Abstract
The paper is devoted to operators given formally by the expression -∂x2+α-141x2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\partial _x^2+\\left( \\alpha -\\frac{1}{4}\\right) \\frac{1}{x^{2}}. \\end{aligned}$$\\end{document}This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real alpha , or closed operator for complex alpha , we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L^2({mathbb {R}}_+), which we denote H_{m,kappa } and H_0^nu , with m^2=alpha , -1<mathrm{Re}(m)<1, and where kappa ,nu in {mathbb {C}}cup {infty } specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0,infty [. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that -1<mathrm{Re}(m)<1 is the maximal region of parameters for which the operators H_{m,kappa } can be defined within the framework of the Hilbert space L^2({mathbb {R}}_+).
Highlights
The family of differential operators x−2 (1.1)is very special
For Re(m) ≥ 1 the operator Hm corresponds to the closure of m2 − 1 4 x−2 (1.2)
We provide formulas for the spectral projections onto parts of the continuous spectrum, and introduce bounded invertible operators that diagonalize Hm,κ and hold: R0ν (−k2)1lR+ (H0ν), except for a small set of parameters that we call exceptional
Summary
We can notice that it is natural to include the non self-adjoint cases, which interpolate in an interesting way between self-adjoint cases, and to study a holomorphic family of closed operators depending on two complex parameters With this idea in mind, let us compare our work with some of the recent papers dealing with the operator (1.2). In [11] the operator (1.2) is thoroughly analyzed in the range m ∈ [0, 1[ but at the end of the day only the Friedrichs self-adjoint extension is considered As mentioned above, these approaches leave plenty of room left for our general approach involving two complex parameters family of operators.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
