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.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Gerald Teschl, Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials, Opuscula Math. 33, no. 3 (2013), 467-563, http://dx.doi.org/10.7494/OpMath.2013.33.3.467

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.