Relative oscillation theory for Sturm-Liouville operators extended. (English) Zbl 1144.34014
The purpose of the paper is to extend relative oscillation theory for two different Sturm-Liouville operators
\[ H_ju=r^{-1}(-(p_ju')'+q_ju)\quad (j=0,1). \]
The authors have shown that the weighted number of zeros of Wronskians of certain solutions equals to the value of Krein’s spectral shift function \(\xi(\lambda,H_1,H_0)\) inside essential spectral gaps. The method of proof is based on finding a continuous path connecting the operators \(H_0\) and \(H_1\) in the metric introduced by the trace norm of resolvent differences.
\[ H_ju=r^{-1}(-(p_ju')'+q_ju)\quad (j=0,1). \]
The authors have shown that the weighted number of zeros of Wronskians of certain solutions equals to the value of Krein’s spectral shift function \(\xi(\lambda,H_1,H_0)\) inside essential spectral gaps. The method of proof is based on finding a continuous path connecting the operators \(H_0\) and \(H_1\) in the metric introduced by the trace norm of resolvent differences.
Reviewer: Aleksandr Lomtatidze (Brno)
MSC:
| 34B24 | Sturm-Liouville theory |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
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