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Isospectral sets for boundary value problems on the unit interval. (English) Zbl 0678.34025

In a series of papers, E. Trubowitz and coauthors have classified isospectral sets of potentials for various self-adjoint boundary conditions for the Sturm-Liouville equation on the unit interval. The present work completes this classification by treatment of the case of generalized boundary conditions. In particular, consider the problem \(L(q)y\equiv -y''+q(x)y=0,\) \(0\leq x\leq 1\) where the potential is real- valued and square integrable, with boundary conditions \[ B\left( \begin{matrix} y(1)\\ y'(1)\end{matrix} \right)=\left( \begin{matrix} y(0)\\ y'(0)\end{matrix} \right), \] where \(B=\left( \begin{matrix} a\quad b\\ c\quad d\end{matrix} \right)\), det B\(=1\). The analysis is carried out by studying the manifold M of all pairs of boundary conditions B and potentials q with a given spectrum and characterizing the critical points of the map from M to the trace \(a+d\). Isospectral sets appear as slices of M determined by the trace function. The main portion of the paper covers the cases where \(b\neq 0\); an appendix by J. Tysk treats the remaining case.
Reviewer: L.Grimm

MSC:

34L99 Ordinary differential operators
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References:

[1] Pöschel, Inverse Spectral Theory (1987)
[2] DOI: 10.1002/cpa.3160300305 · Zbl 0403.34022 · doi:10.1002/cpa.3160300305
[3] DOI: 10.1002/cpa.3160370205 · Zbl 0601.34017 · doi:10.1002/cpa.3160370205
[4] DOI: 10.1002/cpa.3160370102 · Zbl 0552.58024 · doi:10.1002/cpa.3160370102
[5] DOI: 10.1007/BF02566350 · Zbl 0554.34013 · doi:10.1007/BF02566350
[6] DOI: 10.1002/cpa.3160360604 · Zbl 0507.58037 · doi:10.1002/cpa.3160360604
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