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.

Wegner estimates, Lifshitz tails, and Anderson localization for Gaussian random magnetic fields
View Description Hide DescriptionThe Wegner estimate for the Hamiltonian of the Anderson model for the special Gaussian random magnetic field is extended to more general magnetic fields. The Lifshitz tail upper bounds of the integrated density of states as analyzed by Nakamura are reviewed and extended so that Gaussian random magnetic fields can be treated. By these and multiscale analysis, the Anderson localization at low energies is proven.

The initial value problem for a Novikov system
View Description Hide DescriptionIt is shown that the initial value problem for an integrable Novikov system is wellposed in Sobolev spaces H^{s}, s > 3/2, in the sense of Hadamard. Furthermore, it is proved that the dependence on initial data is sharp, i.e., the datatosolution map is continuous but not uniformly continuous. Also, peakon traveling wave solutions are used to prove that the solution map is not uniformly continuous in H^{s} for s < 3/2.

Semiclassical states of pLaplacian equations with a general nonlinearity in critical case
View Description Hide DescriptionWe consider the pLaplacian problem where p ∈ (1, N) and f(s) is of critical growth. In this paper, we construct a single peak solution around an isolated component of the positive local minimum points of V as ε → 0 with a general nonlinearity f. In particular, the monotonicity of f(s)/s ^{p−1} and the socalled AmbrosettiRabinowitz condition are not required.
 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.

On spinors transformations
View Description Hide DescriptionWe begin showing that for even dimensional vector spaces V all automorphisms of their Clifford algebras are inner. So all orthogonal transformations of V are restrictions to V of inner automorphisms of the algebra. Thus under orthogonal transformations P and T—space and time reversal—all algebra elements, including vectors v and spinors φ, transform as v → xvx ^{−1} and φ → xφx ^{−1} for some algebra element x. We show that while under combined PT spinor φ → xφx ^{−1} remains in its spinor space, under P or T separately φ goes to a different spinor space and may have opposite chirality. We conclude with a preliminary characterization of inner automorphisms with respect to their property to change, or not, spinor spaces.
 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.

Applications of rigged Hilbert spaces in quantum mechanics and signal processing
View Description Hide DescriptionSimultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie enveloping algebra and support representations of semigroups. We explicitly construct here basis dependent RHS of the line and halfline and relate them to the universal enveloping algebras of the WeylHeisenberg algebra and su(1, 1), respectively. The complete substructure of both RHS and of the operators acting on them is obtained from their algebraic structures or from the related fractional Fourier transforms. This allows us to describe both quantum and signal processing states and their dynamics. Two relevant improvements are introduced: (i) new kinds of filters related to restrictions to subspaces and/or the elimination of high frequency fluctuations and (ii) an operatorial structure that, starting from fix objects, describes their time evolution.
 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.
 General Relativity and Gravitation

Nonlocal Newtonian cosmology
View Description Hide DescriptionWe explore some of the cosmological implications of the recent classical nonlocal generalization of Einstein’s theory of gravitation in which nonlocality is due to the gravitational memory of past events. In the Newtonian regime of this theory, the nonlocal character of gravity simulates dark matter in spiral galaxies and clusters of galaxies. However, dark matter is considered indispensable as well for structure formation in standard models of cosmology. Can nonlocal gravity solve the problem of structure formation without recourse to dark matter? Here we make a beginning in this direction by extending nonlocal gravity in the Newtonian regime to the cosmological domain. The nonlocal analog of the Zel’dovich solution is formulated and the consequences of the resulting nonlocal Zel’dovich model are investigated in detail.
 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.