P. Kargaev, E. Korotyaev
The Inverse Problem for Hill Operator, the Direct Approach
The paper is published: Invent. Math., 129, 3 (1997), 567-593

MSC:

34A55 Inverse problems

34B30 Special equations (Mathieu, Hill, Bessel, etc.)

Abstract: Let $G_n=(A_n^-, A_n^+) , n \geq 1,$ denote the set of
gaps of the Hill operator $T = -d^2/dx^2+V(x)$ in $L^2({\bf R})$ where $V$
is an even 1-periodic real potential from $L^2(0,1)$
and $h_n$ be heights of the corresponding slits
on the quasimomentum domain, $M_n^{\pm}$ be effective masses associated
with the edges of the gap $G_n$. Let $g_n , n \geq 1,$ denote the gaps of the
operator $T_0=\sqrt {T-N_0} \geq 0 $ where $N_0 $ is the beginning
of the spectrum of $T$ , and $\mu _n^{\pm}$ be the
reduced masses (analog of the effective masses) connected with the gap
$g_n$. We study the inverse problem for the mappings
$V \to \{|g_n|\}, V \to \{h_n\}, V \to \{\mu_n^{\pm}\}$ and
$ V \to \{M_n^{\pm}\}$ by a direct approach.


Keywords: Hill operator, inverse problem