The Bessel differential equation and the Hankel transform

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.