We consider Schrödinger operators Hα given by equation
(1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ)
for λ → 0 and
the L1 →
L∞ dispersive estimates associated to the
evolution operator e−
itHα.
In particular we prove that for positive values of α, the spectral density
E(Hα,λ)
tends to zero as λ →
0 with higher speed compared to the spectral density of Schrödinger
operators with a short-range potential V. We then show how the long time behavior of
e−
itHα
depends on α.
More precisely we show that the decay rate of e−
itHα
for t → ∞ can
be made arbitrarily large provided we choose α large enough and consider a suitable operator
norm.