Abstract
The paper is devoted to operators given formally by the expression
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 \(L^2({\mathbb {R}}_+)\).
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Communicated by Claude Alain Pillet.
J. Dereziński: The financial support of the National Science Center, Poland, under the Grant UMO-2014/15/B/ST1/00126, is gratefully acknowledged.
S. Richard: On leave of absence from Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 November 1918, F-69622 Villeurbanne cedex, France.
S. Richard: Supported by JSPS Grant-in-Aid for Young Scientists A no 26707005.
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Dereziński, J., Richard, S. On Schrödinger Operators with Inverse Square Potentials on the Half-Line. Ann. Henri Poincaré 18, 869–928 (2017). https://doi.org/10.1007/s00023-016-0520-7
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DOI: https://doi.org/10.1007/s00023-016-0520-7