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On sufficient and necessary conditions for the Jacobi matrix inverse eigenvalue problem. (English) Zbl 1058.65046

The authors establish a sufficient condition, and some necessary conditions which complement those previously given by the first author and W. W. Sun [Inverse Probl. 15, 977–987 (1999; Zbl 0942.65042)], for the existence of real numbers \(\beta _1 ,\dots ,\beta_{n-1}\), such that the \(n\times n\) Jacobi matrix with \(i\)th diagonal element \(1+\beta _{i-1}^2\), and \(i\)th superdiagonal element \(\beta _i\), where \(\beta _0 =0\), has a given set of eigenvalues. They also give an algorithm for computing some of the (generally non-unique) solutions.
This matrix arises from the discretization of a certain Sturm–Liouville problem. When such discretizations are used to solve inverse Sturm-Liouville problems numerically, allowance must be made for the different asymptotic behaviour of the eigenvalues of the discrete and continuous problems.
For some recent references on how this may be done, see the reviewer [Inverse Probl. 21, 223-238 (2005)].

MSC:

65F18 Numerical solutions to inverse eigenvalue problems
15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors
65L09 Numerical solution of inverse problems involving ordinary differential equations
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34A55 Inverse problems involving ordinary differential equations
34B24 Sturm-Liouville theory

Citations:

Zbl 0942.65042
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References:

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