On the spectral surface of a model two-parameter Sturm–Liouville problem

Abstract

The model eigenvalue problem d 2 w dr 2 þ ðE þ z sgn rÞw ¼ 0, w(À1) ¼ w(1) ¼ 0, contains two complex parameters E and z. Considering E as a function of z one obtains a spectral Riemann surface. Level crossings between sheets of the spectral surface are given explicitly, and the existence of a double sequence of branch points is proved.

On Sturm–Liouville equations with several spectral parameters

“…We will take one example, the "hyperbolic" refractive profile n(x) = n(0)e (x/b) log(n(b)/n(0)) (30) Figure 7: Logarithmic plots of |R| 2 (solid), (n 2 /n 1 )|T | 2 (dotted) and formula (31) (dashed) as functions of the adimensional magnitude b/λ 2 ∈ [β/(2π), 1], for n(x) given by (30).…”

“…The following example involves consideration of complex eigenvalues. The boundary value problem y ′′ (t) + (E + z sgn t)y(t) = 0, y(−1) = y(1) = 0, (24) where sgn x is the sign of x, was studied in detail in [30]. A spectral surface is formed of pairs…”

“…This produces analytic expressions R(λ 1 , λ 2 ), T (λ 1 , λ 2 ) which, while they are not simple power series, serve conveniently for calculations. We will take one example, the "hyperbolic" refractive profile given by (30).…”