**Opuscula Math. 33, 467-563 (2013)**[DOI: 10.7494/OpMath.2013.33.3.467]

## Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

### Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, and Gerald Teschl

*(a,b) ⊆ ℝ*associated with rather general differential expressions of the type

*τ f = 1/r ( - (p[f' + s f])' + s p[f' + s f] + qf),*

*p*,

*q*,

*r*,

*s*are real-valued and Lebesgue measurable on

*(a,b)*, with

*p≠ 0*,

*r>0*a.e. on

*(a,b)*, and

*p*,

^{-1}*q*,

*r*,

*s ∈ L*, and

^{1}_{loc}((a,b); dx)*f*is supposed to satisfy

*f ∈ AC*

_{loc}((a,b)), p[f' + s f] ∈ AC_{loc}((a,b)).*τ*permits a distributional potential coefficient, including potentials in

*H*.

^{-1}_{loc}((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*, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of

_{min}*T*. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira

_{min}*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 *τ* 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).

** MSC2010:** Primary 34B20, 34B24, 34L05; Secondary 34B27, 34L10, 34L40.

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