A few approaches are derived to calculate three-particle integrals which include spherical Bessel functions of the first and second kind, i.e., the $j_{\ell}(V r)$ and $n_{\ell}(V r)$ functions. Such integrals are important in applications to various problems known in atomic and nuclear physics. In particular, these integrals are needed in accurate computations of the photodetachment cross-section(s) of the negatively charged hydrogen ions.
A Fourier-type integral representation for Bessel's function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson's integral representation for the Bessel function along with a trigonometric integral representation of Gegenbauer's polynomials. This representation lets us express various functions related to the incomplete gamma function in series of Bessel's functions. Neumann series of Bessel functions are also considered and a new closed-form integral representation for this class of series is given. The density function of this representation is simpl…
A new representation for a regular solution of the radial Dirac system of a special form is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to the spectral parameter. For the coefficients of the series convenient for numerical computation recurrent integration formulas are given. Numerical examples are presented.
In a recent paper we unified Bessel functions of different orders .Here we extend the unification to other linairely independant solutions to Bessel equation, Neumann's and Hankel's functions
We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are singular and involve terms proportional to the Dirac delta function. Our approach makes use of the Rayleigh expansion of exp(i p.r) in terms of spherical Bessel functions, and we study a number of integrals, including singular integrals, involving a power of the independent variable times a spherical Bessel function. We work through several examples of three-…
This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identites for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions
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 arXiv:1001.0552. In arXiv:1410.4873 using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators arXiv:1111.4449. In arXiv:1508.02738 a Fourier-Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exa…
We provide a simple analytic formula in terms of elementary functions for the Laplace transform j_{l}(p) of the spherical Bessel function than that appearing in the literature, and we show that any such integral transform is a polynomial of order l in the variable p with constant coefficients for the first l-1 powers, and with an inverse tangent function of argument 1/p as the coefficient of the power l. We apply this formula for the Laplace transform of the memory function related to the Langevin equation in a one-dimensional Debye model.
A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order $m$. The sum allows for values of $|m|$ that are greater than $l$. A generalisation for the associated Legendre functions to allow for any rational $m$ for a specific $l$ is also shown
The integrals $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+1}\,e^{-αr}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ and $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+2}\,e^{-αr}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind.
We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schrödinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.
Non-local interactions are assumed for the deuteron with form factors $g_C(k)= j_0(b_1k)\,j_0(b_2k)$ for the central part responsible for the S-state and $g_T(k)= j_1(b_1k)\,j_1(b_2k)$ for the tensor part responsible for the D-state, where $b_1$ and $b_2$ are range parameters for the proton and neutron, respectively. The analytically obtained wavefunctions in coordinate space have different forms in three different regions. The inner most region is between $r=0$ and $r=b_2-b_1$ (assuming $b_2>b_1$), followed by a region between $r=b_2-b_1$ and $b_1+b_2$ and finally the region $r>b_1+b_2$. The …
Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an integral that involves the Bessel function and associated Laguerre polynomial. The result, which is given by Eq. (19) in the text, is written in terms of the Gegenbauer (ultra-spherical) polynomial.
As is known, the free heat-kernel on the integers (a modified Bessel function) is turned into the periodic free heat-kernel on the discrete circle by factoring, giving a pre-image sum. I generalise existing treatments by making the functions periodic up to a phase, thus introducing an extra parameter into the analysis. Identifying the classical paths form with the conventional eigenfunction expression, I find a combinatorial trace identity which allows various Bessel identities to be extracted, such as a generalisation of the Jacobi-Anger expansion.The free Dirichlet, Neumann and hybrid Dirich…
In this paper, exact solutions to the problem of acoustic scattering by elastic spherical symmetric scatterers are developed. The scatterer may consist of an arbitrary number of fluid and solid layers, and scattering with single Neumann conditions (replacing Neumann-to-Neumann conditions) is added. The solution is obtained by separation of variables, resulting in an infinite series which must be truncated for numerical evaluation. The implemented numerical solution is exact in the sense that numerical error is solely due to round-off errors, which will be shown using the symbolic toolbox in MA…
The Geometric factors in the field commutators and spring constants of the measurement devices in the famous analysis of the measurability of the electromagnetic field by Bohr and Rosenfeld are calculated using a Fourier--Bessel method for the evaluation of folding integrals, which enables one to obtain the general geometric factors as a Fourier--Bessel series. When the space region over which the factors are defined are spherical, the Fourier--Bessel series terms are given by elementary functions, and using the standard Fourier-integral method of calculating folding integrals, the geometric f…
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schrödinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on ra…
New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=ω^{2}u$ are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $ω$. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones from arXiv:1609.06679 and arXiv:1712.…
This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_\k^3$ ($κ<0$), to the standard {\itshape spherical waves} in $E^3$. The curvature $\k$ is considered as a parameter and for any $\k$ we show how the radial Schrödinger equation can be transformed into a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the (spherical) Bessel equation. The s…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric functions.
We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.
We consider integrals of products of Bessel functions and of spherical Bessel functions, combined with a Gaussian factor guaranteeing convergence at infinity. Explicit representations are obtained for the integrals, building on those in the 1992 paper by McPhedran, Dawes and Scott. Attention is paid to those sums with a distributive part arising as the Gaussian tends towards a constant. Taking this limit provides explicit analytic formulae for integrals which are difficult to evaluate numerically or otherwise, and whose interpretation has been contentious both in the literature of quantum mech…
This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for problems involving Fourier transforms of functions with rotational symmetry. The method used to derive the expansion formula is based entirely on differential methods and completely avoids the use of various integral identities commonly used in this context. Some new identities for the Fourier transform are derived and as a byproduct seemingly new recurrence rela…
One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on the using the Abel-Plana summation formula. This allows to derive the regularized quantities by manifestly cutoff independent way and to present them in the form of strongly converging integrals. However the applications of Abel- Plana formula in usual form is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on …
We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $μ$ approaches $0,$ can be expressed in terms of the Riema…
