Dirac particle in generalized Pöschl–Teller field including a Coulomb‐like tensor coupling: super‐symmetric solution

In the presence of pseudo‐spin (p‐spin) and spin symmetries, we use the super‐symmetric formalism to solve the Dirac equation with the generalized Pöschl–Teller potential including the Coulomb‐like tensor interaction with any arbitrary spin‐orbit quantum number κ.. Using the Greene–Aldrich usual approximation scheme to deal with pseudo‐centrifugal or centrifugal rotational kinetic energy l (l + 1) ∕ r² or , we obtain the Pseudo‐spin and spin‐symmetric energy eigenvalue equation and the normalized upper and lower components of the radial wave functions in closed form. The presence of the tensor coupling interaction removes the degeneracy in the p‐spin and spin doublets. Copyright © 2013 John Wiley & Sons, Ltd.     

Spontaneous fermion creation in the coulomb field and Aharonov-Bohm potential in 2+1 dimensions

We find exact solutions of the Dirac equation and the fermion energy spectrum in the Coulomb (vector and scalar) potential and Aharonov-Bohm potential in 2+1 dimensions taking the particle spin into account. We describe the fermion spin using the two-component Dirac equation with the additional (spin) parameter introduced by Hagen. We consider the effect of creation of fermion pairs from the vacuum by a strong Coulomb field in the Aharonov-Bohm potential in 2+1 dimensions. We obtain transcendental equations implicitly determining the electron energy spectrum near the boundary of the lower energy continuum and the critical charge. We numerically solve the equation for the critical charge. We show that for relatively weak magnetic flows, the critical charge decreases (compared with the case with no magnetic field) if the energy of interaction of the electron spin magnetic moment with the magnetic field is negative and increases if this energy is positive. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 250–262, February, 2009.     

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the (3+1)-dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov-Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases , the constant mass and the non-relativistic limits are briefly investigated.     

Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

For a connected graph G, the rth extraconnectivity κ_r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ₀(G) and κ₁(G), respectively. The minimum r-tree degree of G, denoted by ξ_r(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ₁(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κ_r-optimal, if κ_r(G)?ξ_r(G). In this paper, we present some sufficient conditions that guarantee κ_r(G)?ξ_r(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1.     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

Galois stratification over Frobenius fields

New first-order conformally covariant differential operators P_k on spinor-k-forms, i.e., tensor products of contravariant spinors with k-forms, in an arbitrary n-dimensional pseudo-Riemannian spin manifold, are introduced. This provides a series of generalizations of the Dirac operator $\text{?}\text{?}$, in analogy with the series of generalizations (introduced by the author in [1]) of the Maxwell operator and the conformally covariant Laplacian $\text{□}$ on functions. In particular, new intertwining operators for representations of SU(2, 2) and SO(p + 1, q + 1) are found. Related nonlinear covariant operators are also introduced, and mixed nonlinear covariant systems are obtained by coupling to the Yang-Mills-Higgs-Dirac system in dimension 4. The spinor-form bundle is isomorphic with E⁽³⁾ = E ? E ? E, where E is the spin bundle, and the P_k give a covariant operator on sections of E⁽³⁾. This is generalized to a covariant operator on E^{(2l + 1)}. The relation of powers of these operators to higher-order covariant operators on lower spin bundles (analogous to the relation between $\text{?}\text{?}$ and $\text{□}$) is discussed.     

Dirac and Plateau billiards in domains with corners

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C ²-smooth Riemannian metrics g on a smooth manifold X, such that scal _g (x) ≥ κ(x), is closed under C ⁰-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

Spectral analysis of a Dirac operator with a meromorphic potential

We consider an operator Q(V) of Dirac type with a meromorphic potential given in terms of a function V of the form V(z)=λV₁(z)+μV₂(z), z∈C?{0}, where V₁ is a complex polynomial of 1/z, V₂ is a polynomial of z, and λ and μ are nonzero complex parameters. The operator Q(V) acts in the Hilbert space L²(R²;C⁴)=^⊕4L²(R²). The main results we prove include: (i) the (essential) self-adjointness of Q(V); (ii) the pure discreteness of the spectrum of Q(V); (iii) if V₁(z)=z−p and 4?degV₂?p+2, then kerQ(V)≠{0} and dimkerQ(V) is independent of (λ,μ) and lower order terms of ∂V₂/∂z; (iv) a trace formula for dimkerQ(V).     

Spectral stability of the Coulomb-Dirac Hamiltonian with anomalous magnetic moment

We show that the point spectrum of the standard Coulomb-Dirac operator H₀ is the limit of the point spectrum of the Dirac operator with anomalous magnetic moment H_a as the anomaly parameter tends to 0. For negative angular momentum quantum number κ, this holds for all Coulomb coupling constants c for which H₀ has a distinguished self-adjoint realisation. For positive κ, however, there are some exceptional values for c, and in general an index shift between the eigenvalues of H₀ and the limits of eigenvalues of H_a appears, accompanied with additional oscillations of the eigenfunctions of H_a very close to the origin.     

The Green Ring of Drinfeld Double D(H 4)

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double D(H ₄) of Sweedler’s four-dimensional Hopf algebra H ₄. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over D(H ₄). Then we describe the structure of the Green ring r(D(H ₄)) of D(H ₄) and show that r(D(H ₄)) is generated, as a ring, by infinitely many elements subject to a family of relations.     

Operateurs essentiellement reguliers dans les espaces de Banach

Given a closed operatorA acting in a Banach spaceX, we define the regular (respectively the essentialy regular) spectrum σ _r (A) (respectively σ _e,r (A)) ofA. We prove that σ _r (A) and σ _e,r (A) are a closed subsets of the classical spectrum σ(A) ofA. Morever ifA is bounded we prove that σ _r (A) and σ _e,r (A)) satisfies the spectral mapping theorem.     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].     

Arithmetic progressions in multiplicative groups of finite fields

Let G be a multiplicative subgroup of the prime field F _p of size |G| > p^1?κ and r an arbitrarily fixed positive integer. Assuming κ = κ(r) > 0 and p large enough, it is shown that any proportional subset A ? G contains non-trivial arithmetic progressions of length r. The main ingredient is the Szemerédi–Green–Tao theorem.     

Maximally edge-connected and vertex-connected graphs and digraphs: A survey

Let D be a graph or a digraph. If δ(D) is the minimum degree, λ(D) the edge-connectivity and κ(D) the vertex-connectivity, then κ(D)?λ(D)?δ(D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if λ(D)=δ(D) and maximally vertex-connected if κ(D)=δ(D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively.     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Almost Cosymplectic and Almost Kenmotsu (κ, μ, ν)-Spaces

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.