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Levinson's Theorem for the Nonlocal Interaction in One Dimension

Authors:

Institution:

(1) Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ, U.K.;(2) Department of Physics, Cardwell Hall, Kansas State University, Manhattan, Kansas, 66506

Abstract:

The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm–Liouville theorem. The critical case where the Schro;audinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n ₊ (n _–) of bound states with even (odd) parity is related to the phase shift eegr

₊(0)

_–(0)] of the scattering states with the same parity at zero momentum as

${\eta }_ + (0) = \left\{ \begin{gathered} (n_ + - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}){\pi noncritical case} \hfill \\ n_ + {\pi critical case } \hfill \\ \end{gathered} \right.$

and

$\eta _ - \left( 0 \right) = \left\{ \begin{gathered} n_ - \pi noncritical case \hfill \\ \left( {n_ - + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\pi critical case \hfill \\ \end{gathered} \right.$

The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.

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