Confluent Heun Equation Research Articles

Accessory parameters in confluent Heun equations and classical irregular conformal blocks

Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquet type in the Heun equation and some of its confluent versions. We extend this relation to another class of accessory parameter functions that are defined by inverting all-order Bohr-Sommerfeld periods for confluent and biconfluent Heun equation. The relevant conformal blocks involve Nagoya irregular vertex operators of rank 1 and 2 and conjecturally correspond to partition functions of a 4D $\mathcal{N}=2$, $N_f=3$ gauge theory at strong coupling and an Argyres-Douglas theory.

Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation

Categories of Symmetry Groups of the Space of Solutions of the Special Doubly Confluent Heun Equation

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Isomonodromy sets of accessory parameters for Heun class equations

AbstractIn this paper, we consider the monodromy and, in particular, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. The isomonodromy sets of accessory parameters for the Heun class equations are described by Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly‐confluent Heun equation, and the reduced biconfluent Heun equation.

On the deformation of linear Hamiltonian systems

For linear Hamiltonian 2n×2n systems Jy′(x)=(λW(x)+H(x))y(x) we investigate the problem how the eigenvalues λ depend on the entries of the coefficient matrix H. This question turns into a deformation equation for H and a partial differential equation for the eigenvalues λ. We apply our results to various examples, including generalizations of the confluent Heun equation and the Chandrasekhar-Page angular equation. We are mainly concerned with the 2×2 case, and in order to reduce the degrees of freedom in H as much as possible, we will first convert such systems into a complementary triangular form, which is a canonical form with a minimum number of free parameters. Furthermore, we discuss relations to monodromy preserving deformations and to matrix Lax pairs.

Scalar perturbations of black holes in Jackiw-Teitelboim gravity

We study linear scalar perturbations of black holes in two-dimensional (2D) gravity models with a particular emphasis on Jackiw-Teitelboim (JT) gravity. We obtain an exact expression of the quasinormal mode frequencies for single horizon black holes in JT gravity and then verify it numerically using the Horowitz-Hubeny method. For a 2D Reissner-Nordstr\"om like solution, we find that the massless scalar wave equation reduces to the confluent Heun equation using which we calculate the Hawking spectra. Finally, we consider the dimensionally reduced Ba\~nados-Teitelboim-Zanelli (BTZ) black hole and obtain the exterior and interior quasinormal modes. The dynamics of a scalar field near the Cauchy horizon mimics the behavior of the same for the usual BTZ black hole, indicating a possible violation of the strong cosmic censorship conjecture in the near extreme limit. However, quantum effects seem to rescue strong cosmic censorship.

PT -Symmetric Potentials from the Confluent Heun Equation.

We derive exactly solvable potentials from the formal solutions of the confluent Heun equation and determine conditions under which the potentials possess symmetry. We point out that for the implementation of symmetry, the symmetrical canonical form of the Heun equation is more suitable than its non-symmetrical canonical form. The potentials identified in this construction depend on twelve parameters, of which three contribute to scaling and shifting the energy and the coordinate. Five parameters control the function that detemines the variable transformation taking the Heun equation into the one-dimensional Schrödinger equation, while four parameters play the role of the coupling coefficients of four independently tunable potential terms. The potentials obtained this way contain Natanzon-class potentials as special cases. Comparison with the results of an earlier study based on potentials obtained from the non-symmetrical canonical form of the confluent Heun equation is also presented. While the explicit general solutions of the confluent Heun equation are not available, the results are instructive in identifying which potentials can be obtained from this equation and under which conditions they exhibit symmetry, either unbroken or broken.

First-Order Ode Systems Generating Confluent Heun Equations

We study the relation between linear second-order equations that are confluent Heun equations, namely, the biconfluent and triconfluent Heun equations, and first-order linear systems of equations generating Painleve equations. The generation process is interpreted in physical terms as antiquantization. Technically, the study in volves manipulations with polynomials. The complexity of computations sometimes requires using computer algebra systems. Bibliography: 13 titles.

Solitary magnetostrophic Rossby waves in spherical shells

Abstract

The Rabi problem with elliptical polarization

Abstract We consider the solution of the equation of motion of a classical/quantum spin subject to a monochromatical, elliptically polarized external field. The classical Rabi problem can be reduced to third-order differential equations with polynomial coefficients and hence solved in terms of power series in close analogy to the confluent Heun equation occurring for linear polarization. Application of Floquet theory yields physically interesting quantities like the quasienergy as a function of the problem’s parameters and expressions for the Bloch–Siegert shift of resonance frequencies. Various limit cases are thoroughly investigated.

Square Root of the Monodromy Map Associated with the Equation of RSJ Model of Josephson Junction

An explicit representation of the monodromy transformation of the space of solutions to the nonlinear ODE, utilized for the modeling of Josephson junctions, is given. In case of positive integer order, the transformation, interpreted as the square root of the noted monodromy transformation, is derived by making use of a symmetry possessed by the associated double confluent Heun equation.

Evolutionary dynamics and eigenspectrum of confluent Heun equation

We consider a biological population evolving under the joint action of selection, mutation and random genetic drift. The evolutionary dynamics are described by a one-dimensional Fokker–Planck equation whose eigenfunctions obey a confluent Heun equation. These eigenfunctions are expanded in an infinite series of orthogonal Jacobi polynomials and the expansion coefficients are found to obey a three-term recursion equation. Using scaling ideas, we obtain an expression for the expansion coefficients and an analytical estimate of the number of terms required in the series for an accurate determination of the eigenfunction. The eigenvalue spectrum is studied using a perturbation theory for weak selection and numerically for strong selection. In the latter case, we find that the eigenvalue for the first excited state exhibits a sharp transition: for mutation rate below one, the eigenvalue increases linearly with increasing mutation rate and then remains a constant; higher eigenvalues are found to display a more complex behavior.

Group algebras acting on the space of solutions of a special double confluent Heun equation

We study properties of the space $$\boldsymbol{\Omega}$$ of solutions of a special double confluent Heun equation closely related to the model of a overdamped Josephson junction. We describe operators acting on $$\boldsymbol{\Omega}$$ and relations in the algebra $$\mathcal{A}$$ generated by them over the real number field. The structure of $$\mathcal{A}$$ depends on parameters. We give conditions under which $$\mathcal{A}$$ is isomorphic to a group algebra and describe two corresponding group structures.

Analytical and numerical treatment of perturbed black holes in horizon-penetrating coordinates

The deviations of non-linear perturbations of black holes from the linear case are important in the context of ringdown signals with large signal-to-noise ratio. To facilitate a comparison between the two we derive several results of linear perturbation theory in coordinates which may be adopted in numerical work. Specifically, our results are derived in Kerr-Schild coordinates adjusted by a general height function. In the first part of the paper we address the questions: for an initial configuration of a massless scalar field, what is the amplitude of the excited quasinormal mode (QNM) for any observer outside outside the event horizon, and furthermore what is the resulting tail contribution? This is done by constructing the full Green's function for the problem with exact solutions of the confluent Heun equation satisfying appropriate boundary conditions. In the second part of the paper, we detail new developments to our pseudospectral numerical relativity code bamps to handle scalar fields. In the linear regime we employ precisely the Kerr-Schild coordinates treated by our previous analysis. In particular, we evolve pure QNM type initial data along with several other types of initial data and report on the presence of overtone modes in the signal.

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

AbstractThe purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRMs), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H_{\textrm{Rabi}}^{\varepsilon }$ of the AQRM is defined by adding the fluctuation term $\varepsilon \sigma _x$, with $\sigma _x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H_{\textrm{Rabi}}^{\varepsilon }$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $\varepsilon $ is a halfinteger. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of $\mathfrak{s}\mathfrak{l}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

On Spectral Curves and Complexified Boundaries of the Phase-Lock Areas in a Model of Josephson Junction

The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are \(l,\lambda ,\mu \in \mathbb {R}\). Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for μ ≠ 0 this happens exactly when \(l\in \mathbb {N}\) and the parameters (λ, μ) lie on an algebraic curve \({\Gamma }_{l}\subset \mathbb {C}^{2}_{(\lambda ,\mu )}\) called the l-spectral curve and defined as zero locus of determinant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dynamical systems on 2-torus depending on real parameters (B, A; ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in \(\mathbb {R}^{2}_{(B,A)}\) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every \(l\in \mathbb {N}\). We also calculate its genus for \(l\leqslant 20\) and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in \(\mathbb {R}^{3}_{(B,A,\omega ^{-1})}\). We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.

Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations

We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansion coefficients are explicitly calculated in terms of the Euler gamma functions.

Solution Space Monodromy of a Special Double Confluent Heun Equation and Its Applications

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order (L-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the L-operators, called the universal L-operator. We construct composition laws for L-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the L-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset Re z > 0 of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations.

Symmetries of the space of solutions to special double confluent Heun equations of integer order

This paper investigates the triplet of linear operators that determines automorphisms of the set of solutions to special double confluent Heun equations of integer order. Their pairwise composition rules are computed in explicit form. It is shown that, under the conditions motivated by physical applications, these operators generate the group of symmetries of the linear space of solutions that is isomorphic to the dihedral group, provided the monodromy equivalence relation is applied. On the corresponding projective space, the symmetry group reduces to the Klein group. The results presented in this paper have implications for the modeling of Josephson junctions.

Analytic solutions for generalized -symmetric Rabi models**Project supported in part by the National Natural Science Foundation of China (Grant No. 11874251), the Scientific Research Starting Foundation of Shaanxi Normal University, China, and the Scientific Research Plan Funded by the Education Department of Shaanxi Province, China (Grant No. 17JK0786).

We theoretically investigate the exact solutions for generalized parity–time()-reversal-symmetric Rabi models driven by external fields with monochromatic periodic, linear, and parabolic forms, respectively. The corresponding exact solutions are presented in terms of the confluent Heun equations without any approximation. In principle, the analytic solutions derived here are valid in the whole parameter space. Such a kind of study may offer potential coherent control schemes of the -symmetric two-level systems.