TY - JOUR
T1 - The Bessel differential equation and the Hankel transform
AU - Everitt, W
AU - Kalf, H
PY - 2007/11/1
Y1 - 2007/11/1
N2 - This paper studies the classical second-order Bessel differential equation in Liouville form: - y '' (x) + (v(2)-1/4)x(-2) y(x) = lambda y(x) for all x is an element of (0, infinity). Here, the parameter v represents the order of the associated Bessel functions and is the complex spectral parameter involved in considering properties of the equation in the Hilbert function space L-2(0, infinity). Properties of the equation are considered when the order V E [0, 1); in this case the singular end-point 0 is in the limit-circle non-oscillatory classification in the space L-2(0, infinity); the equation is in the strong limit-point and Dirichlet condition at the end-point +infinity. Applying the generalised initial value theorem at the singular end-point 0 allows of the definition of a single Titchmarsh-Weyl m-coefficient for the whole interval (0, infinity). In turn this information yields a proof of the Hankel transform as an eigenfunction expansion for the case when V is an element of [0, 1), a result which is not available in the existing literature. The application of the principal solution, from the end-point 0 of the Bessel equation, as a boundary condition function yields the Friedrichs self-adjoint extension in L-2(0, infinity); the domain of this extension has many special known properties, of which new proofs are presented. (c) 2006 Elsevier B.V. All rights reserved.
AB - This paper studies the classical second-order Bessel differential equation in Liouville form: - y '' (x) + (v(2)-1/4)x(-2) y(x) = lambda y(x) for all x is an element of (0, infinity). Here, the parameter v represents the order of the associated Bessel functions and is the complex spectral parameter involved in considering properties of the equation in the Hilbert function space L-2(0, infinity). Properties of the equation are considered when the order V E [0, 1); in this case the singular end-point 0 is in the limit-circle non-oscillatory classification in the space L-2(0, infinity); the equation is in the strong limit-point and Dirichlet condition at the end-point +infinity. Applying the generalised initial value theorem at the singular end-point 0 allows of the definition of a single Titchmarsh-Weyl m-coefficient for the whole interval (0, infinity). In turn this information yields a proof of the Hankel transform as an eigenfunction expansion for the case when V is an element of [0, 1), a result which is not available in the existing literature. The application of the principal solution, from the end-point 0 of the Bessel equation, as a boundary condition function yields the Friedrichs self-adjoint extension in L-2(0, infinity); the domain of this extension has many special known properties, of which new proofs are presented. (c) 2006 Elsevier B.V. All rights reserved.
UR - http://www.scopus.com/inward/record.url?scp=34547567199&partnerID=8YFLogxK
U2 - 10.1016/j.cam.2006.10.029
DO - 10.1016/j.cam.2006.10.029
M3 - Article
VL - 208
SP - 3
EP - 19
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 1
ER -