Bounds on the Non‐real Spectrum of a Singular Indefinite Sturm‐Liouville Operator on ℝ

Abstract

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained.

Spectral bounds for indefinite singular Sturm–Liouville operators with uniformly locally integrable potentials

“…The more difficult case of singular indefinite Sturm-Liouville operators was so far only studied in very special situations; cf. [4,5] for p = 1, r = sgn, and q ∈ L s (R) for s = 1 or s = ∞ (see also [2,6]). In contrast to the abovementioned contributions here we impose only rather weak assumptions in Hypothesis 2.1 on the coefficients in (1.1), in particular, we treat weight functions r with finitely or infinitely many sign changes within a compact interval, functions 1/p ∈ L η (R) for η ∈ [1, ∞] and uniformly locally integrable…”

Spectral bounds for singular indefinite Sturm-Liouville operators with 𝐿¹-potentials

“…Explicit bounds on non-real eigenvalues for singular Sturm-Liouville operators with L ∞ -potentials were obtained with Krein space perturbation techniques in [5] and under additional assumptions for L 1 -potentials in [6,7], see also [3] for the absence of real eigenvalues and [19] for the accumulation of non-real eigenvalues of a very particular family of potentials. In this paper we substantially improve the earlier bounds in [6,7] and relax the conditions on the potential. More precisely, here we prove for arbitrary real q ∈ L 1 (R) the following bound.…”

An estimate on the non‐real spectrum of a singular indefinite Sturm‐Liouville operator

“…Then the non-real spectrum of A consists only of isolated eigenvalues and every non-real eigenvalue λ of A satisfies |λ| ≤ q 2 L 1 . This result improves the bounds in [6] for certain potentials and is based on the techniques in [1]. For further bounds on the non-real spectrum of indefinite Sturm-Liouville operators we refer to [4] for the case of a bounded potential q and [3,7,8,[11][12][13] for the regular case.…”