Indefinite Sturm-Liouville operators in polar form. (English) Zbl 07812575
Summary: We consider the indefinite Sturm-Liouville differential expression
\[
\mathfrak{a}(f):= -\frac{1}{w}\left(\frac{1}{r}f'\right)',
\]
where \(\mathfrak{a}\) is defined on a finite or infinite open interval \(I\) with \(0\in I\) and the coefficients \(r\) and \(w\) are locally summable and such that \(r(x)\) and \((\mathrm{sgn}\, x) w(x)\) are positive a.e.on \(I\). With the differential expression \(\mathfrak{a}\) we associate a nonnegative self-adjoint operator \(A\) in the Krein space \(L^2_w (I)\) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of \(I\) with the positive and the negative semi-axis. For the operator \(A\) we derive conditions in terms of the coefficients \(w\) and \(r\) for the existence of a Riesz basis consisting of generalized eigenfunctions of \(A\) and for the similarity of \(A\) to a self-adjoint operator in a Hilbert space \(L^2_{|w|}(I)\). These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.
MSC:
| 47B50 | Linear operators on spaces with an indefinite metric |
| 34B24 | Sturm-Liouville theory |
| 46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
Keywords:
self-adjoint extension; symmetric operator; Krein space; Riesz basis; coupling of operators; boundary triple; Weyl function; regular critical pointReferences:
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