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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 018 119 071
Gibson, Peter C.:
Inverse spectral theory of finite Jacobi matrices
Trans. Am. Math. Soc. 354, No. 12, 4703 - 4749 (2002)
15A29Inverse problems
Primary Classification:
47B36Jacobi tridiagonal operators matrices and generalizations
Secondary Classification:
34K29Inverse problems
14D21Applications of vector bundles and moduli spaces in mathematical physics twistor theory, instantons, quantum field theory
Jacobi matrices; Titchmarsh-Weyl m-function; orthogonal polynomials; inverse spectral theory; Green's function; probability distributions; factorization of the inverse problem; parametrization of the fibres; singular solutions; explicit formulas

People who love Jacobi matrices will be excited as well as puzzled by
the presented construction. And there definitely exist quite a few,
even if only for the close connection between the Jacobi matrices J
and physics. For the rest of the population, the text is probably too
thick. Still, its reading is of its own reward: its relationship to
the classical moment problem offers the key sample reason, and for me
it was its character of the generalization of the old relationship of
J to continued fractions and so called Titchmarsh-Weyl m-functions.
An arbitrary prescribed function f(z) is assumed equal to the (i,j)'s
Green's function of some J which is to be re-constructed. The core of
the paper lies in the proposal of a geometric arrangement of the
information about the solution set: In an overall setting using the
language of orthogonal polynomials and probability distribution
functions (p.d.f.s), the first surprise comes at the very start: a
non-diagonal generalization of the problem is considered [for the
first time, as far as I know (and the author claims)]. In fact, the
absolute value of the difference of indices i and j is an invariant
of the solution set, and it makes the difference if it vanishes
(solution - re-derived here - was known) or not (the results are new
and qualitatively different).
In order to factor the map of J on f, the paper describes, roughly
speaking, a bijective parametrization of p.d.f.s and subdeterminants
q of J. This characterizes the solution set in geometric language [as
a fibration over a (connected) coordinate base] and opens the way
towards constructions via the intermediate, so called auxiliary
polynomial of a solution. In this formulation, the inverse problem is
reduced to the construction of the fibres which makes use of the
properties of roots. As a result, every output of the construction is
shown to be a solution (the existence of which is assumed), and one
becomes free to generate formulas for solutions.
People who still hesitate may be definitely persuaded and impressed
by an explicit illustrative example on p. 4746.
Remarks to the editors:

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