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 $[0, \infty)$.