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

Abstract

We provide bounds on the non‐real spectra of indefinite Sturm–Liouville differential operators of the form (Af)(x) = sgn(x)(−f″(x) + q(x)f(x)) on the interval [a, ∞), −∞ < a < 0, with real potential q ∈ L1(a, ∞). The bounds depend only on the L1‐norm of the negative part of q and the boundary condition at the regular endpoint a.

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

“…without the sign function), the existing literature on eigenvalue bounds for operators of type (1.1) seems to be much more sparse even in the one-dimensional setting with real potentials, see e.g. [4,6,8], and a number of interesting questions remain open. One of them is a conjecture of Behrndt in [3] according to which the eigenvalues of singular indefinite Sturm-Liouville operators (i.e.…”

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.…”