Neumann Series Of Bessel Functions Research Articles

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

We consider Gaussian random monochromatic waves u on the plane depending on a real parameter s that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of u is fdσ, where dσ is the Hausdorff measure on the unit circle and the density f is a function on the circle that, roughly speaking, has exactly s−12 derivatives in L2 almost surely. When s=0, one recovers the classical setting for random waves with a translation-invariant covariance kernel. The main thrust of this paper is to explore the connection between the regularity parameter s and the asymptotic behavior of the number N(∇u,R) of critical points that are contained in the disk of radius R≫1. More precisely, we show that the expectation EN(∇u,R) grows like the area of the disk when the regularity is low enough (s<32) and like the diameter when the regularity is high enough (s>52), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, s=0. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is s>5, we show that in fact N(∇u,R) grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.

Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm–Liouville equation with a real‐valued potential . Let be its solution satisfying certain initial conditions for a number of , where , and are some complex numbers. Denote , where . We propose a method for solving the inverse problem of the approximate recovery of the potential and number from the following data . In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

A Dual Integral Equation Approach for Evaluating the Shielding of Thick Circular Disks against a Coaxial Loop

The electromagnetic interaction between a circular disk with finite conductivity and finite thickness and a coaxial circular loop of constant current is addressed here. The finite conductivity and thickness of the material disk lead to the adoption of suitable generalized boundary conditions, and the problem is thereby reduced to the solution of two sets of dual integral equations in the Hankel transform domain. Such equations are then solved by expanding the spectral unknowns in Neumann series of Bessel functions. An alternative formulation that is valid for purely conductive screens with no magnetic properties, which is computationally much faster, is proposed as well. The magnetic shielding effectiveness of the structure is studied in detail, pointing out its dependencies and possible critical situations.

A Jentzsch-Theorem for Kapteyn, Neumann and General Dirichlet Series

Comparing phase plots of truncated series solutions of Kepler’s equation by Lagrange’s power series with those by Bessel’s Kapteyn series strongly suggests that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence in the complex plane is a cluster point of zeros of sections of the series. We prove this result by studying properties of the growth function of a sequence of entire functions. For series, this growth function is computable in terms of the convergence abscissa of an associated general Dirichlet series. The proof then extends, besides including Jentzsch’s classical result for power series, to general Dirichlet series, to Kapteyn, and to Neumann series of Bessel functions. Moreover, sections of Kapteyn and Neumann series generally exhibit zeros close to the real axis which can be explained, including their asymptotic linear density, by the theory of the distribution of zeros of entire functions.

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.

Time-Dependent One-Dimensional Electromagnetic Wave Propagation in Inhomogeneous Media: Exact Solution in Terms of Transmutations and Neumann Series of Bessel Functions

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see [7]. In [5] using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators [8]. In [6] a Fourier–Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exact solution of the problem of the transmission of a normally incident electromagnetic time-dependent plane wave through an arbitrary inhomogeneous layer. The result can be used for efficient computation of the transmitted modulated signals. In particular, it is shown that in the classical situation of a signal represented in terms of a trigonometric Fourier series the solution of the problem can be written in the form of Neumann series of Bessel functions with exact formulas for the coefficients. The representation lends itself to numerical computation.

A method for computation of scattering amplitudes and Green functions of whole axis problems

A method for the computation of scattering data and of the Green function for the one‐dimensional Schrödinger operator with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF). The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator H, which in turn allow one to compute the scattering amplitudes and the Green function of the operator H−λ with .

A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $\omega$ the difference between the exact solution and the approximate one (the truncated NSBF) depends on $N$ (the truncation parameter) and the coefficients of the equation and does not depend on $\omega$. A similar result is valid when $\omega\in\mathbb{C}$ belongs to a strip $|Im\omega|<C$. This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of $\omega$. Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm-Liouville equation is developed and illustrated on a test problem.

On a Neumann-type series for modified Bessel functions of the first kind

In this paper, we are interested in a Neumann-type series for modified Bessel functions of the first kind which arises in the study of Dunkl operators associated with dihedral groups and as an instance of the Laguerre semigroup constructed by Ben Said-Kobayashi-Orsted. We first revisit the particular case corresponding to the group of square-preserving symmetries for which we give two new and different proofs other than the existing ones. The first proof uses the expansion of powers in a Neumann series of Bessel functions, while the second one is based on a quadratic transformation for the Gauss hypergeometric function and opens the way to derive further expressions when the orders of the underlying dihedral groups are powers of two. More generally, we give another proof of De Bie et al.’s formula expressing this series as a Φ 2 \Phi _2 -Horn confluent hypergeometric function. In the course of the proof, we shed light on the occurrence of multiple angles in their formula through elementary symmetric functions and get a new representation of Gegenbauer polynomials.

Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation

A representation for a solution $u(\omega,x)$ of the equation $-u"+q(x)u=\omega^2 u$, satisfying the initial conditions $u(\omega,0)=1$, $u'(\omega,0)=i\omega$ is derived in the form \[ u(\omega,x)=e^{i\omega x}\left( 1+\frac{u_1(x)}{\omega}+ \frac{u_2(x)}{\omega^2}\right) +\frac{e^{-i\omega x}u_3(x)}{\omega^2}-\frac{1}{\omega^2}\sum_{n=0}^{\infty} i^{n}\alpha_n(x)j_n(\omega x), \] where $u_m(x)$, $m=1,2,3$ are given in a closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the coefficients $\alpha_n$ are calculated by a recurrent integration procedure. The following estimate is proved $\vert u(\omega,x) -u_N(\omega,x)\vert \leq \frac{1}{\vert \omega \vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm Im}\omega\,x)}{\mathop{\rm Im}\omega}}$ for any $\omega\in\mathbb{C}\backslash \{0\}$, where $u_N(\omega,x)$ is an approximate solution given by truncating the series in the representation for $u(\omega,x)$ and $\varepsilon_N(x)$ is a nonnegative function tending to zero for all $x$ belonging to a finite interval of interest. In particular, for $\omega\in\mathbb{R}\backslash \{0\}$ the estimate has the form $\vert u(\omega,x)-u_N(\omega,x)\vert \leq \frac{1}{\vert\omega\vert^2}\varepsilon_N(x)$. A numerical illustration of application of the new representation for computing the solution $u(\omega,x)$ on large sets of values of the spectral parameter $\omega$ with an accuracy nondeteriorating (and even improving) when $\omega \rightarrow \pm \infty$ is given.

Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions

A new representation of solutions to the equation −y′′+q(x)y=ω2y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.

Neumann series of Bessel functions

Recently, Pogány and Süli [Integral representation for Neumann series of Bessel functions, Proc. Amer. Math. Soc. 137(7) (2009), pp. 2363–2368] derived a closed-form integral expression for a Neumann series of Bessel functions. In this note, our aim is to establish another kind of integral representations for the Neumann series of Bessel functions of the first kind J ν.

On the coefficients of Neumann series of Bessel functions

Recently Pogány and Süli (Proc. Amer. Math. Soc. 137 (7) (2009) 2363–2368) derived a closed-form integral expression for Neumann series of Bessel functions. In this note we precisely characterize the class of functions α that generate the integral representation of a Neumann series of Bessel functions in the sense that the restriction α | N = ( α n ) of a function α to the set N of all positive integers is the sequence of coefficients of the initial Neumann series.

Integral representation for Neumann series of Bessel functions

A closed integral expression is derived for Neumann series of Bessel functions ― a series of Bessel functions of increasing order ― over the set of real numbers.

Rational approximation to Neumann series of Bessel functions

LetJn(z) be the Bessel function of the first kind and ordern, and letf(z) be an analytic function in|z|≤r (r>0); then it is known that the Bessel expansion $$f(z) = \sum\limits_{n = 0}^\infty {a_n J_n (z) (Neumann series)} $$ converges for|z|≤r. In this paper, we shall be concerned with the construction of “approximating” functions to (1) which are easily computable (rational functions). Namely, making use of the generating function for the family {Jn(z)}, a rational functionfk(z) with prescribed poles can be obtained such thatfk(z) “approximates” tof(z) in the following sense: $$f_k (z) = \sum\limits_{n = 0}^\infty {\bar a_n J_n (z)} with \bar a_n = a_n , n = 0,1,...,k - 1;$$ and it will be said thatfk is an “approximant” of orderk. When orthogonality conditions with respect to a linear functional defined from the sequence {an} are used, then the order of approximation may be increased up to2k. An algebraic approach of these approximants is carried out.

Axially-symmetrical oscillation of a semi-infinite layered medium with complex damping

Abstract A horizontally-layered, semi-infinite, isotropic medium with complex damping (O-damping) is considered to be excited by a prescribed periodic vertical motion at a point on the free surface. The resulting periodic displacement vector of the medium is expanded in a Neumann series of Bessel functions of the radial coordinate r. which ensures that the displacement vanishes as r->oo. Ordinary, linear, differential equations for the coefficients in these series are derived, along with continuity conditions at the interfaces and boundary conditions at the free surface.

A SOLUTION OF TRANTER'S DUAL INTEGRAL EQUATIONS PROBLEM

Tranter (1, 2) has given the solution of the pair of integral equations (1) and (2) below in the form of a Neumann series of Bessel functions whose coefficients are determined by means of an infinite series of linear equations. This paper gives the formal solution as an integral, containing an unknown function which is determined by means of an integral equation of Fredholm's type. This solution appears to have both theoretical and practical advantages over Tranter's solution in that it is (in a sense) in closed form; moreover it does not depend on certain parameters being specified as large or small in order to complete a practical solution. Applications are given to several potential problems involving circular disks, some of these problems being believed to be new.

Neumann Series of Bessel Functions. II

Neumann Series of Bessel Functions. II

Neumann series of Bessel functions. II

Neumann series of Bessel functions. II