2022/01/16 by Fritz Gesztesy, Gesztesy, Fritz, Roger Nichols +1 · 1 citation
Mathematics · #34L05 #34L15 #47E05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Primary: 34B24 #Secondary: 47A10 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2201.05948
openalex publication_date 2022/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 TF of 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 = (1)/(r) ( - (f[1])' + s f[1] + qf) \text with f[1] = p [f' + s f] 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 ∈ L1loc((a,b); dx), and f is supposed to satisfy f ∈ ACloc((a,b)), p[f' + s f] ∈ ACloc((a,b)). This setup is sufficiently general so that τ permits certain distributional potential coefficients q, including potentials in H-1loc((a,b)). As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result, we characterize all lower bounds of TF as those λ∈ ℝ for which the differential equation τu = λu has a strictly positive solution u > 0 on (a,b).