The Bessel differential equation and the Hankel transform

摘要

This paper studies the classical second-order Bessel differential equation in Liouville form:-y″(x)+(ν2-14)x-2y(x)=λy(x)for allx∈(0,∞).Here, the parameter ν represents the order of the associated Bessel functions and λ is the complex spectral parameter involved in considering properties of the equation in the Hilbert function space L2(0,∞).Properties of the equation are considered when the order ν∈[0,1); in this case the singular end-point 0 is in the limit-circle non-oscillatory classification in the space L2(0,∞); the equation is in the strong limit-point and Dirichlet condition at the end-point +∞.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,∞). In turn this information yields a proof of the Hankel transform as an eigenfunction expansion for the case when ν∈[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 L2(0,∞); the domain of this extension has many special known properties, of which new proofs are presented.