Logarithmic Schrödinger operators

In this paper we consider the Schrödinger operator ${\mathcal{L}}_V=-\mathrm{\Delta }+V$ in ${\mathbb{R}}^d$ with a non negative potential $V$, and $V\equiv ̸0$. 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 $$\{\begin{array}{ll}\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=-(\log {\mathcal{L}}_V)u,& \text{in }{\mathbb{R}}^n\times (0,\mathrm{\infty }),\\ u(x,0)=f(x),& x\in {\mathbb{R}}^d\end{array}$$ in terms of the fractional integral associated with ${\mathcal{L}}_V$.