Logarithmic Schrödinger operators

Abstract

In this paper we consider the Schrödinger operator ${\mathcal{L}}_V=-\hat{I}”+V$ in ${\mathbb{R}}^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schrödinger operator $\log {\mathcal{L}}_V$ proving its main properties. We obtain a pointwise representation of $\log {\mathcal{L}}_V$ when $V$ satisfies a reverse Hölder inequality of exponent $q>\frac{d}{2}$ by using the semigroup of operators $\{T_t^V{\}}_{t>0}$ generated by ${\mathcal{L}}_V$. We consider the Lipschitz function space adapted to the Schrödinger setting to solve the initial value problem \[ \begin{cases} \frac{\partial u}{\partial t}=-(\log \mathcal{L}_V)u, & \text{in } \mathbb{R}^n \times (0,\infty), \\ u(x,0)=f(x), & x \in \mathbb{R}^d \end{cases} \] in terms of the fractional integral associated with ${\mathcal{L}}_V$.