Generalized eigenvectors are key tools in the theory of rigged Hilbert spaces. Let H be a Hilbert space and let Φ be a dense subspace of H. Let A be a densely defined linear operator in H such that Φ ⊂ DA and AΦ ⊂ Φ. The generalized eigenvectors of A are the eigenvectors of the algebraic dual of A |Φ. In the case where Φ is endowed with a topology τ finer than the norm topology inherited from H, generalized

We show that the approximate quasi-orthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of the measure of orthogonality in terms of Choi matrices and Kraus operators for completely positive maps. We present examples drawn from different areas of quantum

When solving Einstein's equations for an isolated system of masses, V. Fock introduced a harmonic reference frame and obtained an unambiguous solution. Further, he concluded that there existed a harmonic reference frame which was determined uniquely apart from a Lorentz transformation if suitable supplementary conditions were imposed. It is known that wave equations keep the same form under Lorentz

In this article, we consider and study the order topology in Minkowski spacetime of the special theory of relativity, i.e. the finest locally convex topology τ on spacetime for which every order bounded subset of spacetime is τ-bounded. This order topology that is introduced into spacetime as an ordered vector space proves to be Hausdorff and differs from Zeeman's order topology. Applying the order

In an attempt to generalize Hamilton's principle, an action functional is proposed which, unlike the standard version of the principle, accounts properly for all initial data and the possible presence of dissipation. To this end, the convolution is used instead of the L2 inner product so as to eliminate the undesirable end temporal condition of Hamilton's principle. Also, fractional derivatives are

Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra

In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations.

In this paper, we propose a stochastic version of the Hawking–Penrose black hole model. We describe the dynamics of the stochastic model as a continuous-time Markov jump process of quanta out and in the black hole. The average of the random process satisfies the deterministic picture accepted in the physical literature. Assuming that the number of quanta is finite the proposed Markov process consists

A set of gauge transformations of a relativistic field of quantum harmonic oscillators is studied in a mathematically rigorous manner. Each wave function in the domain of the number operator of a single oscillator generates a Fréchet-differentiable field of wave functions. Starting from a coherent wave function one obtains a two-dimensional differentiable manifold of coherent vector states. As an illustration

The dynamical rules in an auxiliary stochastic process that generate the biased ensemble of rare events are nonlocal. For the systems with one type of particles, it is shown that one can find special cases for which the transition rates of the auxiliary stochastic generator can be local. In this paper, we investigate this possibility for a system of classical hard-core particles with more than one

To capture a multidimensional consistency feature of integrable systems in terms of geometry, we give a condition called geodesic compatibility implying the existence of integrals in involution of the geodesic flow. The geodesic compatibility condition is constructed from a concrete example namely the integrable Calogero's goldfish system through the Poisson structure and the variational principle

In this paper we propose and discuss the fractional diffusion equation with time-dependent diffusion coefficient, considering the Hilfer-type and Weyl fractional derivatives in the time-variable and space-variable, respectively. We apply the similarity method and Mellin transform methodology to find an explicit solution in terms of Fox H-function. We illustrate graphically the diffusive behaviour described

Let ℌ be a complex Hilbert space with dim ℌ ≥ 3 and ℬ(ℋ) the algebra of all bounded linear operators on ℌ. The effect algebra E (ℌ) on ℌ is the set of all positive contractions in ℬ(ℋ). We consider the automorphism of E(ℌ) with respect to convex sequential product ox on E(ℌ) for some λ ∈ [0, 1] defined by A∘λB=λA12BA12+(1−λ)B12AB12,∀A,B∈E(ℋ). We show that an automorphism of E (ℌ) with respect to convex

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside ℝ × ℝ3 = ℝ4, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or ‘linked’ with each other, and they are closely

We formulate the dynamics of an infinitely extended open dissipative quantum system, Σ, in the Schroedinger picture. The generic model on which this is based comprises a C★-algebra, A, of observables, a folium, ℱ, of states on this algebra and a one-parameter semigroup, τ, of linear transformations of ℱ that represents its dynamics and is given by a natural infinite-volume limit of the corresponding

Nonlinear coherent states or f -coherent states are one of the important class of quantum states of light attached to the f -deformed oscillators. They have been introduced in a pioneering work by Manko et al. and have been realized physically as the stationary states of the centre of mass motion of a trapped ion by de Matos Filho et al. To gain insight into the effectiveness of these states in quantum

In this paper the space-fractional Schrödinger equations with singular potentials are studied. Delta like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of ‘weakened’ solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particles

Solutions of the classical and nonlocal Gross–Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second-order nonisospectral Ablowitz–Kaup–Newell–Segur system and reduction technique of double Wronskians. Solutions of the classical GP equation show typical space-time localized characteristics. An interesting dynamics, solitons carrying an oscillating wave, are

In this paper we consider an Ising model with nearest-neighbour interactions with spin space [0, 1] on a Cayley tree. We present a sufficient condition under which the Ising model has a unique splitting Gibbs measure.

The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, GAP(ρβ), for a thermal density operator ρβ at inverse temperature β. More generally, GAP(ρ) is a probability measure on the unit sphere in Hilbert space for any density operator ρ (i.e. a positive operator with trace 1). In this note, we collect the mathematical details

In this paper we consider an SOS model with nearest-neighbour interactions and with three spin values, on a Cayley tree of k ≥ 3. For this model, using the well-known translation-invariant Gibbs measures, some non-translation-invariant Gibbs measures are constructed

A method to derive the matrix spectral problems of the Błaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Błaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Błaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed

We will prove that there is a direct relationship between the Poincaré subgroup of translations, and the group of tetrad transformations LB1 introduced in a previous paper. LB1 is the group composed of SO(1, 1) plus two kinds of discrete transformations. Translations have been extensively studied under the scope of gauge theories. By using the geometric structures built to prove this elementary result

A ℤ2 × ℤ2-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the double-graded quantum superplane. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural

In this work, two space-time fractional nonlinear systems named Drinfeld–Sokolov–Satsuma–Hirota system and Broer Kaup system are considered with fractional derivatives in Riemann–Liouville type. The symmetry approach and power series expansion technique are applied to derive the explicit solutions of both the systems. In addition, the nontrivial conserved vectors are reported using the nonlinear self-adjointness

Topological structure of translation-invariant noncommutative Yang–Mills theories are studied by means of a cohomology theory, the so-called ⋆-cohomology, which plays an intermediate role between de Rham and cyclic (co)homology theory for noncommutative algebras and gives rise to a cohomological formulation comparable to Seiberg–Witten map.

We study hypergraphs which represent finite quantum event structures. We contribute to results of graph theory, regarding bounds on the number of edges, given the number of vertices. We develop a missing one for 3-graphs of girth 4. As an application of the graph-theoretical approach to quantum structures, we show that the smallest orthoalgebra with an empty state space has 10 atoms. Optimized constructions

The geometric phase that appears in the effects of Aharonov–Bohm type is interpreted in the frame of Deligne's version of the Riemann–Hilbert correspondence. We extend also the concept of flat gauge field to B-branes on a complex manifold X, so that such a field on a B-brane turns it into an object of the category of constructible sheaves on X.

In this paper we start the investigation of a metric-related weakening of positive expansiveness. We will show some of its fundamental properties and relation to notions such as weak positive expansiveness and cover expansiveness. Finally, we will see how to use the Bing–Hanner modification to produce examples of positively subexpansive dynamical systems with nonmetrizable phase space.

We prove a local in time existence theorem to a Cauchy problem for the coupled Yang–Mills–Boltzmann system in Bianchi type 1 space-time background.

In this work, we define a spinor covariant derivative for degenerate manifolds with 4-dimensions. To perform this, we have found the principal bundle by using a degenerate spin group. Then, we benefit from a covering map to establish a relationship between the local connection forms of principal bundles. After that, we define a covariant derivative on a degenerate spinor bundle which is an associated

In this study, nonlocal phenomena in quantum mechanics are investigated by making use of fractional calculus. in this context, fractional creation and annihilation operators are introduced and quantum mechanical harmonic oscillator has been generalized as an important tool in quantum field theory. Therefore wave functions and energy eigenvalues of harmonic oscillator are obtained with respect to the

The classical model of the spinning particle is analyzed in detail in two versions – with single spinor and two spinors put on the trajectory. Equations of motion of the first version are easily solvable. The system with two spinors becomes nonlinear. Nevertheless, the equations of motion are analyzed in detail and solved analytically and numerically. In either case the trajectories are illustrated

Exactly-solvable confined model of the nonrelativistic quantum harmonic oscillator is proposed. Its position-dependent effective mass Hamiltonian is defined via the von Roos kinetic energy operator. The confinement effect to harmonic oscillator potential is included as a result of certain behaviour of the position-dependent effective mass. The corresponding Schrödinger equation in the canonical approach

The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and ( D, ℰ)-quasi bases for two fixed biorthogonal sequences {ϕn} and {Ψn}. In a previous paper, we have studied the order structure of the set Cϕ of all constructing operators for a generalized Riesz system {ϕn}, and furthermore we have shown that

We define the center C(A) of a d0-algebra A as a set of self-mappings on A. The center can be regarded as the set of all possible d0-subdirect factors of A which are subalgebras of A. we show that C(A) is always a Boolean algebra. we also show that C(A) admits an embedding in the power Aκ, where κ is the cofinality of A. In case κ = 1 (i.e. A is a D-lattice) we reobtain the well-known fact that C(A)

In this work we applied the Dirac-Bergmann procedure to establish the Dirac brackets, which have regard to interaction for the light-front Yukawa model in D = 1 + 3 dimensions. We made use of a simple matrix equation leading to solution of the set task, wherein the main problem was to calculate the inverse matrix to the array composed of the constraints for enabled interaction. The proposed device

In this paper we study both analytical and combinatorial properties of solutions of the eigenproblem for the Heisenberg s-1/2 model for two deviations. Our analysis uses Chebyshev polynomials, inverse Bethe Ansatz, winding numbers and rigged string configurations. We show some combinatorial aspects of strings in a geometric way. We discuss some exceptions from the connection between the combinatorial

In classical information theory the conditional probability uniquely defines a classical channel. Introducing a quantum analogue of conditional probability we analyze the mutual information between input and output states of a given quantum channel when the compound state is fully quantum-quantum and classical-quantum. We introduce an analogue of the quantum discord of the channel which measures the

Wave equations on flat Minkowski spacetimes are generalized to curved spacetimes. We present two candidates for generalized wave equation, discuss their legitimacy and consider their applications to Klein–Gordon equation.

In this paper we extend the result of [5] to the quantum version. We present 11 examples of f's and discuss f-deformed quantum mechanics in one dimension for each f which possesses the f-deformed translational symmetry.

Formula for the size of the scatterer is derived explicitly in terms of the scattering amplitude corresponding to this scatterer. By the scatterer either a bounded obstacle D or the support of the compactly supported potential is meant.

Controlled time-decaying harmonic potentials decelerate the velocity of charged particles; however, the particles are never trapped by the harmonic potentials. This physical phenomenon changes the threshold between the short-range and long-range classes of the potential through physical wave operators. We herein report that the threshold is 1/(1 − λ) for some 0 ≤ λ < 1/2, as determined using the mass

We introduce a class of novel ℤ2 × ℤ2-graded color superalgebras of infinite dimension. It is done by realizing each member of the class in the universal enveloping algebra of a Lie superalgebra which is a module extension of the Virasoro algebra. Then the complete classification of central extensions of the ℤ2 × ℤ2-graded color superalgebras is presented. It turns out that infinitely many members

Theoretical study of superluminal sources of electromagnetic radiation boosted after the discovery of Cherenkov–Vavilov radiation. Later, the way to create fictitious sources moving superluminally was suggested. Different approaches have been proposed for the research of the distribution of the potential and the fields radiated by the superluminally moving charges. The simplest idealized cases of uniform

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras PG(a,b) of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over PG(a,b) for all complex numbers

The geometric framework of Double Field Theory (DFT) can be constructed on a para-Hermitian manifold. The canonical, generalized-metric connections and the global expression of the corresponding covariant derivative, a generalization of the kinematical structure of DFT, generalized curvature, a corresponding generalized Lie derivative for the Leibniz algebroid on the tangent bundle are constructed

We derive the Simon–Wolff localization criterion from the spectral averaging using an intuitive measure-theoretic lemma.

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras

A two-dimensional observable is a special kind of a σ-homomorphism defined on the Borel σ-algebra of the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra. A two-dimensional spectral resolution is a mapping defined on the real plane with values in a σ-complete MV-algebra or in a monotone σ-complete effect algebra which has properties similar to a two-dimensional

We prove, in this paper, the well-posedness and nonnegativity of the regular solution of the relativistic Boltzmann equation in the presence of a given electromagnetic field, taking as background a Lorentzian space-time which is of type Bianchi I with locally rotational symmetry.

Complex phase factors are viewed not only as redundancies of the quantum formalism but instead as remnants of unitary transformations under which the probabilistic properties of observables are invariant. it is postulated that a quantum observable corresponds to a unitary representation of an abelian Lie group, the irreducible subrepresentations of which correspond to the observable's outcomes. It

The purpose of this article is to give a result of localization in position space of the photons in the ground state for a Hamiltonian describing nuclear magnetic resonance in quantum electrodynamics. We prove that |x|−5 is an upper bound of the photon number density as |x| goes to infinity. We also obtain the asymptotic behaviour of the photon number at infinity and prove in particular that it is

In this paper we consider magnetic Schrödinger operators on the two-dimensional unit disk with a radially symmetric magnetic field which explodes to infinity at the boundary. We prove a bound for the eigenvalue moments and a bound for the number of negative eigenvalues for such operators.

We discuss a few models of quantum universal enveloping algebra Uq(sl(2)) from the q-special matrix functions point of view. These models are in terms of q-dilation operators and matrix q-difference-dilation operators and are connected through a matrix q-integral transformation. In the process, we find new basic matrix function identities involving one and two variable q-special matrix functions.

In this paper we introduce and study three classes of fractional periodic processes. An application to ring polymers is investigated. We obtain closed analytic expressions for the form factors, the Debye functions and their asymptotic decay. The relation between the end-to-halftime and radius of gyration is computed for these classes of periodic processes.

The results for Fisher information of the Morse oscillator have thus far been reported [8] for only the ground state of the system. We make use of the wave functions in [21] to provide a general method for computing Fisher-information values for both bound and excited states of the potential. In order to visualize the effects of anharmonicity, the results of the position- and momentum-space Fisher

We present an application of the Schur–Weyl duality in the one-dimensional Hubbard model in the case of a half-filled system of any number of atoms. We replace the actions of the dual symmetric and unitary groups in the whole 4N-dimensional Hilbert space by the actions of the dual groups in the spin and pseudo-spin spaces. The calculations significantly reduce the dimension of the eigenproblem of the

Based on the general deformation called an f-deformation, the algebraic structure for the f-deformation is investigated. As an example, f-deformed mechanics in one dimension is investigated. The relation between the f-deformed mechanics and the mechanics in a curved space is also investigated.

In this note we obtain sharp Strichartz estimates for the Schrödinger equation associated with the twisted Laplacian on ℂn ≅ ℝ2n. The initial data will be considered in suitable Sobolev spaces associated to the twisted Laplacian.