Almost periodicity of some random potentials

The KDV Hierarchy and Associated Trace Formulas

“…Moreover, (5.30) and (5.33) also extend to certain classes of almost-periodic V (x), see, e.g., [8], [28], [31], [32], Chs. 9, 11.…”

“…The general case j = N appeared in McKean and van Moerbeke [34] and Flaschka [13]. More recent accounts of (5.33) can be found in [8], [28], [31], [32], Chs. 9, 11, [40].…”

On isospectral sets of Jacobi operators

“…Reflectionless conditions such as (3.8) have been used by a variety of authors for particular cases such as almost periodic potentials (see, e.g., [2], [3], [11], [14], [25], [34], [35]) and scattering theoretic situations (cf., e.g., [13], [16]). The following result further illustrates (H.3.1)(iii).…”

“…7. One obtains, a(n) = R λs ± (λ, n, n 0 )s ± (λ, n + 1, n 0 )dω ±,n0 (λ), ±(n − n 0 ) ≥ 1 2 , b(n) = R λs ± (λ, n, n 0 ) 2 dω ±,n0 (λ), ±(n − n 0 ) ≥ 1, 25) where s ± (λ, n, n 0 ), ±(n−n 0 ) ≥ 1 are polynomials (of degree ±(n−n 0 )) orthonormal with respect to dω ±,n0 (λ). This determines H and (4.15).…”

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.…”