Opuscula Math. 33, no. 3 (2013), 467-563
http://dx.doi.org/10.7494/OpMath.2013.33.3.467

Opuscula Mathematica

# Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials

Jonathan Eckhardt
Fritz Gesztesy
Roger Nichols
Gerald Teschl

Abstract. We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $$(a,b) \subseteq \mathbb{R}$$ associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients $$p, q, r, s$$ are real-valued and Lebesgue measurable on $$(a,b)$$, with $$p \neq 0$$, $$r \gt 0$$ a.e. on $$(a,b)$$, and $$p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)$$, and $$f$$ is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*} In particular, this setup implies that $$\tau$$ permits a distributional potential coefficient, including potentials in $$H_{loc}^{-1}((a,b))$$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $$T_{max}$$, or equivalently, all self-adjoint extensions of the minimal operator $$T_{min}$$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $$T_{min}$$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $$T_{min}$$. Finally, in the special case where $$\tau$$ is regular, we characterize the Krein-von Neumann extension of $$T_{min}$$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).

Keywords: Sturm-Liouville operators, distributional coefficients, Weyl-Titchmarsh theory, Friedrichs and Krein extensions, positivity preserving and improving semigroups.

Mathematics Subject Classification: 34B20, 34B24, 34L05, 34B27, 34L10, 34L40.

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• Jonathan Eckhardt
• University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria
• Fritz Gesztesy
• University of Missouri, Department of Mathematics, Columbia, MO 65211, USA
• Roger Nichols
• The University of Tennessee at Chattanooga, Mathematics Department, 415 EMCS Building, Dept. 6956, 615 McCallie Ave, Chattanooga, TN 37403, USA
• Gerald Teschl
• University of Vienna, Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria
• International Erwin Schrödinger, Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria
• Revised: 2013-01-18.
• Accepted: 2013-01-24.