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

Abstract

The spectrum of the singular indefinite Sturm-Liouville operator A = sgn(·) − d 2 dx 2 + q with a real potential q ∈ L 1 (R) covers the whole real line and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound |λ| ≤ q 2 L 1 on the absolute values of the non-real eigenvalues λ of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L 1 -norm of the negative part of q.

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

“…For the potential q in (1.1) we only assume uniform local integrability so that, in particular, potentials in L s (R) for any s ∈ [1, ∞] are allowed. The assumptions in Hypothesis 2.1 naturally generalize the case p = 1, r = sgn and q ∈ L 1 (R) or q ∈ L ∞ (R) studied in [4,5].…”

“…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…”

The non‐real spectrum of a singular indefinite Sturm–Liouville operator with regular left endpoint

“…[13,Lemma 9.37]. Recently, bounds for the non-real spectra of indefinite Sturm-Liouville operators were developed in [3,[8][9][10][11][12] for operators with two regular endpoints and in [4][5][6] for operators with two singular endpoints. By [2, Corollary 3.9] the operator A α has nonempty resolvent set and its essential spectrum coincides with the essential spectrum of JA α , where σ ess (JA α ) = [0, ∞), see Theorem 9.38 and the note below in [13].…”

“…Hence, the definite Sturm-Liouville operator JA α on D(JA α ) = D(A α ) is self-adjoint in the Hilbert space L 2 (a, ∞) and due to (·, ·) = [J·, ·] the self-adjointness of A α with respect to [·, ·] follows. The proof is based on techniques developed in [5,6]. Recently, bounds for the non-real spectra of indefinite Sturm-Liouville operators were developed in [3,[8][9][10][11][12] for operators with two regular endpoints and in [4][5][6] for operators with two singular endpoints.…”

“…Recently, bounds for the non-real spectra of indefinite Sturm-Liouville operators were developed in [3,[8][9][10][11][12] for operators with two regular endpoints and in [4][5][6] for operators with two singular endpoints. The result in Theorem 1.1 addresses singular operators with one regular and one singular endpoint.…”