The Marchenko representation of reflectionless Jacobi and Schrödinger operators (Q2796520)

The authors study 1-D Schrödinger operators on the line, NEWLINE\[NEWLINE(H y)(x)=-y''(x)+V(x)y(x)NEWLINE\]NEWLINE with locally integrable real potentials \(V\) that are limit point at both \(+\infty\) and \(-\infty\) (that is, \(H\) defined on the minimal domain is essentially self-adjoint in \(L^2(\mathbb R)\)), and Jacobi matrices NEWLINE\[NEWLINE(Ju)_n=a_{n-1}u_{n-1}+b_nu_n+a_nu_{n+1},\quad n\in\mathbb N,NEWLINE\]NEWLINE where \(a,b\in\ell^\infty(\mathbb Z)\), \(a_n>0\), \(b_n\in\mathbb R\). These operators admit half-line \(m\)-functions \(m_+\) and \(m_-\). These functions map the upper half-plane into itself, that is, they are Herglotz-Nevanlinna functions. The operator is called \textit{reflectionless} on an open interval \(S=(a,b)\) if NEWLINE\[NEWLINEm_+(x)=-m_-(x)^\ast\text{ for a.e. }x\in S.\eqno{(1)}NEWLINE\]NEWLINE Reflectionless operators have many remarkable properties and they are important for several reasons. For example, multi-soliton profiles for the KdV and Toda equations serve as reflectionless potentials on \((0,\infty)\) and \((-2,2)\), respectively.NEWLINENEWLINELet \(\mathcal M_R\) be the so-called Marchenko class, which consists of Schrödinger (Jacobi) operators that are reflectionless on \((0,\infty)\) (on \((-2,2)\)) and \(\sigma(H)\subseteq [-R^2,\infty)\) for some \(R\leq 0\) (\(\| J\|\leq R\) for some \(R\geq 2\)). In fact, (1) holds for \(S=(a,b)\) if and only if \(m_+\) admits a holomorphic continuation \(M:\mathbb C_+\cup S\cup\mathbb C_-\to\mathbb C_+\). Notice that this continuation is given by \(M(z)=-m_-(z^\ast)^\ast\) for all \(z\in\mathbb C_-\). Then for a simply connected domain \(\Omega:=\mathbb C_+\cup S\cup\mathbb C_-\) one can define a conformal map \(\varphi:\mathbb C_+\to\Omega\) and hence obtain a new Herglotz-Nevanlinna function NEWLINE\[NEWLINEF(z):=M(\varphi(z)),\quad z\in\mathbb C_+.NEWLINE\]NEWLINE The measures from integral representations of these functions \(F\) are then used to parameterize the operators from the set \(\mathcal M_R\) (see Theorems 3.1 and 4.1). As an immediate application of these results in the discrete case, the authors present a simple proof of Theorem 1.2 from [the third author, Proc. Am. Math. Soc. 139, No. 6, 2175--2182 (2011; Zbl 1218.42011)], which states that: if \(J\in \mathcal M_R\) for some \(R\geq 2\), then \(a_n\geq 1\) for all \(n\in\mathbb Z\) and \(a_n=1\) for some \(n\in\mathbb Z\) only if \(a_n\equiv 1\) and \(b_n\equiv 0\). Moreover, these parameterisations in the continuous case are used to investigate properties of potentials \(V\) such that the corresponding half-line Schrödinger operators \(H_+\) satisfy \((0,\infty)\subset \sigma_{\mathrm{ac}}(H_+)\).