Electron. J. Differential Equations, Vol. 2019 (2019), No. 23, pp. 1-19.

Bound states of the discrete Schrodinger equation with compactly supported potentials

Tuncay Aktosun, Abdon E. Choque-Rivero, Vassilis G. Papanicolaou

The discrete Schrodinger operator is considered on the half-line lattice $n\in \{1,2,3,\dots\}$ with the Dirichlet boundary condition at $n=0$. It is assumed that the potential belongs to class $\mathcal{A}_b$, i.e. it is real valued, vanishes when n>b with b being a fixed positive integer, and is nonzero at n=b. The proof is provided to show that the corresponding number of bound states, N, must satisfy the inequalities $0\le N\le b$. It is shown that for each fixed nonnegative integer k in the set $\{0,1,2,\ldots,b\}$, there exist infinitely many potentials in class $\mathcal{A}_b$ for which the corresponding Schrodinger operator has exactly k bound states. Some auxiliary results are presented to relate the number of bound states to the number of real resonances associated with the corresponding Schrodinger operator. The theory presented is illustrated with some explicit examples.

Submitted December 16, 2018. Published February 11, 2019.
Math Subject Classifications: 39A70, 47B39, 81U15, 34A33.
Key Words: Discrete Schrodinger operator; half-line lattice; bound states; resonances; compactly-supported potential; number of bound states.

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Tuncay Aktosun
Department of Mathematics
University of Texas at Arlington
Arlington, TX 76019-0408, USA
email: aktosun@uta.edu
Abdon E. Choque-Rivero
Instituto de Física y Matemáticas
Universidad Michoacana de San Nicolás de Hidalgo
Ciudad Universitaria, C.P. 58048
Morelia, Michoacán, México
email: abdon@ifm.umich.mx
Vassilis G. Papanicolaou
Department of Mathematics
National Technical University of Athens
Zografou Campus, 157 80
Athens, Greece
email: papanico@math.ntua.gr

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