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preprint2020arXiv

Inverse scattering transform for N-wave interaction problem with a dispersive term in two spatial dimensions

In this work, we introduce a dispersive N(=2n)-wave interaction problem involving n velocities in two spatial dimensions and one temporal dimension. Exact solutions of the problem are exhibited. This is a generalization of the N-wave interaction problem and matrix Davey-Stewartson equation with 2+1 dimensions that examines the Benney-type model of interactions between short and long waves. Accordingly, associated with the solutions of two dimensional analog of the Manakov system, a Gelfand-Levitan-Marchenko (GLM)-type, or so-called inversion-like, equation is constructed. It is shown that the presence of the degenerate kernel reads exact soliton-like solutions of the dispersive N-wave interaction problem.We also mention the unique solution of the Cauchy problem on an arbitrary time interval for small initial data.

preprint2020arXiv

On the steady motion of Navier-Stokes flows past a fixed obstacle in a three-dimensional channel under mixed boundary conditions

We analyze the steady motion of a viscous incompressible fluid in a three-dimensional channel containing an obstacle through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet, together with the standard no-slip assumption on the obstacle and on the remaining walls of the domain. Explicit bounds on the inflow velocity guaranteeing existence and uniqueness of such steady motion are provided after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the inlet velocity through the Bogovskii formula. A quantitative analysis of the forces exterted by the fluid over the obstacle constitutes the main application of our results: by deriving a volume integral formula for the drag and lift, explicit upper bounds on these forces are given in terms of the geometrical constraints of the domain.

preprint2020arXiv

Asymptotic shallow models arising in magnetohydrodynamics

In this paper, we derive a new shallow asymptotic model for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equation, vital in describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves in the presence of weakly shared vorticity and magnetic currents. The method is inspired by developed ideas for hydrodynamics flows in by Castro and Lannes (2014) to reduce the $(d+1)$-dimensional dynamics of the problem to a finite cascade of equations which can be closed at the precision of the model.

preprint2020arXiv

Phase transitions for $ϕ^4_3$

We establish a surface order large deviation estimate for the magnetisation of low temperature $ϕ^4_3$. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $ϕ^4_3$ singular stochastic PDE. Our main technical contributions are contour bounds for $ϕ^4_3$, which extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by Bodineau, Velenik, and Ioffe (2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of $ϕ^4_3$ whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $ϕ^4_3$ developed recently by Barashkov and Gubinelli (2019).

preprint2020arXiv

Multispinon excitations in the spin S=1/2 antiferromagnetic Heisenberg model

With the commutation relations of the spin operators, we first write out the equations of motion of the spin susceptibility and related correlation functions that have a hierarchical structure, then under the "soft cut-off" approximation, we give a set of equations of motion of spin susceptibilities for a spin S=1/2 antiferromagnetic Heisenberg model, that is independent of whether or not the system has a long range order in the low energy/temperature limit. Applying for a chain, a square lattice and a honeycomb lattice, respectively, we obtain the upper and the lowest boundaries of the low-lying excitations by solving this set of equations. For a chain, the upper and the lowest boundaries of the low-lying excitations are the same as that of the exact ones obtained by the Bethe ansatz, where the elementary excitations are the spinon pairs. For a square lattice, the spin wave excitation (magnons) resides in the region close to the lowest boundary of the low-lying excitations, and the multispinon excitations take place in the high energy region close to the upper boundary of the low-lying excitations. For a honeycomb lattice, we have one kind of "mode" of the low-lyin

preprint2020arXiv

Characterization of quantum entanglement via a hypercube of Segre embeddings

A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension n-1, where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the n-1 edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with $q$-partitions for q>1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement.

preprint2020arXiv

Spectral sum rules for the Schrödinger equation

We study the sum rules of the form $Z(s) = \sum_n E_n^{-s}$, where $E_n$ are the eigenvalues of the time--independent Schrödinger equation (in one or more dimensions) and $s$ is a rational number for which the series converges. We have used perturbation theory to obtain an explicit formula for the sum rules up to second order in the perturbation and we have extended it non--perturbatively by means of a Padé--approximant. For the special case of a box decorated with one impurity in one dimension we have calculated the first few sum rules of integer order exactly; the sum rule of order one has also been calculated exactly for the problem of a box with two impurities. In two dimensions we have considered the case of an impurity distributed on a circle of arbitrary radius and we have calculated the exact sum rules of order two. Finally we show that exact sum rules can be obtained, in one dimension, by transforming the Schrödinger equation into the Helmholtz equation with a suitable density.

preprint2020arXiv

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler--Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler--Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The~existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of

preprint2020arXiv

Complete Sobolev Type Inequalities

We establish Sobolev type inequalities in the noncommutative settings by generalizing monotone metrics in the space of quantum states, such as matrix-valued Beckner inequalities. We also discuss examples such as random transpositions and Bernoulli-Laplace models.

preprint2020arXiv

Energy cutoff, effective theories, noncommutativity, fuzzyness: the case of O(D)-covariant fuzzy spheres

Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff $\overline{E}$ may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well $V$ with a very sharp minimum on a submanifold $N$ of the original space(time) $M$ may induce a dimensional reduction to a noncommutative quantum theory on $N$. Here in particular we briefly report on our application of this procedure to spheres $S^d\subset\mathbb{R}^D$ of radius $r=1$ ($D=d\!+\!1>1$): making $\overline{E}$ and the depth of the well depend on (and diverge with) $Λ\in\mathbb{N}$ we obtain new fuzzy spheres $S^d_Λ$ covariant under the {\it full} orthogonal groups $O(D)$; the commutators of the coordinates depend only on the angular momentum, as in Snyder noncommutative spaces. Focusing on $d=1,2$, we also discuss uncertainty relations, localization of states, diagonalization of the space coordinates and construction of coherent states. As $Λ\to\infty$ the Hilbert space dimension diverges, $S^d_Λ\to S^d$, and we recover ordinary quantum mechanics on $S^d$. These models might be suggestive for effective models in quantum field theor

preprint2020arXiv

Nonlocal homogenisation theory for curl-div-systems

We study the curl-div-system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal $H$-convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non-periodic coefficients in curl-div systems to model highly oscillatory behaviour of the coefficients on small scales. We address curl-div systems under various boundary conditions and analyse the limit of the ratio of small scale over large scale tending to zero. Already for standard Dirichlet boundary conditions and local coefficients the limit system is nontrivial and unexpected. Furthermore, we provide an analysis of highly oscillatory local coefficients for a curl-div system with impedance type boundary conditions relevant in scattering theory for Maxwell's equations and relate the abstract findings to local $H$-convergence and weak$*$-convergence of the coefficients.

preprint2020arXiv

Loop-erased random walk as a spin system observable

The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.

preprint2020arXiv

The Saito determinant for Coxeter discriminant strata

Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in $\mathcal{M}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.

preprint2020arXiv

Spectral analysis for a multi-dimensional split-step quantum walk with a defect

This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analysed in the previous papers. To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-defect model. We also derive the time-averaged limit measure for one-dimensional case as an application of the spectral analysis.

preprint2020arXiv

Stability estimate for scalar image velocimetry

In this paper we analyse the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors ${u}$ from images of a passive scalar field $ψ$ by minimising a cost functional, that penalises the difference between the reconstructed scalar field $ϕ$ and the measured scalar field $ψ$, under the constraint that $ϕ$ is advected by the reconstructed velocity field ${u}$, which again is governed by the Navier-Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence, that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two dimensional case, smooth solutions of the Navier-Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field $ψ$ to the reconstructed velocity field $u$, on any interior subset, is Hölder continuous.

preprint2020arXiv

Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2π$ perimeter and of the unit disk

We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.

preprint2020arXiv

Projections, modules and connections for the noncommutative cylinder

We initiate a study of projections and modules over a noncommutative cylinder, a simple example of a noncompact noncommutative manifold. Since its algebraic structure turns out to have many similarities with the noncommutative torus, one can develop several concepts in a close analogy with the latter. In particular, we exhibit a countable number of nontrivial projections in the algebra of the noncommutative cylinder itself, and show that they provide concrete representatives for each class in the corresponding $K_0$ group. We also construct a class of bimodules endowed with connections of constant curvature. Furthermore, with the noncommutative cylinder considered from the perspective of pseudo-Riemannian calculi, we derive an explicit expression for the Levi-Civita connection and compute the Gaussian curvature.

preprint2020arXiv

Perturbative Quantum Field Theory and Homotopy Algebras

We review the homotopy algebraic perspective on perturbative quantum field theory: classical field theories correspond to homotopy algebras such as $A_\infty$- and $L_\infty$-algebras. Furthermore, their scattering amplitudes are encoded in minimal models of these homotopy algebras at tree level and their quantum relatives at loop level. The translation between Lagrangian field theories and homotopy algebras is provided by the Batalin--Vilkovisky formalism. The minimal models are computed recursively using the homological perturbation lemma, which induces useful recursion relations for the computation of scattering amplitudes. After explaining how the homological perturbation lemma produces the usual Feynman diagram expansion, we use our techniques to verify an identity for the Berends--Giele currents which implies the Kleiss--Kuijf relations.

preprint2020arXiv

Embedding Galilean and Carrollian geometries I. Gravitational waves

The aim of this series of papers is to generalise the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this first part, we propose a generalisation of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalisation by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves - dubbed Dodgson waves - is introduced and geometrically characterised. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows to generalise the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike fo

preprint2020arXiv

Two-temperatures overlap distribution for the 2D discrete Gaussian free field

In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.

preprint2020arXiv

Homotopy Algebras in Higher Spin Theory

Motivated by string field theory, we explore various algebraic aspects of higher spin theory and Vasiliev equation in terms of homotopy algebras. We present a systematic study of unfolded formulation developed for the higher spin equation in terms of the Maurer-Cartan equation associated to differential forms valued in L-infinity algebras. The elimination of auxiliary variables of Vasiliev equation is analyzed through homological perturbation theory. This leads to a closed combinatorial graph formula for all the vertices of higher spin equations in the unfolded formulation. We also discover a topological quantum mechanics model whose correlation functions give deformed higher spin vertices at first order.

preprint2020arXiv

Transport in boundary-driven quantum spin systems: One-way street for the energy current

We study transport properties in boundary-driven asymmetric quantum spin chains given by $\mathit{XXZ}$ and $\mathit{XXX}$ Heisenberg models. Our approach exploits symmetry transformations in the Lindblad master equation associated to the dynamics of the systems. We describe the mathematical steps to build the unitary transformations related to the symmetry properties. For general target polarizations, we show the occurrence of the one-way street phenomenon for the energy current, namely, the energy current does not change in magnitude and direction as we invert the baths at the boundaries. We also analyze the spin current in some situations, and we prove the uniqueness of the steady state for all investigated cases. Our results, involving nontrivial properties of the energy flow, shall interest researchers working on the control and manipulation of quantum transport.

preprint2020arXiv

Rotating fermions inside a spherical boundary

We apply the cannonical quantization procedure to the Dirac field inside a spherical boundary with rotating coordinates. The rotating quantum states with two kinds of boundary conditions, namely, spectral and MIT boundary conditions, are defined. To avoid faster-than-light, we require the speed on the surface to be less than the speed of light. For this situation, the definition of vacuum is unique and identical with the Minkowski vacuum. Finally, we calculate the thermal expectation value of the fermion condensate in a thermal equilibrium rotating fermion field and find it depends on the boundary condition.

preprint2020arXiv

An approach to BPS black hole microstate counting in an N=2 STU model

We consider four-dimensional dyonic single-center BPS black holes in the $N=2$ STU model of Sen and Vafa. By working in a region of moduli space where the real part of two of the three complex scalars $S, T, U$ are taken to be large, we evaluate the quantum entropy function for these BPS black holes. In this regime, the subleading corrections point to a microstate counting formula partly based on a Siegel modular form of weight two. This is supplemented by another modular object that takes into account the dependence on $Y^0$, a complex scalar field belonging to one of the four off-shell vector multiplets of the underlying supergravity theory. We also observe interesting connections to the rational Calogero model and to formal deformation of a Poisson algebra, and suggest a string web picture of our counting proposal.