F. Gesztesy

Using the Davies-Helffer-Sjostrand functional calculus based on almost analytic extensions, we address the following problem: Given a self-adjoint operator $S$ in $\mathcal H$, and functions $f$ in an appropriate class, for instance, $f \in C_0^{\infty}(\mathbb R)$, how to control the norm $\|f(S)\|_{\mathcal B(\mathcal H)}$ in terms of the norm of the resolvent of $S$, $\|(S - z_0 I_{\mathcal H})^{-1}\|_{\mathcal B(\mathcal H)}$, for some $z_0 \in \mathbb C\backslash\mathbb R$. We are particularly interested in the case where $\mathcal B(\mathcal H)$ is replaced by a trace ideal, $\mathcal B_p(\mathcal H)$, $p \in [1,\infty)$.

We determine square root domains for non-self-adjoint Sturm-Liouville operators of the type $$ L_{p,q,r,s} = - \frac{d}{dx}p\frac{d}{dx}+r\frac{d}{dx}-\frac{d}{dx}s+q $$ in $L^2((c,d);dx)$, where either $(c,d)$ coincides with the real line $\mathbb R$, the half-line $(a,\infty)$, $a \in \mathbb R$, or with the bounded interval $(a,b) \subset \mathbb R$, under very general conditions on the coefficients $q, r, s$. We treat Dirichlet and Neumann boundary conditions at $a$ in the half-line case, and Dirichlet and/or Neumann boundary conditions at $a,b$ in the final interval context. (In the particular case $p=1$ a.e. on $(a,b)$, we treat all separated boundary conditions at $a, b$.)