Embedded eigenvalues and Neumann–Wigner potentials for relativistic Schrödinger operators
Jozsef Lorinczi
Itaru Sasaki
2134/24699
https://repository.lboro.ac.uk/articles/journal_contribution/Embedded_eigenvalues_and_Neumann_Wigner_potentials_for_relativistic_Schr_dinger_operators/9386726
The existence of potentials for relativistic Schrodinger operators allowing eigenvalues em
bedded in the essential spectrum is a long-standing open problem. We construct Neumann-Wigner type potentials for the massive relativistic Schrodinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfunctions, for which an
eigenvalue equal to zero or a zero-resonance exists, dependent on the rate of decay of
the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay
behaviours due to the non-locality of the operator.
2017-04-11 13:06:09
Relativistic Schrodinger operator
Non-local operator
Neumann-Wigner potentials
Embedded eigenvalues
Resonances
Mathematical Sciences not elsewhere classified