Irregular Singular Point Research Articles

Behavior of a vacuum and naked singularity under a smooth gauge function in Lyra geometry

Lyra geometry is a conformal geometry that originated from Weyl geometry. In this article, we derive the exterior field equation under a spherically symmetric gauge function x0(r) and metric in Lyra geometry. When we impose a specific form of the gauge function x0(r), the radial differential equation of the metric component g00 will possess an irregular singular point (ISP) at r = 0. Moreover, we can apply the method of dominant balance to get the asymptotic behavior of the new space–time solution. The significance of this work is that we can use a series of smooth gauge functions x0(r) to modulate the degree of divergence of the singularity at r = 0, which will become a naked singularity under certain conditions. Furthermore, we investigate the physical meaning of this novel behavior of space–time in Lyra geometry and find out that no spaceship with finite integrated acceleration can arrive at this singularity at r = 0. The physical meaning of the gauge function and integrability is also discussed.

Linear differential equations on the Riemann sphere and representations of quivers

Our interest in this paper is a generalization of the additive Deligne-Simpson problem which is originally defined for Fuchsian differential equations on the Riemann sphere. We shall extend this problem to differential equations having an arbitrary number of unramified irregular singular points and determine the existence of solutions of the generalized additive Deligne-Simpson problems. Moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the Riemann sphere. Namely, open embedding of the moduli spaces into quiver varieties is given and the non-emptiness condition of the moduli spaces is determined. Furthermore the connectedness of the moduli spaces is shown.

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in ϵ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

On the existence of monodromies for the Rabi model

We discuss the existence of monodromies associated with the singular points of the eigenvalue problem for the Rabi model. The complete control of the full monodromy data requires the taming of the Stokes phenomenon associated with the unique irregular singular point. The monodromy data, in particular, the composite monodromy, are written in terms of the parameters of the model via the isomonodromy method and the τ function of the Painlevé V. These data provide a systematic way to obtain the quantized spectrum of the Rabi model.

Quasinormal modes of extremal black holes

The continued fraction method (also known as Leaver's method) is one of the most effective techniques used to determine the quasinormal modes of a black hole. For extremal black holes, however, the method does not work (since, in such a case, the event horizon is an irregular singular point of the associated wave equation). Fortunately, there exists a modified version of the method, devised by Onozawa et al. [Phys. Rev. D 53, 7033 (1996)], which works for neutral massless fields around an extremal Reissner-Nordstrom black hole. In this paper, we generalize the ideas of Onozawa et al. to charged massless perturbations around an extremal Reissner-Nordstrom black hole and to neutral massless perturbations around an extremal Kerr black hole. Similarities and differences between the results of the original continued fraction method for near extremal black holes and the results of the new continued fraction method for extremal black holes are discussed.

Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlevé equations, using expansions of irregular conformal blocks at an irregular singular point.

Reccurent sequenses in solving the Schrödinger equation

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

An exactly solvable system from quantum optics

We investigate a generalisation of the Rabi system in the Bargmann–Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in terms of differential Galois group of the system is also given.

Holomorphic solutions of the double confluent Heun equation associated with the RSJ model of the Josephson junction

This work is a continuation of research on a first-order nonlinear differential equation applied in the overshunted model of the Josephson junction. The approach is based on the relation between this equation and the double confluent Heun equation, which is a second-order linear homogeneous equation with two irregular singular points. We describe the conditions on the equation parameters under which its general solution is an analytic function on the Riemann sphere except at 0 and ∞. We construct an explicit basis of the solution space. One of the functions in this basis is regular everywhere except 0, and the other is regular everywhere except ∞. We show that in the framework of the RSJ model of Josephson junction dynamics, the described situation corresponds to the condition that the Shapiro step vanishes if all the solutions of the double confluent Heun equation are single-valued on the Riemann sphere without 0 and ∞.

General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue NG (H)/H.

Partially invariant solutions in gas dynamics and implicit equations

This paper studies a nonbarochronic, regular, partially invariant solution (submodel) of rank one and defect two to the equations of gas dynamics which describes spatial unsteady gas motion. The equations of gas dynamics are reduced to an implicit ordinary differential equation of the first order for an auxiliary function and to an integrable system. A complete classification of the irregular singular points of the key equation according to a parameter characterizing the gas flow is given, and transformations of the irregular singular points with variation in the parameter are obtained. Qualitative properties of the solution are investigated and physically interpreted in terms of gas motion. It is shown that there are two modes of motion, one of which is supersonic, and in the second modes, a continuous transition through the speed of sound is possible.

Moduli Spaces of Irregular Singular Connections

In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the condition that a connection has regular semisimple leading term to connections with non-integer slope. Finally, we construct a symplectic moduli space of meromorphic connections on the projective line that contain a regular stratum at each singular point.

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x ( n ) + a 1 x ( n − 1 ) + ⋯ + a n x = 0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients a j = a j ( t ; λ ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara–Levelt–Turritin theorem about formal reduction of a linear differential system near an irregular singular point t = 0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.

On the well-posedness of a weighted boundary-value problem for systems of ODEs with singularity

We consider a system of linear second-order ODEs with an irregular singular point. The matrix that relates to the equation of this system is nonnormal. That implies some complications with respect to the statement of the weighted boundary-value condition at the singular point. We propose here a well-posed weighted boundary-value problem to the considered system of ODEs and prove the existence and uniqueness of its solution.

Isomonodromic confluence of singular points

We consider the problem of confluence of singular points under isomonodromic deformations of linear systems. We prove that a system with irregular singular points is a result of isomonodromic confluence of singular points with minimal Poincare ranks, i.e., of singular points whose Poincare rank does not decrease under gauge transformations.

Painlevé V and time-dependent Jacobi polynomials

In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w0(x) (supported on or subsets of ) by w0(x) e−tx. It is a well-known fact that under such a deformation the recurrence coefficients denoted as αn and βn evolve in t according to the Toda equations, giving rise to the time-dependent orthogonal polynomials and time-dependent determinants, using Sogo's terminology. If w0 is the normal density , or the gamma density xα e−x, , α > −1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w0 is the beta density (1 − x)α(1 + x)β, x ∊ [ − 1, 1], α, β > −1, the resulting ‘time-dependent’ Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm–Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painlevé V. In particular we show that , where is the coefficient of zn−1 of the monic orthogonal polynomials associated with the ‘time-dependent’ Jacobi weight, satisfies, up to a translation in t, the Jimbo–Miwa σ-form of the same PV; while a recurrence coefficient αn(t) is up to a translation in t and a linear fractional transformation PV(α2/2, − β2/2, 2n + 1 + α + β, − 1/2). These results are found from combining a pair of nonlinear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlevé equation. The case with α = β = −1/2 arose from a certain integrable system and this was brought to our attention by A P Veselov.

Asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point

We consider a singularly perturbed system of differential equations with irregular singular point. We prove that, in the general case of multiple elementary divisors, the corresponding asymptotic expansions can be constructed in the form of double series in fractional powers of the parameter and the ratio of the independent variable to the parameter.

Irregularity for Singular System of Ordinary Differential Equations in Complex Domain

An irregularity of an ordinary differential equation with an irregular singular point is defined by the maximal rate of exponential growth of all solutions of the associated homogeneous equation. We shall study first order singular systems of Poincare rank p (≥ 0) and characterize its irregularity from two different viewpoints; the first one is from the order of zeros of coefficient matrices, and the second one is from the comparison of indices of the system on formal Gevrey spaces.

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.

Construction of asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point in the case of a multiple spectrum of the boundary matrix

We construct asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point. We consider the case where the main matrix has a multiple eigenvalue associated with one or several elementary divisors of the same multiplicity. We establish that, in the case of multiple elementary divisors, the corresponding asymptotic expansions can be constructed in the form of double series in fractional powers of the parameter and the ratio of the independent variable to the parameter.