• DOI:10.1051/MMNP/20149511
• Corpus ID: 119577062

Schr\"odinger operators on a half-line with inverse square potentials

@article{Kovak2014SchrodingerOO,
  title={Schr\"odinger operators on a half-line with inverse square potentials},
  author={Hynek Kovař{\'i}k and Françoise Truc},
  journal={arXiv: Mathematical Physics},
  year={2014}
}

• H. Kovařík, F. Truc
• Published 14 March 2014
• Mathematics
• arXiv: Mathematical Physics

We consider Schr\^odinger operators $H_\alpha$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_\alpha, \lambda)$ when $\lambda$ goes to $0$ and the $L^1\to L^\infty$ dispersive estimates associated to the evolution operator $e^{-i t H_\alpha}$. In particular we prove that for positive values of $\alpha$, the spectral density tends to zero as $\lambda\to 0$ with higher speed compared to the spectral density of Schr\"odinger operators with a short-range potential $V$. We then show how the long time behavior of $e^{-i t H_\alpha}$ depends on $\alpha$. More precisely we show that the decay rate of $e^{-i t H_\alpha}$ for $t\to\infty$ can be made arbitrarily large provided we choose $\alpha$ large enough and consider a suitable operator norm.