2013/03/22 by Fritz Gesztesy, Gesztesy, Fritz, Rudi Weikard +1 · 1 citation
Mathematics · Physics and Astronomy · #34C10 #34C25 #34K13 #34L05 #34L25 #34L40 #35Q58 #37K10 #47A10 #47A63 #47A75 #47E05 #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Primary 34B24 #Quantum Mechanics and Non-Hermitian Physics #Secondary 34B20 #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP #msc:34B20 #msc:34B24 #msc:34C10 #msc:34C25 #msc:34K13 #msc:34L05 #msc:34L25 #msc:34L40 #msc:35Q58 #msc:37K10 #msc:47A10 #msc:47A63 #msc:47A75 #msc:47E05
paper · pdf · doi:10.48550/arxiv.1303.5793
44 pages
arxiv created 2013/03/22 · openalex publication_date 2013/03/22 · arxiv updated 2013/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider left-definite eigenvalue problems A ψ= λB ψ, with A ≥ ε I for some ε > 0 and B self-adjoint, but B not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa-Holm hierarchy. In fact, we will treat a more general version where A represents a positive definite Schrödinger or Sturm-Liouville operator T in L2(\bbR; dx) associated with a differential expression of the form τ= - (d/dx) p(x) (d/dx) + q(x), x ∈ \bbR, and B represents an operator of multiplication by r(x) in L2(\bbR; dx), which, in general, is not a weight, that is, it is not nonnegative a.e. on \bbR. Our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients q and r and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space L2(\bbR; dx). Our approach relies on rewriting the eigenvalue problem A ψ= λB ψ in the form A-1/2 B A-1/2 χ= λ-1 χ, χ= A1/2 ψ, and a careful study of (appropriate realizations of) the operator A-1/2 B A-1/2 in L2(\bbR; dx). In the course of our treatment we employ a supersymmetric formalism which permits us to factor the second-order operator T into a product of two first-order operators familiar from (and inspired by) Miura's transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients q and r, where q may be a distribution and r generates a measure and hence no smoothness is assumed for q and r.