On $m$-order logarithmic Schrödinger operator

Abstract

In this paper we study the logarithm of order $m$ of the Schrödinger operator ${\mathcal{L}}_V$ in ${\mathbb{R}}^d$, for certain nonnegative potentials $V$. First, the operator ${\log }^m{\mathcal{L}}_V$, $m\in \mathbb{N}$, is defined by using the spectral measure associated with the self-adjoint operator ${\mathcal{L}}_V$ on a suitable subspace of $L^2({\mathbb{R}}^d)$. Then, the semigroup of operators $\{T_t^V{\}}_{t>0}$ generated by ${\mathcal{L}}_V$ allows us to extend the definition of ${\log }^m{\mathcal{L}}_V$ to a wider class of Lipschitz functions. By using logarithmic operators ${\log }^m{\mathcal{L}}_V$, $m\in \mathbb{N}$, we prove Taylor expansions for the fractional powers ${\mathcal{L}}_V^s$ and ${\mathcal{L}}_V^{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p({\mathbb{R}}^d)$, $1<p<\infty$.