Transmutation Operators Research Articles

On soft factors and transmutation operators

The well known soft theorems state the specific factorizations of tree level gravitational (GR) amplitudes at leading, sub-leading and sub-sub-leading orders, with universal soft factors. For Yang-Mills (YM) amplitudes, similar factorizations and universal soft factors are found at leading and sub-leading orders. Then it is natural to ask if the similar factorizations and soft factors exist at higher orders. In this note, by using transformation operators proposed by Cheung, Shen and Wen, we reconstruct the known soft factors of YM and GR amplitudes, and prove the nonexistence of higher order soft factor of YM or GR amplitude which satisfies the universality.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations

Abstract In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform ℬ α {\mathscr{B}_{\alpha}} in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space ℱ α ⁢ ( ℂ d ) {\mathscr{F}_{\alpha}(\mathbb{C}^{d})} .

Transmutation Operators for Eigenfunctions of Certain Differentiation Operators and Their Fractional Powers

Transmutation Operators for Eigenfunctions of Certain Differentiation Operators and Their Fractional Powers

Transmutation operators intertwining first-order and distributed-order derivatives

Transmutation operators intertwining first-order and distributed-order derivatives

Solution of the Goursat Problem for the Boussinesq–Love Equation with the Bessel Operator by the Method of Transmutation Operators

Solution of the Goursat Problem for the Boussinesq–Love Equation with the Bessel Operator by the Method of Transmutation Operators

Solution of a Characteristic Problem for the Third Order Pseudoparabolic Equation with the Bessel Operator by the Method of Transmutation Operators

Solution of a Characteristic Problem for the Third Order Pseudoparabolic Equation with the Bessel Operator by the Method of Transmutation Operators

Solution of the Cauchy problem for a hyperbolic equation of the fourth order with the Bessel operator by the method of transmutation operators

Solution of the Cauchy problem for a hyperbolic equation of the fourth order with the Bessel operator by the method of transmutation operators

Transmutation Operators Based on Homotope Analysis for Nonlocal Boundary Value Problems

Transmutation Operators Based on Homotope Analysis for Nonlocal Boundary Value Problems

Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed.

Sonine-Dimovski transform and spectral synthesis associated with the hyper-Bessel operator on the complex plane

We consider the hyper-Bessel operator of order $$r\ge 2$$ : $$\begin{aligned} B_{\alpha }:= \frac{1}{z^{r-1}}\prod _{i=1}^{r-1} \big ( zD_z+({ r \alpha _{i} + 1})\big )D_z, \end{aligned}$$ where $$\alpha =(\alpha _1,\ldots ,\alpha _{r-1})$$ is a real multi-index such that $$\alpha _k \ge - 1 + {k}/{r}$$ for $$k=1,...,r - 1$$ and $$D_z$$ is the usual derivative in complex plane. We characterize the transmutation operators between two hyper-Bessel operators, namely from $$B_\beta $$ into $$B_\alpha $$ on the space $$H_r(\mathbb {C})$$ of r-even and entire functions with the help of the Sonine-Dimovski transform and we prove the spectral synthesis property associated with the operator $$B_\alpha $$ for the space $$H_r(\mathbb {C})$$ . Let us note that the hyper-Bessel operator $$B_{\alpha }$$ and the related transmutation operators can be also represented as operators of the generalized fractional calculus.

TRANSMUTATION OPERATORS METHOD FOR STURM-LIOUVILLE EQUATIONS IN IMPEDANCE FORM II: INVERSE PROBLEM

The inverse spectral problem for the Sturm-Liouville equation in impedance form is considered. A corresponding Gelfand-Levitan integral equation is derived. A Fourier-Legendre series expansion for the transmutation operator kernel combined with the Gelfand-Levitan equation leads to a simple direct method for solving the inverse problem of recovering the impedance function from spectral data by solving a system of linear algebraic equations, such that the impedance function is recovered from the first element of the solution vector. The stability of the method is proved. Its numerical performance is illustrated by several examples.

GENERALIZED EXPONENTIAL BASIS FOR EFFICIENT SOLVING OF HOMOGENEOUS DIFFUSION FREE BOUNDARY PROBLEMS: RUSSIAN OPTION PRICING

This paper develops a method for solving free boundary problems for time-homogeneous diffusions. We combine the complete exponential system of solutions for the heat equation, transmutation operators and recently discovered Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations to construct a complete system of solutions for the considered partial differential equations. The conceptual algorithm for the application of the method is presented. The valuation of Russian options with finite horizon is used as a numerical illustration. The solution under different horizons is computed and compared to the results that appear in the literature.

TRANSMUTATION OPERATORS METHOD FOR STURM-LIOUVILLE EQUATIONS IN IMPEDANCE FORM I: CONSTRUCTION AND ANALYTICAL PROPERTIES

The construction of integro-differential transmutation operators for the Sturm-Liouville equation in impedance form is presented. Their analytical properties of boundedness and invertibility in appropriate functional spaces are studied. A Fourier-Legendre series for the integral transmutation kernel in terms of the Legendre polynomials is obtained together with a representation of the solutions of the Sturm-Liouville equation in impedance form as Neumann series of Bessel functions.

Transmutation operators and expansions for one-loop Feynman integrands

In this paper, the connections among $1$-loop Feynman integrands of a large variety of theories with massless external states are further investigated. The work includes two parts. First, we construct a new class of differential operators which transmute the $1$-loop gravitational Feynman integrands to $1$-loop Yang-Mills Feynman integrands. The new operators are commutable with the integration of loop momentum, thus the corresponding transmutational relations hold at not only the integrands level, but also the $1$-loop amplitudes level. Secondly, by using $1$-loop level transmutational relations, together with some general requirements such as gauge and Lorentz invariance, we derive the expansions of the Feynman integrands of one theory to those of other theories. The unified web for expansions is established, including a wide range of theories which are gravitational theory, Einstein-Yang-Mills theory, Einstein-Maxwell theory, Born-Infeld theory, pure Yang-Mills theory, Yang-Mills-scalar theory, special Yang-Mills-scalar theory, Dirac-Born-Infeld theory, extended Dirac-Born-Infeld theory, special Galileon theory, non-linear sigma model. The systematic rules for evaluating coefficients in the expansions are provided, and the duality between transmutational relations and expansions is shown.

Transmutation Operators and Liouville Normal Form, Leading to Harmonic Solutions with Variable Components

Transmutation Operators and Liouville Normal Form, Leading to Harmonic Solutions with Variable Components

Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

Bessel-Wright transform in the setting of quantum calculus

This work is devoted to the study of a $q$-harmonic analysis related to the $q$-analog of the Bessel-Wright integral transform [6]. We establish some important properties of this transform and we focalise our attention in studying the associated transmutation operator.

Analytic approximation of transmutation operators for one-dimensional stationary Dirac operators and applications to solution of initial value and spectral problems

A method for approximate solution of initial value and spectral problems for one dimensional Dirac equation based on an analytic approximation of the transmutation operator is presented. In fact the problem of numerical approximation of solutions is reduced to approximation of the potential matrix by a finite linear combination of matrix valued functions related to generalized formal powers introduced in [16]. Convergence rate estimates in terms of smoothness of the potential are proved. The method allows one to compute both lower and higher eigendata with an extreme accuracy.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.