep-th↗

arXiv2013

Stability of symmetric vortices for two-component Ginzburg-Landau systems

We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.

math.AP↗

arXiv2013

Fluctuation Relations in Stochastic Thermodynamics

Fluctuation relations are identities, holding in non-equilibrium systems, that have attracted a lot of interest in the last 20 years. This is a series of 4 lectures discussing various aspects of such relations for stochastic equations modeling non-equilibrium processes.

math-ph↗

arXiv2013

Quadratic control of quantum processes

Within the framework of the Accardi-Fagnola-Quaegebeur (AFQ) representation free calculus of \cite{b}, we consider the problem of controlling the size of a quantum stochastic flow generated by a unitary stochastic evolution affected by quantum noise. In the case when the evolution is driven by first order white noise (which includes quantum Brownian motion) the control is shown to be given in terms of the solution of an algebraic Riccati equation. This is done by first solving the problem of controlling (by minimizing an associated quadratic performance criterion) a stochastic process whose evolution is described by a stochastic differential equation of the type considerd in \cite{b}. The solution is given as a feedback control law in terms of the solution of a stochastic Riccati equation.

math-ph↗

arXiv2013

Optimal Hamiltonian of Fermion Flows

After providing a general formulation of Fermion flows within the context of Hudson-Parthasarathy quantum stochastic calculus, we consider the problem of determining the noise coefficients of the Hamiltonian associated with a Fermion flow so as to minimize a naturally associated quadratic performance functional. This extends to Fermion flows results of the authors previously obtained for Boson flows .

math-ph↗

arXiv2013

Application of Quantum Stochastic Calculus to Feedback Control

The basic aspects of the Hudson-Parthasarathy quantum stochastic calculus and of the Accardi-Fagnola-Quaegebeur representation free stochastic calculus are presented. The basic features of the stochastic calculus for the square of white noise recently developed by Accardi-Boukas are described. The feedback control problem for stochastic processes driven by quantum noise is solved

math-ph↗

arXiv2013

Quantum Black Holes, Elliptic Genera and Spectral Partition Functions

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology.

hep-th↗

arXiv2013

Universality in spectral statistics of "open" quantum graphs

The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their spectral statistics exhibit universality at the "soft" edges of the spectrum. The same spectral behavior is observed in many classical non-unitary ensembles of random matrices with rotationally invariant measures.

nlin.CD↗

arXiv2013

Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.

math.PR↗

arXiv2013

On the leading term of asymptotics of the $n$ like-charged quantum particles scattering problem solution

An ansatz describing in terms of formal asymptotic decompositions a leading term of asymptotics of the $n$ three-dimensional like-charged quantum particles scattering problem solution is suggested. The description of the solution in those asymptotic configurations in which it was known earlier (for example, $n=3$), coincides with the previously known constructions \cite{BBK,z1,am92,BL2,KL}. It is shown that the Schredinger equation discrepancy for the suggested ansatz decreases faster than the potential uniformly in all angle variables at infinity in configuration space. An assumption is made about the structure of the leading term of the asymptotics of the scattering problem solution related to the $n$ three-dimensional quantum particles interacting by a broad class of slowly decreasing pair potentials.

math-ph↗

arXiv2013

A Modification on the Ivanenko-Landau-Kähler Equation

In this paper, we illustrated one scenario to modify the Ivanenko-Landau-Kähler equation. Since Ivanenko and Landau introduced the equation in 1928, the equation has been regarded as having a certain role as a fermion in particular in the discrete Lattice. Also, although it correctly is formulated as an alternative classical field equation by the Ideal projection for the Dirac equation in the Minkowski space-time, so does it only in that flat geometry. I. M. Benn and R. W. Tucker in 1985 and Yu. N. Obukhov and S. N. Solodukhin in 1994 suggested two resolutions respectively. They modified the equation in order for it to make senses as an alternative for the Dirac equation in the general space-time. This paper advances a still another approach, however in the Minkowski space yet, as the first stage toward the generalization. Two ingredients for the modifications are essential. One is the restriction of the space of anti-symmetric tensor field to only its subalgebras, not to its Ideals. The other is the modification on the mass term for the equation to mean the eigenvalue problem generating physical states. The Vector U(1) and the Axial U(1) symmetries built in the modified equation are described.

hep-th↗

arXiv2013

The Fundamental Group of a Spatial Section Represented by a Topspin Network

We present an algorithm which determines the fundamental group of a spatial section using topspin networks. Tracking the topology of the spatial section is a unique feature of this approach, which is not possible in standard Loop Quantum Gravity. This leads to an example of spatial topology change in a smooth 4-manifold represented by a topspin foam.

gr-qc↗

arXiv2013

Double-dimers, the Ising model and the hexahedron recurrence

We define and study a recurrence relation in ${\mathbb Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition sums for edge configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.

math-ph↗

arXiv2013

Local scaling asymptotics in phase space and time in Berezin-Toeplitz quantization

This paper deals with the local semiclassical asymptotics of a quantum evolution operator in the Berezin-Toeplitz scheme, when both time and phase space variables are subject to appropriate scalings in the neighborhood of the graph of the underlying classical dynamics. Global consequences are then drawn regarding the scaling asymptotics of the trace of the quantum evolution as a function of time.

math.SG↗

arXiv2013

Quasi-local characteristics of dynamical extreme black holes

Introducing the concept of the extreme trapping horizon, we discuss geometric features of dynamical extreme black holes in four dimensions and then derive the integral identities which hold for the dynamical extreme black holes. We address the causal/geometrical features too.

gr-qc↗