Abstract
We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations.
Highlights
Coupling problem Find a pair of real entire functions ( −, +) of exponential type zero such that the three conditions listed below are satisfied. (C) Coupling condition
The unique solution ( −, +) of the coupling problem with data η is given by 1 − min ±(z) = 1 − z
First and foremost, the coupling problem is essentially equivalent to an inverse spectral problem for second-order ordinary differential equations or two-dimensional first-order systems with trace class resolvents
Summary
We readily see that some η ∈ Rσ is admissible if and only if the coupling constant η(λ0) is not a negative real number In this case, the unique solution ( −, +) of the coupling problem with data η is given by 1 − min ±(z) = 1 − z. First and foremost, the coupling problem is essentially equivalent to an inverse spectral problem for second-order ordinary differential equations or two-dimensional first-order systems with trace class resolvents This circumstance indicates that it is not likely for a simple elementary proof of our theorem to exist, as the uniqueness part allows one to effortlessly deduce (generalizations of) results in [3,7,13,14,19], which had to be proven in a more cumbersome way before. The stability result for the coupling problem enables us to derive long-time asymptotics for solutions of such nonlinear wave equations [18]
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