Spectral and Scattering Theory for Ordinary Differential Equations

Minimization of lowest positive periodic eigenvalue for the Camassa–Holm equation with indefinite potential

“…For the lowest positive eigenvalue, we have the following minimization characterization, which is a consequence of general results for the usual Courant-Fischer and can also be deduced from [20,Section 3]. Since problem (1.5) can be understood as a self-adjoint operator/linear relation, the proof of the following variational characterization can be obtained by similar arguments to those in [3,4,5].…”

“…This space is suitable to obtain the self-adjoint realization of (2.2). For self-adjoint relations, we refer to the monograph [5] for a detailed discussion in the setting of ordinary differential equations, and to [19,20] for the Camassa-Holm equations.…”

Series representation for the Jost solution of the Sturm‐Liouville equation in impedance form

“…A g(𝜆)𝜑(𝜆, x)d𝜈(𝜆), where the limit converges in  p (the scheme of the proof can be found in Bennewitz et al 34 , Th. 4.37 and Akhiezer and Glazman 22,Th.…”

Inverse scattering problem with Levinson formula for eigenparameter‐dependent discrete Sturm–Liouville equation

“…Therefore, it is extremely important to analyze the various studies that underlie these problems. [1][2][3][4][5][6][7][8][9] Let the following boundary value problem (BVP)…”

“…Especially, examining the behavior of wave functions at infinity contributed greatly to the development of the inverse scattering theory of discrete Sturm–Liouville equations. Therefore, it is extremely important to analyze the various studies that underlie these problems 1–9 …”