Logarithmic Schr\"odinger operators

Abstract

In this paper we consider the Schr\"odinger operator $\mathcal L_V= -\Delta + V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schr\"odinger operator $\log \mathcal L_V$ proving its main properties. We obtain a pointwise representation of $\log \mathcal L_V$ when $V$ satisfies a reverse H\"older 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 Schr\"odinger 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 $\mathcal L_V$.