Strict domain monotonicity of the principal eigenvalue and a characterization of lower boundedness for the Friedrichs extension of four-coefficient Sturm–Liouville operators

Using the variational characterization of principal (i.e., smallest) eigenvalue below essential spectrum a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing finite interval length) Friedrichs extension $$ T_F$$ minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In more general singular context, these expressions act according $$\tau f= \frac{1}{r}(-(f^{[1]})'+sf^{[1]}+qf)\, {\rm with}\, f^{[1]}=p[f'+sf]\, on}\, (a,b) \subseteq \mathbb{R}, where coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p > 0, r 0 a.e. p^{-1},q,r,s \in L^{1}_{loc}((a,b);dx)$$ , f$$ is supposed satisfy $$f\in AC_{loc}((a,b)),p[f'+sf] AC_{loc}((a,b)). This setup sufficiently so that permits certain distributional potential including potentials in $$H^{-1}_{loc}((a,b)) . As consequence case, basis oscillation theory our main result, characterize all bounds $$T_F as those $$\lambda \mathbb{R} which equation u =\lambda has strictly positive solution b).