An expansion theorem for eigenvalue problems with several turning points

Indefinite Eigenvalue Problems with Several Singular Points and Turning Points

“…This paper continues the investigations made by Eberhard, Freiling and Schneider (see [6], [7], [8]) who considered (1.1) in the case of several zeros of φ 2 and a bounded and integrable function χ; it contains and extends the main results of the dissertation [24] of the third author, written under the guidance of the two first authors.…”

“…The proof of this lemma is essentially the same as the proof of [7], Lemma 1, (where l ν ∈ IN). Lemma 6.6.…”

“…Therefore we substitute the fundamental system derived by Langer by another fundamental system which is more suitable for the matching process. The idea of the construction comes from [8], where all turning points were zeros of φ 2 ; in the presence of turning points being poles we get asymptotic estimates for a special fundamental system by combining the ideas of [13], [14] and [7]. Since the proofs of the results in Sections 2 -5 are very technical, we give here only the main idea of the proofs and refer the interested reader to [24] for further details.…”

“…Nevertheless a rigorous treatment of such spectral problems is up to the present time only available in special cases (see [1], [7], [8], [10], [20], [21] and [25] for symmetric problems and for problems with turning points being zeros and also [2], [4], [19] where a Krein/Pontrjagin-space approach is used). The main reason for this gap in the literature is the fact that up to now there existed no adequate asymptotic estimates for differential equations of the general form (1.1); we hope to fill this gap with our results.…”

Inverse spectral problems for differential pencils on the half-line with turning points

“…The functions q j (x) are complex-valued, q 1 (x) is absolutely continuous, and (1 + x)q (ν) j (x) ∈ L(0, ∞) for 0 ν j 1. Differential equations with nonlinear dependence on the spectral parameter and with turning points arise in various problems of mathematics as well as in applications (see [1][2][3][4][5][6][7][8][9] for details). In this paper we study the inverse problem for singular non-selfadjoint indefinite pencil (1).…”