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 the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension TFof the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In the more general singular context, these four-coefficient differential expressions act according to τf=1r(-(f[1])′+sf[1]+qf)withf[1]=p[f′+sf]on(a,b)⊆R,where the coefficients 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∈Lloc1((a,b);dx), and fis supposed to satisfy f∈ACloc((a,b)),p[f′+sf]∈ACloc((a,b)).This setup is sufficiently general so that τpermits certain distributional potential coefficients q, including potentials in Hloc-1((a,b)).