Fractional powers of a Kontorovich–Lebedev-type Sturm–Liouville operator

We study a second-order Sturm–Liouville operator of modified Bessel-type naturally associated with the Kontorovich–Lebedev transform. This operator arises as a particular case of a Whittaker-type family and possesses strongly singular coefficients at the origin. We develop a complete spectral theory for this operator in L2 spaces, including an explicit description of its generalized eigenfunctions and the associated modified Kontorovich–Lebedev transform. A central result of the paper is the construction of an explicit intertwining operator between the modified Bessel operator and the shifted one-dimensional Laplacian. This intertwining relation allows us to transfer spectral and analytic properties from the Laplacian to the singular operator under consideration. As an application, we establish positivity of the corresponding heat kernel and derive integral representations based on the Kontorovich–Lebedev convolution structure. We then develop a fractional calculus associated with the operator. Fractional powers are defined spectrally and shown to admit both a Bochner subordination formula and a singular integral representation. The associated Lévy kernel is described explicitly and its precise asymptotic behaviour is obtained using Tauberian methods. These results provide a nonlocal characterization of the fractional operator analogous to that of the classical fractional Laplacian.