On a simple identity for eigenvalues of second order differential operator

Determining the potential of a Sturm–Liouville operator from its Dirichlet and Neumann spectra

“…The endpoints of a gap are either a pair of periodic or of anti-periodic eigenvalues of Q. Therefore the Dirichlet and Neumann eigenvalues of − This periodic spectrum, {4µ n , 4ν n , 4ν 0 } determines ∆(λ) by Proposition (6). From ∆(λ) we find the anti-periodic spectrum as the roots of ∆(λ) + 2.…”

“…Its and Matveev [10], Gelfand [5], Gelfand and Levitan [6], McKean [15], Garnett [4], Trubowitz [17], and Buslaev and Faddeev [2] illustrate that for periodic potentials the periodic, anti-periodic, and Dirichlet spectra determine the potential.…”

“…Let {λ j } denote the joint periodic and anti-periodic spectra of this operator. Note that the periodic spectrum determines the anti-periodic spectrum (see Proposition (6)). The advantage of an even potential is that its periodic and anti-periodic spectra are also its Dirichlet and Neumann spectra.…”

Absolute summability of the trace relation for certain Schrödinger operators

“…Further explicit examples can be found I U, Λ "C U in Remark 2.5 in the context of reflectionless (7V-soliton) potentials and in (4.18)- (4.20) in connection with periodic potentials. In fact, historically, after the pioneering work by GeΓfand and Levitan [12] on regularized traces for Schrodinger operators on a compact interval, the trace formula (4.19) for periodic (and certain classes of almost periodic) potentials was one of the two previously systematically studied trace formulae of the type (1.6) for Schrodinger operators on the whole real line (see, e.g., [8,11,22,30,34] and more recently [5,24,27,28]). The other case studied in detail by Deift and Trubowitz [7] in 1979 was concerned with shortrange potentials V(x) decaying sufficently fast as \x\ -> oo under the assumption that H = -j^2 + y na s no eigenvalues.…”

A trace formula for the Sturm-Liouville type equation with retarded argument

“…Gelfand and Levitan [1] …rstly obtained a trace formula for a self adjoint SturmLiouville di¤erential equation. This investigation was continued in many directions, such as Dirac systems [2][3][4], the case of continuous [5][6][7][8][9][10][11], discontinuous [12,13] or matrix Sturm-Liouville operator [14][15][16] and also Sturm-Liouville problems with retarded argument [17][18][19].…”