Sturm-liouville operators with singular potentials

Reconstruction of energy-dependent Sturm-Liouville equations from two spectra. II

“…Therefore we need to define the action of ℓ(y) more rigourously. To do this we use the regularization procedure due to Savchuk and Shkalikov (see [19,20]) based on the notion of quasi-derivatives. Namely, for every absolutely continuous function y we denote by y [1] := y ′ − ry its quasi-derivative and define ℓ(y) as…”

“…Denote by n(t) the number of zeros ofψ in the disk |z| ≤ t; then, in view of (19), the Jensen's formula gives…”

“…q = r ′ with a real-valued r ∈ L 2 (0, 1). We consider this equation under two types of boundary conditions: the Dirichlet ones y(0) = y(1) = 0 (2) and the so-called mixed conditions y(0) = y [1] (1) + Hy(1) = 0, where H ∈ R is some constant and y [1] := y ′ − ry is a quasi-derivative of the function y used in the regularization procedure due to Savchuk and Shkalikov (see [19,20] and the next section for details). Since primitive of q is defined only up to an additive constant, by replacing r with r − H we reduce the above mixed boundary conditions to the following ones:…”

On spectra of non-self-adjoint Sturm–Liouville operators

“…The precise definition of a Sturm-Liouville operator (1.1) with distributional potential q ∈ W −1 2 (0, 1) goes as follows [27]. We take σ ∈ L 2 (0, 1) to be the distributional primitive of q of zero mean and denote by T the operator in L 2 (0, 1) given by T y = l(q)y := −(y − σy) − σy on the domain dom T = {y ∈ W 1 1 (0, 1) | y [1] := y − σy ∈ W 1 1 (0, 1), l(q)y ∈ L 2 (0, 1)}.…”

“…We take σ ∈ L 2 (0, 1) to be the distributional primitive of q of zero mean and denote by T the operator in L 2 (0, 1) given by T y = l(q)y := −(y − σy) − σy on the domain dom T = {y ∈ W 1 1 (0, 1) | y [1] := y − σy ∈ W 1 1 (0, 1), l(q)y ∈ L 2 (0, 1)}. We denote by T (q, ∞) the restriction of T by the Dirichlet boundary conditions y(0) = y(1) = 0 and recall [13,27] that T (q, ∞) has discrete spectrum accumulating at +∞ and that the sequence (λ n ) n∈N of its eigenvalues obeys the asymptotics λ n = (πn +λ n ) 2 , (λ n ) ∈ 2 .…”