Geometric Phase, Curvature and the Monodromy Group

Exact solutions of the Klein–Gordon–Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr–Newman–(anti) de Sitter black hole are investigated. In the general case of a rotating, charged, cosmological black hole the solution of the KGF equation with the method of separation of variables results in Fuchsian differential equations for the radial and angular parts which for most of the parameter space contain more than three finite singularities and thereby generalise the Heun differential equations. For particular values of the physical parameters (i.e. mass of the scalar particle) these Fuchsian equations reduce to the case of the Heun equation and the closed form analytic solutions we derive are expressed in terms of Heun functions. For other values of the parameters some of the extra singular points are false singular points. We derive the conditions on the coefficients of the generalised Fuchsian equation such that a singular point is a false point. In such a case the exact solution of the Fuchsian equation can in principle be simplified and expressed in terms of Heun functions. This is the generalisation of the case of a Heun equation with a false singular point in which the exact solution of Heun’s differential equation is expressed in terms of Gauß hypergeometric function. We also derive the exact solutions of the radial and angular equations for a charged massive scalar particle in the Kerr–Newman spacetime. The analytic solutions are expressed in terms of confluent Heun functions. Moreover, we derived the constraints on the parameters of the theory such that the solution simplifies and expressed in terms of confluent Kummer hypergeometric functions. We also investigate the radial solutions in the KN case in the regions near the event horizon and far from the black hole. Finally, we construct several expansions of the solutions of the Heun equation in terms of generalised hypergeometric functions of Lauricella–Appell.

We present a novel representation for generalized hypergeometric functions of type p+1Fp which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type p+1Fp. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this section, we study the behavior of solutions of a linear Pfaffian system in a neighborhood of a singular point of the coefficients, where we restrict singular points to ones corresponding to regular singular points in ordinary differential equations. Note that a singular point of analytic functions in several variables is not a point but a set of codimension one (a hypersurface).

In this chapter, we shall first explain the existence of solutions of initial value problems for differential equations and then fundamental theorems for linear differential equations in the complex domain. We explain the definition of regular and irregular singularities of linear differential equations, and the behavior of of local solutions near those singular points.

We consider the equation , where , h>0 is a parameter, and is a given analytic function. We get asymptotics of its analytic solutions as . The asymptotic formulas contain an analog of the geometric (Berry) phase well known in the quasiclassical analysis of differential equations.

We study the integrability properties of nonlinearly perturbed Euler equations (linear ordinary differential equations with one regular singular point in the complex plane plus a nonlinear perturbation) near the singular point. We allow for first integrals with essential singularities and give sufficient conditions for the nonintegrability of the equations in the complex domain. We extend normal form theorems for singular equations and argue that equivalence to normal forms captures the spirit of the poly-Painlevé test and is a powerful tool for a rigorous approach to nonintegrability.

The geometric phase (GP) is a fundamental quantum effect arising from conical intersections (CIs), with profound consequences for vibronic energy levels. Standard imaginary-time path integral molecular dynamics (PIMD) based on the Born-Oppenheimer approximation does not account for the GP, potentially leading to significant errors in low-temperature thermodynamic properties. In this Perspective, we demonstrate that the multi-electronic-state path integral (MES-PI) formulation in imaginary time─developed in J. Chem. Phys. 2018, 148, 102319─naturally captures the GP effect through the electronic trace of the product of statistically weighted overlap matrices between successive imaginary-time slices. This crucial capability was already implicit in the benchmark MES-PIMD simulations in that foundational work. To isolate this topological effect from other nonadiabatic effects, we introduce a geometric signature matrix (for the CI) and a winding-number-induced phase factor, constructing an ad hoc GP-excluded MES-PI method. Comparing this ad hoc baseline against the rigorous MES-PI approach allows us to unambiguously quantify the impact of the GP on thermodynamic properties. While simpler approximations exist when only the ground electronic state is considered, MES-PIMD is the most general and accurate approach applicable to real complex systems, where the location and topology of CI seams are often not known a priori.

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.

Moduli space for generic unfolded differential linear systems

<p style='text-indent:20px;'>We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a <i>global holonomy group</i>. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.</p>

Algebraic microlocalization and modules with regular singularities over filtered rings

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.

In introduction we claimed that for p > 1 a Fuchsian equation of pth order with singularities a 1,..., a n contains fewer parameters than the set of classes of conjugate representations χ : Δ → GL(p, ℂ). This set depends on N r = p 2(n − 2) + 1 parameters (cf. (5.6.1)). Let us calculate the number of parameters for a Fuchsian equation of the pth order with n singular points a 1,..., a n.KeywordsSingular PointEntire FunctionFundamental MatrixRiemann SphereJordan BlockThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Efficient accelero-summation of holonomic functions