Long-time tails in the parabolic Anderson model with bounded potential

We consider the parabolic Anderson problem $partial_t u=kappaDelta u+xi u$ on $(0,infty)times Z^d$ with random i.i.d. potential $xi=(xi(z))_{zinZ^d}$ and the initial condition $u(0,cdot)equiv1$. Our main assumption is that $esssupxi(0)=0$. Depending on the thickness of the distribution $prob(xi(0)incdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $ttoinfty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrodinger operator $-kappaDelta-xi$ at the bottom of its spectrum. In our class of $xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $infty$; the power law is typically accompanied by lower-order corrections.