On a class of Schrödinger operators exhibiting spectral transition

Abstract

We show that the operator for $\lambda <\frac{{\pi }^2}{4}$ is bounded from below and has purely discrete spectrum, while for $\lambda >\frac{{\pi }^2}{4}$ its spectrum contains the real line. In the critical case $\lambda =\frac{{\pi }^2}{4}$ we prove that the spectrum coincides with the half line $[0,\infty )$.