The Bessel differential equation and the Hankel transform

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 νν represents the order of the associated Bessel functions and λλ is the complex spectral parameter involved in considering properties of the equation in the Hilbert function space L2(0,∞)L2(0,∞).Properties of the equation are considered when the order ν∈[0,1);ν∈[0,1); in this case the singular end-point 00 is in the limit-circle non-oscillatory classification in the space L2(0,∞);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 00 allows of the definition of a single Titchmarsh–Weyl mm-coefficient for the whole interval (0,∞)(0,∞). In turn this information yields a proof of the Hankel transform as an eigenfunction expansion for the case when ν∈[0,1)ν∈[0,1), a result which is not available in the existing literature.The application of the principal solution, from the end-point 00 of the Bessel equation, as a boundary condition function yields the Friedrichs self-adjoint extension in L2(0,∞);L2(0,∞); the domain of this extension has many special known properties, of which new proofs are presented.