TY  - JOUR
ID  - Lotoreichik2011
LB  - Lotoreichik2011
AU  - Lotoreichik, Vladimir
TI  - Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set
ST  - Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 4
SP  - 615
EP  - 628
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.4.615
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.4.615
AB  - We say that a discrete set \(X = \{ x_n \}_{n\in \mathbb{N}_0}\) on the half-line \[0 = x_0 \lt x_1 \lt x_2 \lt x_3 \lt ... \lt x_n \lt ... \lt +\infty \] is sparse if the distances \(\Delta x_n = x_{n+1}- x_n\) between neighbouring points satisfy the condition \(\frac{\Delta x_n}{\Delta x_{n-1}} \to +\infty\). In this paper half-line Schrödinger operators with point \(\delta\)- and \(\delta'\)-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to \(\infty\) we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with \(\mathbb{R}_+\).
KW  - half-line Schrödinger operators
KW  - \(\delta\)-interactions
KW  - \(\delta '\)-interactions
KW  - singular continuous spectrum
SN  - 1232-9274
ER  -