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Jacobi flow on SMP matrices and Killip-Simon problem on two disjoint intervals. (English) Zbl 1372.47041
A seminal result of R. Killip and B. Simon [Ann. Math. (2) 158, No. 1, 253–321 (2003; Zbl 1050.47025)] provides a complete spectral characterization of Jacobi matrices \(J\) such that \(J-J_0\) are in the Hilbert-Schmidt class, where \(J_0\) is the discrete Laplacian. The authors establish a counterpart of this result in the following setting.
Let \(E\) be a union of two disjoint intervals, properly normalized, \[ E=[x_0,y_0]\cup [x_1,y_1]=V^{-1}([-2,2]), \quad V(z):=\alpha z+\beta-\frac1z\,, \quad \alpha>0, \;\;\beta\in\mathbb{R}. \] Let \(\sigma\) be a measure on \(E\cup X\), where \(X=\{x_k\}\) is an at most countable set of nonzero points in \(\mathbb{R}\backslash E\). \(\sigma\) satisfies the Killip-Simon (KS) condition if \[ \int_E |\log\sigma'(x)|\,\sqrt{\text{dist}(x,\mathbb{R}\backslash E)}\,dx+\sum_{x_k\in X} (\text{dist}(x,E))^{3/2}<\infty. \] The main result of the paper concerns a complete parametric representation of the class \(KS(E)\) of Jacobi matrices with the spectral measures satisfying the KS condition.
A key ingredient of the construction is a class of \(5\)-diagonal structured matrices, called the SMP matrices, and a discrete dynamical system on this class, called the Jacobi flow. Each \(J\in KS(E)\) is obtained via such Jacobi flow with the initial matrix \(A\) being an arbitrary SMP matrix so that \(V(A)-S^2-S^{-2}\) is in the Hilbert-Schmidt class, where \(S\) is the shift operator.

MSC:
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
58J53 Isospectrality
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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