Perturbation and spectral theory for singular indefinite Sturm-Liouville operators (Q6592813)

Consider two singular indefinite Sturm-Liouville expressions \(l_j = \frac{1}{r_j}(-\frac{d}{dx} p_j \frac{d}{dx} + q_j)\) for \(j=0,1\) where \(\frac{1}{p_j}, q_j, r_j \in L^1_{loc}(a,b)\) are real coefficients with \(p_j > 0\) and \(-\infty \leq a < b \leq \infty\). It is assumed that the weight functions \(r_j\) satisfy the sign condition \(r_j < 0\) on \((a, \alpha)\) and \(r_j > 0\) on \((\beta, b)\) for some numbers \(a < \alpha < \beta < b\) and that the endpoints \(a\) and \(b\) are in the limit point case. Then, the associated maximal operators \(K_j\) are self-adjoint in the Krein spaces \(L^2((a,b); r_j)\) with the indefinite inner products \([f,g] = \int_a^b f(x) \overline{g(x)} r_j(x) \; dx\) \((j=0,1)\). The authors study the structure of the spectrum of \(K_0\) and \(K_1\) where \(K_1\) is regarded as a perturbation of \(K_0\). For each \(j=0,1\) the space is split up by \(L^2((a,b); r_j) = L^2((a,\alpha); r_j) \oplus L^2((\alpha,\beta); r_j) \oplus L^2((\beta,b); r_j)\) and -- imposing Dirichlet boundary conditions at \(\alpha\) and \(\beta\) -- the differential expression induces three operators \(-H_{j,-}, K_{j,\alpha \beta}, H_{j,+}\) where \(H_{j,-}, H_{j,+}\) are (singular) self-adjoint Hilbert space operators and \(K_{j,\alpha \beta}\) is a regular indefinite Sturm-Liouville operator. This construction leads to the first main result: Conditions on the coefficients are presented such that the essential spectrum satisfies \(\sigma_{\mathrm{ess}}(K_0) = \sigma_{\mathrm{ess}}(K_1) \subset {\mathbb R}\) (and this is determined by \(H_{j,-}\) and \(H_{j,+}\)). Under some additional conditions this essential spectrum appears as the whole real axis except of a certain gap. In the next step the discrete spectrum is studied, in particular in view of accumulation properties of the (real and non-real) eigenvalues. A Kneser type result is obtained for \(K_0\) allowing also implications for a certain perturbation \(K_1\). Finally, the case of periodic coefficients for \(K_0\) is considered where \(\sigma_{\mathrm{ess}}(K_0)\) consists of the union of infinitely many intervals. If \(\frac{1}{p_1}, q_1, r_1\) are \(L^1\)-perturbations of \(\frac{1}{p_0}, q_0, r_0\) then, it turns out that again \(\sigma_{\mathrm{ess}}(K_0) = \sigma_{\mathrm{ess}}(K_1)\) and accumulation results for the (real and non-real) eigenvalues can be obtained.