The Bessel differential equation and the Hankel transform

This paper studies the classical second-order Bessel differential equation in Liouville form: - y ″ ( x ) + ( ν 2 - 1 4 ) x - 2 y ( x ) = λ y ( x ) for all x ∈ ( 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 L 2 ( 0 , ∞ ) . ties 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 L 2 ( 0 , ∞ ) ; the equation is in the strong limit-point and Dirichlet condition at the end-point + ∞ . ng 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. plication 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 , ∞ ) ; the domain of this extension has many special known properties, of which new proofs are presented.