Volume 57, Issue 7, July 2016

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ)^{1/2}, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into nonoverlapping, orbitally labelled E (k,l) series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth series eigenvalues. Each of them is 2l + 1degenerate. The l = 0 eigenvalues series E (k,0) are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E (k,0)(d = 3) = E 2k(d = 1). Likewise, the eigenfunctions ψ (k,0)(d = 3) and ψ 2k(d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).
 ARTICLES

 Partial Differential Equations

Largetime behavior for the Vlasov/compressible NavierStokes equations
View Description Hide DescriptionWe establish the largetime behavior for the coupled kineticfluid equations. More precisely, we consider the Vlasov equation coupled to the compressible isentropic NavierStokes equations through a drag forcing term. For this system, the largetime behavior shows the exponential alignment between particles and fluid velocities as time evolves. This improves the previous result by Bae et al. [Discrete Contin. Dyn. Syst. 34, 4419–4458 (2014)] in which they considered the Vlasov/NavierStokes equations with nonlocal velocity alignment forces for particles. Employing a new Lyapunov functional measuring the fluctuations of momentum and mass from the averaged quantities, we refine assumptions for the largetime behavior of the solutions to that system.
 Representation Theory and Algebraic Methods

Actions of the quantum toroidal algebra of type sl2 on the space of vertex operators for modules
View Description Hide DescriptionHighest weight modules for are endowed with a structure of modules for the quantum toriodal algebra of type sl 2. Using this, we define actions on the space of vertex operators for irreducible highest weight modules. Highest or lowest weight vectors of the thus obtained modules are expressed in terms of an intertwiner for modules and an extra boson. The submodules generated by these vectors are investigated.

Black holes, information, and the universal coefficient theorem
View Description Hide DescriptionGeneral relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article, I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by the demanding invariance of the laws of physics to the change of coefficient structures in cohomology.
 ManyBody and Condensed Matter Physics

On the third critical speed for rotating BoseEinstein condensates
View Description Hide DescriptionWe study a twodimensional rotating BoseEinstein condensate confined by an anharmonic trap in the framework of the GrossPitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε^{−4}, with ε^{−2} denoting the coefficient of the nonlinear term in the GrossPitaevskii functional and ε ≪ 1 (ThomasFermi regime). In this paper, we identify a finite value Ωc such that if Ω = Ω0/ε^{4} with Ω0 > Ωc, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any GrossPitaevskii minimizer.

Fundamental limitations in the purifications of tensor networks
View Description Hide DescriptionWe show a fundamental limitation in the description of quantum manybody mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics
View Description Hide DescriptionAn analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., BogoliubovKuboMori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, encov, and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter h. Due to the sum rules for the frequency moments of the dynamic structure factor, new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator S are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.
 Quantum Mechanics

Slowly changing potential problems in Quantum Mechanics: Adiabatic theorems, ergodic theorems, and scattering
View Description Hide DescriptionWe employ the recently developed multitime scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. We prove a new uniform ergodic theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness.

Wegner estimate for Landaubreather Hamiltonians
View Description Hide DescriptionWe consider Landau Hamiltonians with a weak coupling random electric potential of breather type. Under appropriate assumptions we prove a Wegner estimate. It implies the Hölder continuity of the integrated density of states. The main challenge is the problem how to deal with nonlinear dependence on the random parameters.

Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels
View Description Hide DescriptionTo each hyperbolic Landau level of the Poincaré disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its atomic decomposition as a perturbation of the negative binomial distribution by a finitely supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we derive a LévyKhintchinetype representation of its characteristic function when the latter does not vanish and deduce that it is quasiinfinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasiLévy measure, we introduce a new infinitely divisible distribution for which we derive the characteristic function.

Schrödinger equation for nonpure dipole potential in 2D systems
View Description Hide DescriptionIn this work, we analytically study the Schrödinger equation for the (nonpure) dipolar ion potential V(r) = q/r + Dcosθ/r ^{2}, in the case of 2D systems (systems in twodimensional Euclidean plane) using the separation of variables and the Mathieu equations for the angular part. We give the expressions of eigenenergies and eigenfunctions and study their dependence on the dipole moment D. Imposing the condition of reality on the energies E n,m implies that the dipole moment must not exceed a maximum value, otherwise the corresponding bound state disappears. We also find that the s states (m = 0) can no longer exist in the system as soon as the dipole term is present.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Consistency of multitime Dirac equations with general interaction potentials
View Description Hide DescriptionIn 1932, Dirac proposed a formulation in terms of multitime wave functions as candidate for relativistic manyparticle quantum mechanics. A wellknown consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spincoupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multitime formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

Ultrarelativistic bound states in the spherical well
View Description Hide DescriptionWe address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ)^{1/2}, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into nonoverlapping, orbitally labelled E (k,l) series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth series eigenvalues. Each of them is 2l + 1degenerate. The l = 0 eigenvalues series E (k,0) are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E (k,0)(d = 3) = E 2k(d = 1). Likewise, the eigenfunctions ψ (k,0)(d = 3) and ψ 2k(d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).

The fermionic projector in a timedependent external potential: Mass oscillation property and Hadamard states
View Description Hide DescriptionWe give a nonperturbative construction of the fermionic projector in Minkowski space coupled to a timedependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of the Hadamard form, provided that the time integral of the spatial supnorm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasifree Hadamard state.
 Classical Mechanics and Classical Fields

Novel isochronous Nbody problems featuring N arbitrary rational coupling constants
View Description Hide DescriptionA novel class of Nbody problems is identified, with N an arbitrary positive integer (N ≥ 2). These models are characterized by Newtonian (“accelerations equal forces”) equations of motion describing N equal pointparticles moving in the complex zplane. These highly nonlinear equations feature N arbitrary coupling constants, yet they can be solved by algebraic operations and if all the N coupling constants are real and rational the corresponding Nbody problem is isochronous: its generic solutions are all completely periodic with an overall period T independent of the initial data (but many solutions feature subperiods T/p with p integer). It is moreover shown that these models are Hamiltonian.

First kind symmetric periodic solutions of the generalized van der Waals Hamiltonian
View Description Hide DescriptionThe aim of this paper is to prove the existence of a new symmetric family of periodic solutions of the generalized van der Waals Hamiltonian. In fact, we prove the existence of several families of first kind symmetric periodic solutions as continuation of circular orbits of the Kepler problem in the spatial case.
 Statistical Physics

The radiative transport equation in flatland with separation of variables
View Description Hide DescriptionThe linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in threedimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singulareigenfunction approach in flatland or twodimensional space.
 Methods of Mathematical Physics

Coherent states, quantum gravity, and the Born Oppenheimer approximation. II. Compact Lie groups
View Description Hide DescriptionIn this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravitytype models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional BornOppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the SegalBargmannHall “coherent state” transform for compact Lie groups G, which we prove for G = U(1)^{n} and support by numerical evidence for G = SU(2). The reason for conjoining this conjecture with the main topic of this article originates in the observation that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups G. But, as Weyl and Berezin quantisation for ℝ^{2d} are intimately related by heat kernel evolution, it is natural to ask whether a similar connection exists for compact Lie groups as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the SegalBargmannHall transform, can serve as basis of the former.

Monopoles, instantons, and the Helmholtz equation
View Description Hide DescriptionIn this work we study the dimensional reduction of smooth circle invariant YangMills instantons defined on 4manifolds which asymptotically become circle fibrations over hyperbolic 3space. A suitable choice of the 4manifold metric within a specific conformal class gives rise to singular and smooth hyperbolic monopoles. A large class of monopoles is obtained if the conformal factor satisfies the Helmholtz equation on hyperbolic 3space. We describe simple configurations and relate our results to the JackiwNohlRebbi construction, for which we provide a geometric interpretation.

The category of supermanifolds
View Description Hide DescriptionIn physics and in mathematics gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the “scalar product” of the involved degrees. The supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any nfold vector bundle has a canonical “superization” to a supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the context.