An inverse problem for weighted Paley-Wiener spaces (Q2835438)

A classical Hamiltonian system of order two on \([0,\ell]\) can be expressed as the \(2\times 2\) matrix equation \(JX'(r,z)=z\mathcal{H}(r) X(r,z)\) (\(r\in[0,\ell]\), \(z\in\mathbb{C}\)), where \(J\) is the square root of \(I\) obtained by interchanging the columns of \(I\) and \(\mathcal{H}\) maps \([0,\ell]\) to \(2\times 2\) real, non-negative matrices. One can define a fundamental matrix \(M\) such that \(JM'=z\mathcal{H}M\), \(M(0,z)=I\) and \(M(r,z)\) has columns \(\Theta_{\mathcal{H}}(r,z)\), \(\Phi_{\mathcal{H}}(r,z)\). If \(\mathcal{H}_0=I\), then \(\Theta_{\mathcal{H}_0}=(\cos(rz),-\sin(rz))^T\). One defines a linear map \(S_{\mathcal{H}}\) that sends \(\Theta_{\mathcal{H}_0}(\cdot,z)\to \Theta_{\mathcal{H}}(\cdot,z)\) (\(z\in\mathbb{C}\)) thus defining \(S_{\mathcal{H}}\) on the linear span of \(\{\Theta_{\mathcal{H}_0}(\cdot,z)\}_{z\in\mathbb{C}}\), a dense subspace of \(L^2([0,a],\mathbb{C}^2)\). For \(r\in [0,\ell]\), the Hilbert space \(L^2(\mathcal{H},r)\) is defined as those vectors \(X\) such that \(\langle \mathcal{H}(s)X(s),X(s)\rangle\in L^2([0,r])\) modulo those vectors a.e.\ in the kernel of \(\mathcal{H}\). If \((\Theta^{-}_{\mathcal{H}},\Phi^{-}_{\mathcal{H}})\) denotes the bottom row of \(M\), then the Weyl function \(\Theta^{-}_{\mathcal{H}}(\ell,\cdot)/ \Phi^{-}_{\mathcal{H}}(\ell,\cdot)\) has positive imaginary part in the upper half plane and there is a measure \(\mu\), called the principal spectral measure of \(\mathcal{H}\), satisfying \(\int\frac{d\mu(t)}{1+t^2}<\infty\) and representing the Weyl function as \(\frac{1}{\pi}\int_{\mathbb{R}}\bigl(\frac{1}{t-z}-\frac{t}{1+t^2}\bigr)\, d\mu(t)+bz+c\) (\(\operatorname{Im}z>0\)) for some \(b\geq 0\) and \(c\in \mathbb{R}\). The inverse problem is to infer \(\mathcal{H}\) from \(\mu\).NEWLINENEWLINEThe authors consider \(\mu\) such that (i) \(\mu(\{0\})>0\) and \(\int\frac{d\mu(t)}{1+t^2}<\infty\); (ii) \(\|f\|_{L^2_\mu}\) and \(\|f\|_{L^2(\mathbb{R})}\) are comparable for \(f\in \text{PW}_a\), the \(L^2\)-functions bandlimited to \([-a,a]\); and (iii) \(\text{PW}_a\) is dense in \(L^2_\mu\). The main theorem states that \(\mu\) satisfying these conditions is equivalent to \(\mu\) being the principal spectral measure for a Hamiltonian \(\mathcal{H}\) on \([0,\ell]\) such that \(S_{\mathcal{H}}\) extends to a bounded, continuously invertible operator between \(L^2(\mathcal{H}_0,[0,a])\) and \(L^2(\mathcal{H},[0,\ell])\). As remarked, it remains an open problem to find tractable descriptions of Hamiltonians of principal spectral measures.NEWLINENEWLINEKey tools from de Branges spaces are used. The first, Lemma 3.1, gives a condition for a chain of de Branges spaces associated with Hamiltonians to be isomorphic to a comparable chain of Paley-Wiener spaces. The second main tool, Lemma 3.2, gives a condition under which \(\mu\) is a principal spectral measure of a Hamiltonian \(\mathcal{H}\) in terms of the restriction of a projection from a de Branges space associated with \(\mathcal{H}\) being a unitary operator on \(L^2_\mu\). The authors also construct a bounded, continuously invertible Hamiltonian \(\mathcal{H}\) such that no associated de Branges space is isomorphic to \(\text{PW}_a\).