Regular Singularities Research Articles

Equivalence Between the Eigenvalue Problem of Non-Commutative Harmonic Oscillators and Existence of Holomorphic Solutions of Heun Differential Equations, Eigenstates Degeneration, and the Rabi Model

The initial aim of the present paper is to provide a complete description for the eigenvalue problem of the non-commutative harmonic oscillator (NcHO), which is given by a two-by-two system of paritypreserving ordinary differential operator [19], in terms of Heun's ordinary differential equations, the second order Fuchsian differential equations with four regular singularities in a complex domain. This description has been achieved for odd eigenfunctions in Ochiai [16] nicely but missing for even eigenfunctions up to now. As a by-product of this study, using the monodromy representation of Heun's equation, we prove that the multiplicity of the eigenvalue of the NcHO is at most two. Moreover, we give a condition for the existence of

Inverse problems for variable order differential operators with regular singularities on graphs

Abstract We study inverse spectral problems for ordinary differential equations with regular singularities on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.

No regularity singularities exist at points of general relativistic shock wave interaction between shocks from different characteristic families.

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C0,1 to C1,1 in a neighbourhood of points of shock wave collision in general relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier Proc. R. Soc. A publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Thus, our result implies that regularity singularities (a type of mild singularity introduced in our Proc. R. Soc. A paper) do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original Proc. R. Soc. A publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein-Euler equations remains open.

An analysis of the wave equation for the U(1)2 gauged supergravity black hole

We study the massless Klein–Gordon equation in the background of the most general rotating dyonic anti-de Sitter black hole in , supergravity in D = 4, originally presented by Chow and Compère (2014 Phys. Rev. D 89 065003). The angular part of the separable wave equation is of Heun type, while the radial part is a Fuchsian equation with five regular singularities. The radial equation is further analyzed and written in a specific form, which reveals the pole structure of the horizon equation, whose residua are expressed in terms of the surface gravities and angular velocities associated with the respective horizons. The near-horizon (near-)extremal limits of the solution are also studied, where the expected hidden conformal symmetry is revealed. Furthermore, we present the retarded Green’s functions for these limiting cases. We also comment on the generality of the charge-dependent parts of the metric parameters and address some further examples of limiting cases.

SYSTEMS OF DIFFERENTIAL EQUATIONS ON THE LINE WITH REGULAR SINGULARITIES

Tamkang University, Taiwan, ctshieh@mail.tku.edu.twNon-selfadjoint second order differential systems on the line having a non-integrable regular singularity are studied. We constructspecial fundamental systems of solutions with prescribed analytic and asymptotic properties. Asymptotics of the correspondingStockes multipliers is established.Key words: differential systems, singularity, spectral analysis.

Inverse problem for dirac system with singularities in interior points

We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.

Points of general relativistic shock wave interaction are ‘regularity singularities’ where space–time is not locally flat

We show that the regularity of the gravitational metric tensor in spherically symmetric space–times cannot be lifted from C 0,1 to C 1,1 within the class of C 1,1 coordinate transformations in a neighbourhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel9s theorem, which states that such coordinate transformations always exist in a neighbourhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of regularity singularity for perfect fluids evolving in space–time, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the space–time is not locally Minkowskian under any coordinate transformation. In particular, at regularity singularities, delta function sources in the second derivatives of the metric exist in all coordinate systems of the C 1,1 -atlas, but due to cancellation, the full Riemann curvature tensor remains supnorm bounded .

Pullback of regular singular stratified bundles and restriction to curves

A stratified bundle is a vector bundle which is a D-module. We show that regular singularity of stratified bundles on smooth varieties in positive characteristic is preserved by pullback and that regular singularity can be checked on curves, if the ground field is large enough.

Parametric Borel summability for some semilinear system of partial differential equations

In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \(n\) independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.

Semi Random Position Based Steganography for Resisting Statistical Steganalysis

Steganography is the branch of information hiding for secret communication. The simplest and widely used steganography is the LSB based approach due to its visual quality with high embedding capacity. However, LSB based steganography techniques are not secure against statistical steganalysis mainly χ2 attack and Regular Singular (RS) attack. These two staganalysis can easily estimate the hidden message length. This work propose a LSB based steganography technique where first a location is obtained randomly based on the bit pattern (except LSB) of a cover pixel using linear probing and embed a secret bit into LSB. This technique makes the stego-image completely secure against both χ2 attack and RS attack.

Lattice Green functions: the seven-dimensional face-centred cubic lattice

We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d = 7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional fcc lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in . This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a ‘decomposition’ of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as d−2 as the dimension d goes to infinity.

Regular singular stratified bundles and tame ramification

Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. $\mathcal {O}_X$-coherent $\mathscr {D}_{X/k}$-modules) on $X$ without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional $k$-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to $X$.

Microlocal Cauchy problem for distribution solutions to systems with regular singularities

For systems of analytic linear differential equation with regular singularities in the sense of Kashiwara–Oshima, the microlocal Cauchy problem for distribution solutions in the framework of algebraic analysis is formulated. Moreover, under a hyperbolicity condition the solvability theorem is obtained.

Pointwise Lower Bounds for Solutions of Semilinear Elliptic Equations and Applications

AbstractWe consider the semilinear elliptic problem −Δu = f (x, u), posed in a smooth bounded domain Ω of ℝNwith Dirichiel data u|∂Ω = 0, where f : Ω × [0, αf) → ℝ+(0 < αf≤ +∞) is a function of appropriate regularity which blows up at αf. We give pointwise lower bounds for the supersolutions under some appropriate conditions on f , and apply them to eigenvalue problem −Δu = λ f (x, u), by giving upper and lower bounds for the extremal parameter λ∗ and the extremal solution u∗. To demonstrate the sharpness of our results, we consider the eigenvalue problem −Δu = λ f (up) (p ≥ 1) with Dirichlet boundary condition, and show that for every increasing, convex and superlinear C2function f: ℝ+→ℝ+with, where ψΩ is the maximum of the torsion function of Ω. Also, we consider the eigenvalue problem −Δu = λρ(x) f (u), where f is either a regular singularity such as f (u) = eu, or a singular one such asand give explicit estimates on λ∗ and u∗, that improve and extend several results in the literature, by Payne[17], Sperb [21], Brezis-Vasquez [3], Guo-Pan-Ward [11], Ghoussoub-Guo [10], Cowan-Ghoussoub [6], and others.

Addentum to: Conformal invariance and near-extreme rotating AdS black holes

We obtained retarded Green's functions for massless scalar fields in the background of near-extreme, near-horizon rotating charged black hole of five-dimensional minimal gauged supergravity in Phys. Rev. D84, 044018 (2011). For general nonextreme black holes, we also derived the radial part of the massless Klein-Gordon equation with zero azimuthal-angle eigenvalues, and showed that it is a general Heun's equation with a regular singularity at each horizon $u_k$ ($k=1,2,3$) and at infinity. We derived explicitly that the residuum of a pole at each $u_k$ is associated with the surface gravity there. In this addendum, probing regular singularities at each $u_k$ we complete the derivation of the full radial equation with nonzero azimuthal-angle eigenvalues. The residua now include modifications by the angular velocities at respective horizons. This result completes the analysis of the wave equation for the massless Klein-Gordon equation for the general rotating charged black hole of five-dimensional minimal gauged supergravity.

Inverse problems for differential systems on graphs with regular singularities

Inverse problems for differential systems on graphs with regular singularities

Wave equation for the Wu black hole

Wu black hole is the most general solution of maximally supersymmetric gauged supergravity in D=5, containing $U(1)^{3}$ gauge symmetry. We study the separability of the massless Klein-Gordon equation and probe its singularities for a general stationary, axisymmetric metric with orthogonal transitivity, and apply the results to the Wu black hole solution. We start with the zero azimuthal-angle eigenvalues in the scalar field Ansatz and find that the residuum of a pole in the radial equation is associated with the surface gravity calculated at this horizon. We then generalize our calculations to nonzero azimuthal eigenvalues and probing each horizon singularity, we show that the residua of the singularities for each horizon are in general associated with a specific combination of the surface gravity and the angular velocities at the associated horizon. It turns out that for the Wu black hole both the radial and angular equations are general Heun's equations with four regular singularities.

Location-Based Image Steganography

The objective of steganography is to conceal the presence of a secret communication. Nevertheless, with the development of steganalytic attack, also known as steganalysis, there are many statistical methods to estimate the presence and ratio of secret messages. The chi square (χ2) attack and regular-singular (RS) attack are two well recognized and widely applied statistical steganalysis schemes. In this paper, a new image steganography scheme with location selection is proposed to resist statistical steganalysis. The experiment results indicate that the proposed method can resist both χ2 attacks as well as RS attacks.

ON MULTISERVER RETRIAL QUEUES: HISTORY, OKUBO-TYPE HYPERGEOMETRIC SYSTEMS AND MATRIX CONTINUED-FRACTIONS

In this paper, we study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiserver M/M/∞ Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a"minimal/nondominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (a) and (b), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest.The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: (i) the irregular singularity at 0; (ii) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and (iii) the point 1, whose Taylor series coefficients are the factorial moments.

Geometric Phase, Curvature and the Monodromy Group

The geometric phase requires the multivaluedness of solutions to Fuchsian second-order equations. The angle, or its complement, is given by half the area of a spherical triangle in the case of three singular points, or half the area of a lune in the case of two singular points. Both are fundamental regions where the automorphic function takes a value only once, and a linear-fractional transformation tessellates the plane in replicas of the fundamental region. The condition that the homologues of the poles, representing vertices, be angles places restrictions on quantum numbers which are no longer integers, for, otherwise, the phase factors would become unity. Restriction must be made to regular singular points for only then will solutions to the differential equation be rational functions so that the covering group will be cyclic and the covering space be a ‘spiral staircase’. Many of the equations of mathematical physics, with essential singularities, become Fuchsian differential equations, with regular singularities, at zero kinetic energy. Examples of geometric phase include the phasor, the Pancharatnam phase of beams of polarized light in different states, the Aharanov-Bohm phase, and angular momentum with centripetal ‘attraction’. In the latter example, the phase is one-half the area of the lune, which disappears when the pole at infinity becomes an essential singularity thereby recovering the Schrodinger equation. The behavior of an automorphic function at a limit point on the boundary is analogous to the confluence of two regular singularities in a linear second-order differential equation to produce an essential singularity at infinity. ∗ info@bernardhlavenda.com; www.bernardhlavenda.com 1 ar X iv :1 41 1. 56 03 v1 [ ph ys ic s. ge nph ] 1 5 M ay 2 01 3