ALBERT

Beschreibung: We study the continuity of an abstract generalization of the maximum-entropy inference—a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.

Beschreibung: A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the Dirac operator. Examples include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are investigated in detail, and it is shown that the fuzzy analogues correspond to two spinor fields on the commutative sphere. In some cases, it is necessary to add a mass mixing matrix to the commutative Dirac operator to get a precise agreement for the eigenvalues.

Beschreibung: By using the method of planar dynamical systems, we solve exactly a nonlinear Schrödinger (NLS) equation discussed by Khare and Saxena [J. Math. Phys. 56 , 032104 (2015)], and give the exact explicit parametric representations of all the traveling wave solutions.

Beschreibung: Unitary representations of the Temperley–Lieb algebra TL N ( Q ) on the tensor space (ℂ n ) ⊗ N are considered. Two criteria are given for determining when an orthogonal projection matrix P of a rank r gives rise to such a representation. The first of them is the equality of traces of certain matrices and the second is the unitary condition for a certain partitioned matrix. Some estimates are obtained on the lower bound of Q for a given dimension n and rank r . It is also shown that if 4 r 〉 n 2 , then Q can take only a discrete set of values determined by the value of n 2 / r . In particular, the only allowed value of Q for n = r = 2 is Q = 2 . Finally, properties of the Clebsch–Gordan coefficients of the quantum Hopf algebra U q ( su 2 ) are used in order to find all r = 1 and r = 2 unitary tensor space representations of TL N ( Q ) such that Q depends continuously on q and P is the projection in the tensor square of a simple U q ( su 2 ) module on the subspace spanned by one or two joint eigenvectors of the Casimir operator C and the generator K of the Cartan subalgebra.

Beschreibung: Let

Beschreibung: This paper is concerned with the existence and some qualitative properties of entire solutions for a reaction-advection-diffusion equation in infinite cylinders with time periodic bistable nonlinearity. Here, an entire solution means a solution defined in the whole space and for all time t ∈ ℝ. By the comparison principle coupled with the supersolution and subsolution technique, it is proved that there exists an entire solution. Furthermore, it is shown that such an entire solution is unique and Liapunov stable. Unlike the reaction-diffusion equation without advection, the lack of symmetry between increasing and decreasing traveling fronts caused by the advection affects the construction of supersolutions and subsolutions.

Beschreibung: We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13 (7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n , we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z -rotation by π n . We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of π n z -rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z [ e i π n , 1 / 2 ] . We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.

Beschreibung: In the study of the relation between complexified classical and non-Hermitian quantum mechanics, physicists found that there are links to quaternionic and split quaternionic mechanics, and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics. This paper, by means of real representation of a split quaternion matrix, studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics.

Beschreibung: Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols, and related quantities are analyzed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are also obtained.

Beschreibung: This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, u tt − Δ p u − Δ u t = f ( u ), in a bounded domain Ω ⊂ ℝ 3 and subject to Dirichlét boundary conditions. The operator Δ p , 2 〈 p 〈 3, denotes the classical p -Laplacian. The nonlinear term f ( u ) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W 0 1 , p ( Ω ) into L 2 (Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

Beschreibung: We discuss a procedure to simplify the Landau potential, based on Michel’s reduction to orbit space and Poincaré normalization procedure, and illustrate it by concrete examples. The method makes use, as in Poincaré theory, of a chain of near-identity coordinate transformations with homogeneous generating functions; using Michel’s insight, one can work in orbit space. It is shown that it is possible to control the choice of generating functions so to obtain a (in many cases, substantial) simplification of the Landau polynomial, including a reduction of the parameters it depends on. Several examples are considered in detail.

Beschreibung: The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as “master dispersionless systems” in four and three dimensions, respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note, we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar partial differential equations for a general anti-self-dual conformal structure in neutral signature.

Beschreibung: We consider an overdetermined elliptic problem known as the hollow vortex problem. We prove that the solutions to this problem are in 1:1 correspondence with minimal graphs bounded by horizontal symmetry lines. We use this correspondence to give various examples of domains with hollow vortices.

Beschreibung: The discrete Schrödinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the corresponding transmission eigenvalues is analyzed. The Marchenko method and the Gel’fand-Levitan method are used to solve the inverse problem uniquely, except in one “unusual” case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential. It is shown that in the unusual case, there may be a unique potential corresponding to a given set of transmission eigenvalues, there may be a finite number of distinct potentials for a given set of transmission eigenvalues, or there may be infinitely many potentials for a given set of transmission eigenvalues. The theory presented is illustrated with several explicit examples.

Beschreibung: We find that in presence of noncommutative Poisson brackets, the relation between Lagrangian and Hamiltonian is modified. We discuss this property by using the path integral formalism for non-relativistic systems. We apply this procedure to the harmonic oscillator with a minimal length.

Beschreibung: We show that a real eigenfunction of the Schrödinger operator changes sign near some point in ℝ n under a suitable assumption on the potential.

Beschreibung: Orthogonal projections in ℂ n ⊗ ℂ n of rank one and rank two that give rise to unitary tensor space representations of the Temperley–Lieb algebra TL N ( Q ) are considered. In the rank one case, a complete classification of solutions is given. In the rank two case, solutions with Q varying in the ranges [2 n /3, ∞) and [ n / 2 , ∞ ) are constructed for n = 3 k and n = 4 k , k ∈ ℕ, respectively.

Beschreibung: The additivity of the minimal output entropy and that of the χ -capacity are known to be equivalent for finite-dimensional irreducibly covariant quantum channels. In this paper, we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper [Giovannetti et al. , Commun. Math. Phys. 334 (3), 1553-1571 (2015)] to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the “threshold condition,” obtained for single mode in Schäfer et al. [Phys. Rev. Lett. 111 , 030503 (2013)], and the proof of the fact that under this condition the classical “Gaussian capacity” resulting from optimization over only Gaussian inputs is equal to the full classical capacity. Complex structures correspond to different squeezings, each with its own normal modes, vacuum and coherent states, and the gauge. Thus our results apply, e.g., to multimode channels with a squeezed Gaussian noise under the standard input energy constraint, provided the squeezing is not too large as to violate the generalized threshold condition. We also investigate the restrictiveness of the gauge-covariance condition for single- and multimode bosonic Gaussian channels.

Beschreibung: Following the ideas of Hudson [J. Funct. Anal. 34 (2), 266-281 (1979)] and Parthasarathy and Sinha [Probab. Theory Relat. Fields 73 , 317-349 (1987)], we define a quantum stopping time (QST, for short) τ in the interacting Fock space (IFS, for short), Γ, over L 2 (ℝ + ), which is actually a spectral measure in [0, ∞] such that τ ([0, t ]) is an adapted process. Motivated by Parthasarathy and Sinha [Probab. Theory Relat. Fields 73 , 317-349 (1987)] and Applebaum [J. Funct. Anal. 65 , 273-291 (1986)], we also develop a corresponding quantum stopping time stochastic integral (QSTSI, for abbreviations) on the IFS over a subspace of L 2 (ℝ + ) equipped with a filtration. As an application, such integral provides a useful tool for proving that Γ admits a strong factorisation, i.e., Γ = Γ τ ] ⊗ Γ [ τ , where Γ τ ] and Γ [ τ stand for the part “before τ ” and the part “after τ ,” respectively. Additionally, this integral also gives rise to a natural composition operation among QST to make the space of all QSTs a semigroup.

Beschreibung: In a previous paper, we have discussed how the Landau potential (entering in Landau theory of phase transitions) can be simplified using the Poincaré normalization procedure. Here, we apply this approach to the Landau-deGennes functional for the isotropic-nematic transitions, and transitions between different nematic phases, in liquid crystals. We give special attention to applying our method in the region near the main transition point, showing in full detail how this can be done via a suitable simple modification of our Poincaré-like method. We also consider the question if biaxial phases can branch directly off the fully symmetric state; some partial results in this direction are presented.

Beschreibung: The triple product system associated with a rational R -matrix is investigated. We prove that there is the Yangian covariant structure for this triple product system. We present a 3-coalgebraic of the Yangian Y ( sl 2 ) associated with the triple system. Moreover, we discuss a 3-Hopf algebraic structure of the Yangian Y ( sl 2 ).

Beschreibung: We introduce a new representation for the discrete Painlevé equations. It is based on the trihomographic form previously used in conjunction with discrete Painlevé equations associated with the affine Weyl group E 8 ( 1 ) . We show here that all discrete Painlevé equations do possess a trihomographic form. The latter is often more convenient than the usual representation in particular when it comes to the application of the singularity confinement integrability criterion. We show how trihomographic forms can be combined in order to construct generic discrete Painlevé equations or limits thereof. A trihomographic form for the generic elliptic Painlevé equation is also derived.

Beschreibung: An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on unbounded domains is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the H 2 -norm for small initial data are proven.

Beschreibung: We report results of an analysis of the dynamics of magnetic flux solitons in the system of three long Josephson junctions between three bulk superconductors that form a prism. The system is modeled by coupled sine-Gordon equations for the phases of the junctions. The Aharonov-Bohm constraint takes into account the axial magnetic flux enclosed by the prism and reduces the system from three independent phases to two. The equations of motion for the phases include dissipative terms, and a control parameter δ which accounts for the deviation of the enclosed flux from half a quantum. Analyzing the effective potential of the coupled equations, we identify different species of topological and non-topological phase solitons (fluxons) in this system. In particular, subkinks with fractional topological charges ±1/3 and ±2/3, confined inside integer-charge fluxons, may be mapped onto the root diagrams for mesons and baryons in the original quark model of hadrons. Solutions for straight-line kinks and for two types of non-topological solitons are obtained in an explicit analytical form. Numerical tests demonstrate that the former species is unstable against breakup into pairs of separating single-fluxon kinks. The non-topological kinks feature metastability, eventually breaking up into fluxon-antifluxon pairs. Free fractional-fluxon kinks, that connect different potential minima and are, accordingly, pulled by the potential difference, are also considered. Using the momentum-balance method, we predict the velocity at which these kinks should move in the presence of the dissipation. Numerical tests demonstrate that the analysis predicts the velocity quite closely. Higher-energy static solutions for all of the stable kink types mentioned above, as well as kinks connecting false vacua, are found by means of the shooting method. Inelastic collisions among the stable fractional and single-fluxon kinks are investigated numerically.

Beschreibung: In this paper, we first consider the existence of tempered random attractor for second-order non-autonomous stochastic lattice dynamical system of nonlinear Boussinesq equations effected by time-dependent coupled coefficients and deterministic forces and multiplicative white noise. Then, we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Beschreibung: We study the stationary scattering theory for the matrix Schrödinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space, the subspace of absolute continuity of the matrix Schrödinger operator, and final space L 2 ((0, ∞)). We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We also construct the spectral shift function and we give its high-energy asymptotics. Furthermore, assuming that the potential also has a finite first moment, we prove a Levinson’s theorem for the spectral shift function.

Beschreibung: We show the existence of multi-bump solutions to a magnetic nonlinear Schrödinger equation with subcritical superlinear nonlinearity, − ( ∇ − i λ A ) 2 u = f ( u 2 ) u on ℝ N under some assumptions. Moreover, we show that the solutions are localized near every prescribed connected component of the interior of the zero set of the magnetic field B = dA as λ → ∞.

Beschreibung: We consider the Cauchy problem for the inhomogeneous fourth-order nonlinear Schrödinger equations. We find the large time asymptotics of solutions to the Cauchy problem. We use the factorization technique similar to that developed for the Schrödinger equation.

Beschreibung: The Schwinger-Keldysh nonequilibrium theory allows the description of various transport phenomena involving bosons (fermions) embedded in bosonic (fermionic) environments. The retarded Green’s function obeys the Dyson equation and determines via its non-vanishing asymptotic behavior the dissipationless open dynamics. The appearance of this regime is conditioned by the existence of the solution of a general class of transcendental equations in complex domain that we study. Particular cases consist in transcendental equations containing exponential, hyperbolic, power law, logarithmic, and special functions. The present analysis provides an analytical description of the thermal and temporal correlation function of two general observables of a quantum system in terms of the corresponding spectral function. Special integral properties of the spectral function guarantee non-vanishing asymptotic behavior of the correlation function.

Beschreibung: Although the origins of electromagnetic theory and the fluid mechanics are completely different, the compressible ideal fluid equations can be expressed in a form analogous to that of electromagnetism. In this paper, the Maxwell type fluid equations are rewritten in terms of octons. Using the analogies between electromagnetism and fluid mechanics, the compact and elegant formulations are derived. It is proved that single octonic expression describes Maxwell-type fluid equations. The octonic fluid wave equations and fluid Poynting theorem are also formulated in a similar form to electromagnetism and linear gravity.

Beschreibung: We obtain the exact solution of the Schrödinger equation for a particle (galaxy) moving in a Newtonian universe with a cosmological constant, which is given in terms of the biconfluent Heun functions. The first six Heun polynomials of the biconfluent function are written explicitly. The energy spectrum which resembles the one corresponding to the isotropic harmonic oscillator is also obtained. The wave functions as well as the energy levels codify the role played by the cosmological constant.

Beschreibung: Conformal transformations are applied in the context of general relativistic elasticity. Expressions relating relativistic elastic quantities and tensors are obtained for two conformal spacetimes, whose material metrics are also conformally related. Non-static shear-free spherically symmetric elastic solutions of the Einstein field equations are constructed by performing conformal transformations of spacetime and material metrics of a known non-static shear-free spherically symmetric elastic solution.

Beschreibung: Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2 j . The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.

Beschreibung: We discuss an extension of the (massless) Thirring model describing interacting fermions in one dimension which are coupled to phonons and where all interactions are local. This fermion-phonon model can be solved exactly by bosonization. We present a construction and solution of this model which is mathematically rigorous by treating it as a continuum limit of a Luttinger-phonon model. A self-contained account of the mathematical results underlying bosonization is included, together with complete proofs.

Beschreibung: For a quantum measurement process described by a quantum instrument I and a system observable corresponding to a positive-operator valued measure (POVM) E , I is said to conserve the information of E if the joint successive measurement of I followed by E is equivalent to a single measurement of E . We show that for any quantum instrument I we can construct a POVM conserved by I . Intuitively, the construction gives the infinite joint successive measurement of I . We also show that the constructed POVM is the least informative observable among POVMs conserved by I , i.e., the constructed POVM can be realized by a classical post-processing of any POVM conserved by I . As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.

Beschreibung: We establish simple connections between response functions of the dynamical Dirac systems and A -amplitudes and Weyl functions of the spectral Dirac systems. Using these connections, we propose a new and rigorous procedure to recover a general-type dynamical Dirac system from its response function as well as a procedure to construct explicit solutions of this problem.

Beschreibung: We study the effects of three-body collisions in the physical properties of a two-mode Bose-Einstein condensate. The model introduced here includes two-body and three-body elastic and mode-exchange collisions and can be solved analytically. We will use this fact to show that three-body interactions can produce drastic changes in the probability distribution of the ground state and the dynamics of the relative population. In particular, we find that three-body interactions under certain circumstances may inhibit the collapse of the relative population.

Beschreibung: A universal inequality that bounds the charge of a body by its size is presented and is proven as a consequence of the Einstein equations in the context of initial data sets which satisfy an appropriate energy condition. We also present a general sufficient condition for the formation of black holes due to concentration of charge and discuss the physical relevance of these results.   

Beschreibung: A hierarchy of soliton equations is constructed from an extended spectral problem associated with the three-dimensional special orthogonal real Lie algebra so(3,ℝ) by means of zero curvature equations. The bi-Hamiltonian structure of this hierarchy is also presented via the trace identity.

Beschreibung: We consider single particle Schrödinger operators with a gap in the energy spectrum. We construct a complete, orthonormal basis function set for the invariant space corresponding to the spectrum below the spectral gap, which are exponentially localized around a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential decay rate and a discussion of the geometry of these surfaces are included.

Beschreibung: Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to the works of Lindblad [Commun. Math. Phys. 48 , 119-130 (1976)], Stinespring [Proc. Am. Math. Soc. 6 , 211-216 (1955)], and Kraus [Ann. Phys. 64 , 311-335 (1970)] imply that the generator of the semigroup has the form L ( A ) = ∑ n = 1 ∞ V n ∗ A V n + G A + A G ∗ , where V n and G are elements of the underlying operator algebra. In the present paper, we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus’ result. Finally, we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.

Beschreibung: We study a two-component asymmetric simple exclusion process (ASEP) that is equivalent to the ASEP with second-class particles. We prove self-duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra U q [

Beschreibung: We present an L 2 -stability of the kinetic Landau equation for a single species charged plasma with an inverse power-law interaction force in the perturbative regime of global Maxwellians. Our result demonstrates that the L 2 -distance between two classical solutions to the Landau equation can be controlled by that between corresponding initial data in a Lipschitz manner. The Coulomb interaction is known to be the singular and marginal case of the theory of the Boltzmann equation where the grazing collisions are the dominant. For some class of classical solutions, we show that our L 2 -stability result can provide a uniform L 2 -stability.

Beschreibung: We develop a method to construct various classes of one-dimensional periodic potentials with two intersecting energy bands. Analytical exact results for the zero-gap states are presented in an explicit form under certain parameter conditions. The position of the energies of these zero-gap states in the energy bands is identified numerically.

Beschreibung: Simple proofs of uniqueness of the thermodynamic limit of KMS states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures. New quantum correlation inequalities for general Heisenberg models are described. Finally, a simplified derivation of a general result on power-law decay of correlations in 2D quantum lattice systems with continuous symmetries is given, extending results of McBryan and Spencer for the 2D classical XY model.

Beschreibung: A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant, the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

Beschreibung: In this paper, we study the families of solitary-wave solutions to the inhomogeneous coupled nonlinear Schrödinger equations with space- and time-modulated coefficients and source terms. By means of the similarity reduction method and Möbius transformations, many types of novel temporal solitary-wave solutions of this nonlinear dynamical system are analytically found under some constraint conditions, such as the bright-bright, bright-dark, dark-dark, periodic-periodic, W-shaped, and rational wave solutions. In particular, we find that the localized rational-type solutions can exhibit both bright-bright and bright-dark wave profiles by choosing different families of free parameters. Moreover, we analyze the relationships among the group-velocity dispersion profiles, gain or loss distributions, external potentials, and inhomogeneous source profiles, which provide the necessary constraint conditions to control the emerging wave dynamics. Finally, a series of numerical simulations are performed to show the robustness to propagation of some of the analytically obtained solitary-wave solutions. The vast class of exact solutions of inhomogeneous coupled nonlinear Schrödinger equations with source terms might be used in the study of the soliton structures in twin-core optical fibers and two-component Bose-Einstein condensates.

Beschreibung: We study a linearly coupled Schrödinger system in ℝ N ( N ≤ 3). Assume that the potentials in the system are continuous functions satisfying suitable decay assumptions, but without any symmetry properties, and the parameters in the system satisfy some suitable restrictions. Using the Liapunov-Schmidt reduction methods two times and combining localized energy method, we prove that the problem has infinitely many positive synchronized solutions, which extends result Theorem 1.2 about nonlinearly coupled Schrödinger equations in Ao and Wei [Calculus Var. Partial Differ. Equations 51 , 761-798 (2014)] to our linearly coupled problem.

Beschreibung: In this paper, we study a class of nonlocal Schrödinger-Kirchhoff problems involving only continuous functions. Using a minimization argument and a quantitative deformation lemma, we find a least energy nodal (or sign-changing) solution to this problem. Moreover, when the problem presents symmetry, we show that it has infinitely many nontrivial solutions.

Beschreibung: We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra

Beschreibung: Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q .

Beschreibung: We consider L 2 -constraint minimizers of mass critical Hartree energy functionals in ℝ N with N ≥ 3. We prove that minimizers exist if and only if the parameter a 〉 0 satisfies a 〈 a ∗ = Q 2 2 , where Q is a positive radially symmetric ground state of Δ u − u + ∫ R N u ( y ) 2 x − y 2 d y u = 0 in ℝ N . The blow-up behavior of minimizers as a approaches a ∗ is also analyzed, for which all the mass concentrates at a global minimum point x 0 of the external potential V ( x ).

Beschreibung: We consider a two-dimensional (2D) jet by van der Waals gas streaming in parallel supersonic flow out of a duct into the atmosphere. We assume that the pressure p 0 of the oncoming uniform parallel flow is greater than the atmospheric pressure p A and belongs to ( p 1 e , p 2 i ) . Then at the corners at exit the oncoming flow expands in two symmetric jump-fan (jf) composite waves to the atmospheric pressure. These two jf composite waves interact and emerge as simple waves from their zone of penetration. We present a mathematical analysis of the interaction of the jf composite waves. To construct the flow in the interaction region, we consider a discontinuous Goursat problem for the 2D isentropic irrotational steady Euler equations. The existence of global piecewise C 1 solution to the discontinuous Goursat problem is obtained constructively.

Beschreibung: The 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.

Beschreibung: Simultaneous 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 half-line and relate them to the universal enveloping algebras of the Weyl-Heisenberg algebra and su (1, 1), respectively. The complete sub-structure 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.

Beschreibung: We consider the p -Laplacian problem − ε p Δ p u + V ( x ) u p − 2 u = f ( u ) , u ∈ W 1 , p ( R N ) , 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 so-called Ambrosetti-Rabinowitz condition are not required.

Beschreibung: It is shown that the initial value problem for an integrable Novikov system is well-posed 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 data-to-solution 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.

Beschreibung: Our discussion here mainly focuses on the formation of singularities for solutions to the N -dimensional Euler-Poisson equations with attractive forces, in radial symmetry. Motivated by the integration method of Yuen, we prove two blow-up results under the conditions that the solutions have compact radius R ( t ) or have no compact support restriction, which generalize the ones Yuen obtained in 2011 [M. W. Yuen, “Blowup for the C 1 solution of the Euler-Poisson equations of gaseous stars in R N ,” J. Math. Anal. Appl. 383 , 627-633 (2011)].

Beschreibung: In this paper, we provide a new method for establishing the blowup of C 2 solutions for the pressureless Euler-Poisson system with attractive forces for R N ( N ≥ 2) with ρ (0, x 0 ) 〉 0 and Ω 0 i j ( x 0 ) = 1 2 ∂ i u j ( 0 , x 0 ) − ∂ j u i ( 0 , x 0 ) = 0 at some point x 0 ∈ R N . By applying the generalized Hubble transformation   div   u ( t , x 0 ( t ) ) = N a ̇ ( t ) a ( t ) to a reduced Riccati differential inequality derived from the system, we simplify the inequality into the Emden equation a ̈ ( t ) = − λ a ( t ) N − 1 , a ( 0 ) = 1 , a ̇ ( 0 ) =   div   u ( 0 , x 0 ) N . Known results on its blowup set allow us to easily obtain the blowup conditions of the Euler-Poisson system.

Beschreibung: Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated.

Beschreibung: The paper focuses on studying the Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. First, the Hamilton equations of nonholonomic nonconservative systems on time scales are established, which is based on the Lagrange equations for nonholonomic systems on time scales. Then, based upon the quasi-invariance of Hamilton action of systems under the infinitesimal transformations with respect to the time and generalized coordinate on time scale, the Noether identity and the conserved quantity of nonholonomic nonconservative systems on time scales are obtained. Finally, an example is presented to illustrate the application of the results.

Beschreibung: General 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.

Beschreibung: The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional 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 singular-eigenfunction approach in flatland or two-dimensional space.

Beschreibung: The BPS Skyrme model is a model containing an SU (2)-valued scalar field, in which a Bogomol’nyi-type inequality can be satisfied by soliton solutions (skyrmions). In this model, the energy density of static configurations is the sum of the square of the topological charge density plus a potential. The topological charge density is nothing else but the pull-back of the Haar measure of the group SU (2) on the physical space by the field configuration. As a consequence, this energy expression has a high degree of symmetry: it is invariant to volume preserving diffeomorphisms both on physical space and on the target space. We demonstrate here that in the BPS Skyrme model such solutions exist that a fraction of its charge and energy densities is localised, and the remaining part can be far away, not interacting with the localised part.

Beschreibung: Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. There exist very few results for steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle for diffusion processes due to the technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loève expansion of complex-valued Ornstein-Uhlenbeck process.

Beschreibung: Using an appropriate norm on the space of entire functions, we extend to the complex plane the renormalization group method as developed by Bricmont et al. The method is based upon a multiscale approach that allows for a detailed description of the long time asymptotics of solutions to initial value problems. The time evolution equation considered here arises in the study of iterations of the block spin renormalization group transformation for the hierarchical N -vector model. We show that, for initial conditions belonging to a certain Fréchet space of entire functions of exponential type, the asymptotics is universal in the sense that it is dictated by the fixed point of a certain operator acting on the space of initial conditions.

Beschreibung: In this paper, we study the existence of positive solution for the following class of fractional elliptic equation ϵ 2 s ( − Δ ) s u + V ( z ) u = λ u q − 2 u + u 2 s ∗ − 2 u in R N , where ϵ , λ 〉 0 are positive parameters, q ∈ ( 2 , 2 s ∗ ) , 2 s ∗ = 2 N N − 2 s , N 〉 2 s , s ∈ ( 0 , 1 ) , ( − Δ ) s u is the fractional Laplacian, and V is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.

Beschreibung: We consider a system of equations governing the motion of a viscous, compressible, and heat conducting liquid-like fluid, with a general equation of state (EOS) of Mie-Grüneisen type. In addition, we suppose that the viscosity coefficients may decay to zero for large values of the temperature. We show the existence of global-in-time weak solution, derive a relative energy inequality, and compare the weak solutions with strong one emanating from the same initial data—the weak strong uniqueness property.

Beschreibung: In the present study, we are interested in the Davey-Stewartson equations (DSE) that model packets of surface and capillary-gravity waves. We focus on the elliptic-elliptic case, for which it is known that DSE may develop a finite-time singularity. We propose three systems of non-viscous regularization to the DSE in a variety of parameter regimes under which the finite-time blow-up of solutions to the DSE occurs. We establish the global well-posedness of the regularized systems for all initial data. The regularized systems, which are inspired by the α -models of turbulence and therefore are called the α -regularized DSE, are also viewed as unbounded, singularly perturbed DSE. Therefore, we also derive reduced systems of ordinary differential equations for the α -regularized DSE by using the modulation theory to investigate the mechanism with which the proposed non-viscous regularization prevents the formation of the singularities in the regularized DSE. This is a follow-up of the work [Cao et al. , Nonlinearity 21 , 879–898 (2008); Cao et al. , Numer. Funct. Anal. Optim. 30 , 46–69 (2009)] on the non-viscous α -regularization of the nonlinear Schrödinger equation.

Beschreibung: We study an important system of Schrödinger equations with linear and nonlinear couplings arising from Bose-Einstein condensates. We use the Nehari manifold to prove the existence of a ground state solution; moreover, we give the sign of the solutions depending on linear coupling; by using index theory and Nehari manifold, we prove that there exist infinitely many positive bound state solutions.

Beschreibung: We 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 non-linear dependence on the random parameters.

Beschreibung: 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 non-overlapping, orbitally labelled E ( k , l ) series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive l th series eigenvalues. Each of them is 2 l + 1-degenerate. 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 2 k ( d = 1). Likewise, the eigenfunctions ψ ( k ,0) ( d = 3) and ψ 2 k ( 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).

Beschreibung: The 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.

Beschreibung: The correspondence between local unitary equivalence of bipartite quantum states and simultaneous orthogonal equivalence is thoroughly investigated and strengthened. It is proved that local unitary equivalence can be studied through simultaneous similarity under projective orthogonal transformations, and four parametrization independent algorithms are proposed to judge when two density matrices on ℂ d 1 ⊗ ℂ d 2 are locally unitary equivalent in connection with trace identities, Kronecker pencils, Albert determinants and Smith normal forms.

Beschreibung: The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network and, more specifically, as a linear map from the input space to the output space. The quantum max-flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min-cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.

Beschreibung: In this paper, we establish the eigenvalue asymptotics for non-self-adjoint Dirac–Bessel operators on (0, 1) with arbitrary real angular momenta and square integrable potentials, which gives the first step for solution of the related inverse problem. The approach is based on a careful examination of the corresponding characteristic functions and their zero distribution.

Beschreibung: An adaptation of the Wentzel–Kramers–Brilluoin method in the deformation quantization formalism is presented with the aim to obtain an approximate technique of solving the eigenvalue problem for energy in the phase space quantum approach. A relationship between the phase σ ( r → ) of a wave function exp i ħ σ ( r → ) and its respective Wigner function is derived. Formulas to calculate the Wigner function of a product and of a superposition of wave functions are proposed. Properties of a Wigner function of interfering states are also investigated. Examples of this quasi–classical approximation in deformation quantization are analysed. A strict form of the Wigner function for states represented by tempered generalised functions has been derived. Wigner functions of unbound states in the Poeschl–Teller potential have been found.

Beschreibung: Quantum Stein’s lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states ( ρ or σ ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n ) identical copies of the state ( ρ ⊗ n or σ ⊗ n ) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability α n of erroneously inferring the state to be σ , the probability β n of erroneously inferring the state to be ρ decays exponentially in n , with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures, and quasi-free states of fermionic lattice gases.

Beschreibung: We consider two non-interacting infinite quantum spin chains immersed in a common thermal environment and undergoing a local dissipative dynamics of Lindblad type. We study the time evolution of collective mesoscopic quantum spin fluctuations that, unlike macroscopic mean-field observables, retain a quantum character in the thermodynamical limit. We show that the microscopic dissipative dynamics is able to entangle these mesoscopic degrees of freedom, through a purely mixing mechanism. Further, the behaviour of the dissipatively generated quantum correlations between the two chains is studied as a function of temperature and dissipation strength.

Beschreibung: In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

Beschreibung: A functional realization of the Lie algebra s l 3 , R as a Vessiot–Guldberg–Lie algebra of second order differential equation (SODE) Lie systems is proposed. It is shown that a minimal Vessiot–Guldberg–Lie algebra L V G is obtained from proper subalgebras of s l 3 , R for each of the SODE Lie systems of this type by particularization of one functional and two scalar parameters of the s l 3 , R -realization. The relation between the various Vessiot–Guldberg–Lie algebras by means of a limiting process in the scalar parameters further allows to define a notion of contraction of SODE Lie systems.

Beschreibung: We introduce a generalization of the Jaynes-Cummings model and study some of its properties. We obtain the energy spectrum and eigenfunctions of this model by using the tilting transformation and the squeezed number states of the one-dimensional harmonic oscillator. As physical applications, we connect this new model to two important and novelty problems: the relativistic parametric amplifier and the quantum simulation of a single trapped ion.

Beschreibung: We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in Euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.

Beschreibung: We study affine fusion with the adjoint representation. For simple Lie algebras, elementary and universal formulas determine the decomposition of a tensor product of an integrable highest-weight representation with the adjoint representation. Using the (refined) affine depth rule, we prove that equally striking results apply to adjoint affine fusion. For diagonal fusion, a coefficient equals the number of nonzero Dynkin labels of the relevant affine highest weight, minus 1. A nice lattice-polytope interpretation follows and allows the straightforward calculation of the genus-1 1-point adjoint Verlinde dimension, the adjoint affine fusion tadpole. Explicit formulas, (piecewise) polynomial in the level, are written for the adjoint tadpoles of all classical Lie algebras. We show that off-diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non-dominant representations.

Beschreibung: We study the constrained Ostrogradski-Hamilton framework for the equations of motion provided by mechanical systems described by second-order derivative actions with a linear dependence in the accelerations. We stress out the peculiar features provided by the surface terms arising for this type of theories and we discuss some important properties for this kind of actions in order to pave the way for the construction of a well defined quantum counterpart by means of canonical methods. In particular, we analyse in detail the constraint structure for these theories and its relation to the inherent conserved quantities where the associated energies together with a Noether charge may be identified. The constraint structure is fully analyzed without the introduction of auxiliary variables, as proposed in recent works involving higher order Lagrangians. Finally, we also provide some examples where our approach is explicitly applied and emphasize the way in which our original arrangement results in propitious for the Hamiltonian formulation of covariant field theories.

Beschreibung: Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincaré duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree k of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincaré duality for the new cohomology groups.

Beschreibung: We introduce the notion of a universal odd generalized Poisson superalgebra associated with an associative algebra A , by generalizing a construction made in the work of De Sole and Kac [Jpn. J. Math. 8 , 1–145 (2013)]. By making use of this notion we give a complete classification of simple linearly compact (generalized) n -Nambu-Poisson algebras over an algebraically closed field of characteristic zero.

Beschreibung: The higher-order superintegrability of the two-dimensional isotonic oscillator (noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator) with rational ratio of frequencies is directly related with the existence of some complex functions with interesting Poisson bracket properties. First the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi-Hamiltonian complex structure. In the second part several real symplectic structures are obtained and the properties of the recursion operators are studied.

Beschreibung: We study finite W -algebras associated to even regular (principal) nilpotent elements for the family of simple exceptional Lie superalgebras D (2, 1; α ) and for the universal central extension of

Beschreibung: We investigate the spectrum of the self-similar Laplacian, which generates the so-called “ pq random walk” on the integer half-line ℤ + . Using the method of spectral decimation, we prove that the spectral type of the Laplacian is singularly continuous whenever p ≠ 1 2 . This serves as a toy model for generating singularly continuous spectrum, which can be generalized to more complicated settings. We hope it will provide more insight into Fibonacci-type and other weakly self-similar models.

Beschreibung: In this paper, we study Lie super-bialgebra and quantization of the super Virasoro algebra, whose even part is the centerless twisted Heisenberg-Virasoro algebra. By using some Liu-Pei-Zhu’s results, we prove that all Lie super-bialgebra structures on the super Virasoro algebra are triangular coboundary. Furthermore, we quantize the super Virasoro algebra by the Drinfel’d twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.

Beschreibung: The paper is motivated by the importance of the rephasing invariance of the CKM (Cabibbo-Kobayashi-Maskawa) matrix observables. These observables appear in the discussion of the CP violation in the standard model (Jarlskog invariant) and also in the renormalization group equations for the quark Yukawa couplings. Our discussion is based on the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the main theorem we show that a general rephasing invariant monomial can be expressed as a product of at most five factors: four of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements. We also show that the imaginary part of any rephasing invariant monomial is proportional to the Jarlskog’s invariant J or is 0.

Beschreibung: In this work we study the dimensional reduction of smooth circle invariant Yang-Mills instantons defined on 4-manifolds which asymptotically become circle fibrations over hyperbolic 3-space. A suitable choice of the 4-manifold 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 3-space. We describe simple configurations and relate our results to the Jackiw-Nohl-Rebbi construction, for which we provide a geometric interpretation.

Beschreibung: We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii 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 Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ω c such that if Ω = Ω 0 /ε 4 with Ω 0 〉 Ω c , 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 Gross-Pitaevskii minimizer.

Beschreibung: Highest weight modules for U q ( g l 2 ̂ ) are endowed with a structure of modules for the quantum toriodal algebra U κ of type sl 2 . Using this, we define U κ actions on the space of vertex operators for irreducible highest weight U q ( g l 2 ̂ ) modules. Highest or lowest weight vectors of the thus obtained U κ modules are expressed in terms of an intertwiner for U q ( s l 2 ̂ ) modules and an extra boson. The submodules generated by these vectors are investigated.

Beschreibung: We show a fundamental limitation in the description of quantum many-body 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.

Beschreibung: An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g., Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova, C ̆ 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.

Beschreibung: In physics and in mathematics Z 2 n -gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the “scalar product” of the involved Z 2 n -degrees. The Z 2 n -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 Z 2 n -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 Z 2 • -supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n -fold vector bundle has a canonical “superization” to a Z 2 n -supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z 2 n -context.

Beschreibung: To 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 Lévy-Khintchine-type representation of its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-Lévy measure, we introduce a new infinitely divisible distribution for which we derive the characteristic function.

Beschreibung: In this work, we analytically study the Schrödinger equation for the (non-pure) dipolar ion potential V ( r ) = q / r + D cos θ / r 2 , in the case of 2D systems (systems in two-dimensional 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.