On the point spectrum of discrete Schrödinger operator
Consider the discrete Schrödinger operator \(L\) defined by \[ Lu_ n= u_{n+ 1}+ u_{n- 1}+ q_ n u_ n,\qquad n= 1,2,\dots\;. \] The equation for eigenvalues \(\lambda\) of \(L\), \(Lu= \lambda u\), means that a certain sequence \(u_ \lambda= \{u_ n\}^ \infty_{n= 1}\in \ell_ 2\) satisfies the relation \[ Lu_ n= \lambda u_ n,\qquad n= 1,2,\dots \] with \(u_ 0= 0\). In the present paper the authors have obtained two results giving a precise condition for an eigenvalue of \(L\) to appear in the interval \((-2,2)\), depending on the following two parameters: the decay law of the potential \(q\) and the distance from \(\lambda\) to the boundary of the continuous spectrum.
scientific article; zbMATH DE number 525656