Abstract
This paper presents a thorough analysis of one-dimensional Schrödinger operators whose potential is a linear combination of the Coulomb term 1 / r and the centrifugal term \(1/r^2\). We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two-parameter holomorphic family of closed operators on \(L^2({\mathbb {R}}_+)\) is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial part of the Schrödinger operator with a Coulomb potential.
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Communicated by Krzysztof Gawȩdzki.
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 novembre 1918, F-69622 Villeurbanne cedex, France. Partially supported by the grant Topological invariants through scattering theory and noncommutative geometry from Nagoya University.
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Dereziński, J., Richard, S. On Radial Schrödinger Operators with a Coulomb Potential. Ann. Henri Poincaré 19, 2869–2917 (2018). https://doi.org/10.1007/s00023-018-0701-7
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DOI: https://doi.org/10.1007/s00023-018-0701-7