PERTURBATION AND SPECTRAL THEORY FOR SINGULAR INDEFINITE STURM–LIOUVILLE OPERATORS

. We study singular Sturm-Liouville operators of the form


Introduction
The purpose of this paper is to develop perturbation methods and to study spectral properties of singular Sturm-Liouville operators K 0 and K 1 associated with the differential expressions on some interval (a, b), where −∞ ≤ a < b ≤ ∞.As usual, we impose the standard assumptions 1/p j , q j , r j ∈ L 1 loc (a, b) real, r j = 0, p j > 0 a. e., and furthermore the endpoints a and b are assumed to be singular and in the limit point case.We will be interested in so-called indefinite Sturm-Liouville operators, i.e., we consider sign changing weight functions r j ; more precisely, here we treat the case r j < 0 in a neighbourhood of a and r j > 0 in a neighbourhood of b, j = 0, 1.In this situation the maximal operators K j , j = 0, 1, associated to ℓ j in the weighted L 2 -spaces L 2 ((a, b); r j ) are self-adjoint with respect to the corresponding Krein space inner products Various aspects in the spectral theory of indefinite Sturm-Liouville operators have been studied intensively in the mathematical literature and we refer the reader to [6,9,10,11,17,23,24,33,45,59,60,62] for different types of eigenvalue estimates and to [19,20,26,27,41,42,43,47,48,49,69] and the references therein for a discussion of so-called critical points, similarity, and special cases as, e.g., left definite problems.
A natural and intuitive approach to the spectral theory of indefinite Sturm-Liouville operators is to reduce a part of the analysis to the definite case near the singular endpoints via Glazman's decomposition method and to apply perturbation techniques; this idea appears already in the fundamental paper [25] and has been further applied and developed in, e.g., [3,8,12,14,18,44].More precisely, fix a < α < β < b such that both weights r j are negative on (a, α) and positive on (β, b) and view the operators K j as finite rank perturbations in resolvent sense of the block diagonal operators in the Krein spaces L 2 ((a, b); r j ) = L 2 ((a, α); r j ) ⊕ L 2 ((α, β); r j ) ⊕ L 2 ((β, b); r j ). (1.3) Note that sign changes of the weight functions are possible only inside the finite interval [α, β].Hence K j,αβ are regular indefinite Sturm-Liouville operators associated to ℓ j in the Krein spaces L 2 ((α, β); r j ), and H j,± are definite singular Sturm-Liouville operators associated with the differential expressions in the weighted L 2 -Hilbert spaces L 2 ((a, α); −r j ) and L 2 ((β, b); r j ).By imposing Dirichlet boundary conditions at α and β the operators H j,± become self-adjoint in the corresponding Hilbert spaces and K j,αβ becomes self-adjoint in the Krein space L 2 ((α, β); r j ).Due to the diagonal form it is clear that the spectra of H j coincide with the union of the spectra of the diagonal entries.It is well-known that the spectrum of K j,αβ is purely discrete and hence the essential spectrum of the block diagonal operators H j in (1.2) is given by the union of the essential spectra of the Hilbert space self-adjoint operators ±H j,± .
In order to conclude spectral properties of the operators K j from (1.2) a careful analysis of the underlying rank two perturbation (the functions in the domains of H j satisfy Dirichlet boundary conditions at α and β) is necessary, which is particularly subtle due to the indefinite nature of the problem as self-adjoint operators in Krein spaces may have a rather arbitrary spectral structure.A priori it is not even clear if the resolvent sets of K j are nonempty, even rank one perturbations may lead to nonreal eigenvalues accumulating towards the essential spectrum, and other spectral effects can appear.
An additional substantial difficulty when comparing the indefinite Sturm-Liouville operators K 0 and K 1 stems from the fact that the operators act in different Krein spaces (as the weight functions r 0 and r 1 are different in general) and hence H 0,± and H 1,± act in different Hilbert spaces; at the same time also the regular indefinite Sturm-Liouville operators K 0,αβ and K 1,αβ act in different Krein spaces.We emphasize that perturbation theory for Sturm-Liouville operators with different weights r 0 = r 1 has not obtained much attention and to the best of our knowledge there is only the contribution [15] for the definite case that contains (nontrivial) results on the invariance of the essential spectrum.
In this paper we use recent results from perturbation theory of definite Sturm-Liouville operators from our publications [15,16] and abstract perturbation results for self-adjoint operators in Krein spaces from [4,10,11] together with the above methodology to obtain a number of spectral results for singular indefinite Sturm-Liouville operators.In Theorem 2.5 we provide conditions on the coefficients r j , p j , q j such that the essential spectra of K 0 and K 1 coincide, which can be considered as one of the main results.In Corollary 2.6 we illustrate this general result in a more explicit situation, where it is assumed that (a, b) = R and the coefficients admit limits at the singular endpoints ±∞.
The accumulation of nonreal eigenvalues towards certain regions of the real axis and the accumulation of discrete real eigenvalues towards the essential spectrum is investigated in Section 3. In Theorem 3.1 and Theorem 3.4 these problems are treated for K 0 in a general setting in terms of the essential spectra of the operators H 0,± ; the concept behind is the so-called local definitizability of selfadjoint operators in Krein spaces and the stability of this property under finite rank perturbations in resolvent sense; cf.[4,10,38,39,40].We pay special attention to the accumulation of real discrete eigenvalues in the case (a, b) = R, where the coefficients r j , p j , q j admit limits at ±∞ such that a gap arises in the essential spectra of K j .This allows to conclude Kneser type results in the spirit of [15,51,52] in the indefinite setting in Theorem 3.9 and Theorem 3.12; cf.[46] and also [9,29,31,35,37,63,64,67].
Another interesting situation appears in Section 4 in the periodic setting under L 1 -perturbations of the periodic coefficients of K 0 : The band structure of the periodic operators ±H 0,± is preserved and leads to a band structure of the perturbed periodic operators ±H 1,± , and hence to a band structure of K 1 ; cf.Theorem 4.1.An additional finite first moment condition on the coefficient differences together with [16,Theorem 2.3] combined with the results in Section 3 allows us to prove finiteness of eigenvalues in the spectral band gaps of the perturbed periodic operator K 1 outside a certain compact region; this can be viewed as an extension of a seminal result by Rofe-Beketov from the 1960s to general indefinite Sturm-Liouville operators; cf.[22,30,51,64].We also refer the reader to [9,28,50,61] for other related studies on indefinite Sturm-Liouville operators in the periodic setting.
For the convenience of the reader the paper contains a short appendix on operators in Krein spaces, where some spectral properties and perturbation results for self-adjoint operators with finitely many negative squares and locally definitizable self-adjoint operators from the mathematical literature are recalled.
Throughout the paper we shall use the notions of essential and discrete spectrum for operators that are not necessarily self-adjoint in a Hilbert space.To avoid possible confusion we recall that for a closed operator T in a Hilbert space λ ∈ C is a discrete eigenvalue if λ ∈ σ p (T ) is an isolated point in the spectrum σ(T ) and the corresponding Riesz projection is a finite rank operator.The essential spectrum σ ess (T ) is the complement of the discrete eigenvalues in σ(T ).We emphasize that under our assumptions the essential spectra of the Sturm-Liouville operators appearing in this paper is automatically real and remains invariant under compact perturbations in resolvent sense.
Acknowledgements.J.B. gratefully acknowledges financial support by the Austrian Science Fund (FWF): P 33568-N.J.B. is also most grateful for the stimulating research stay and the hospitality at the University of Auckland in February and March 2023, where some parts of this paper were written.This publication is also based upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.

Essential Spectrum
Let −∞ ≤ a < b ≤ ∞ and let ℓ j and τ j , j = 0, 1, be the Sturm-Liouville expressions on (a, b) in (1.1) and (1.4), respectively, and assume that the coefficients satisfy the standard assumptions 1/p j , q j , r j ∈ L 1 loc (a, b) real, r j = 0, and p j > 0 almost everywhere.The next hypothesis on the different signs of the weight functions near a and b is central for the present paper.
The Hilbert spaces of measurable complex valued functions f on (a, b) such that r j f 2 ∈ L 1 (a, b) are denoted by L 2 ((a, b); |r j |) and are equipped with the standard scalar products Besides these scalar products we shall also consider the inner products which are both indefinite by Hypothesis 2.1.The spaces L 2 ((a, b); |r j |) equipped with [•, •] become Krein spaces and will be denoted by L 2 ((a, b); r j ).The corresponding fundamental symmetries J = sgn(r j ) connect the inner products via [f, g] = (Jf, g).Note also that the differential expressions ℓ j are formally symmetric with respect to the indefinite inner products [•, •] and that ℓ j = Jτ j , j = 0, 1.For the basic properties of indefinite inner product spaces and linear operators therein we refer to [1,21,32].Next we will define various Sturm-Liouville operators associated with ℓ j and τ j .For our purposes the following hypothesis on the definite Sturm-Liouville expression τ 0 is appropriate.
Hypothesis 2.2.The endpoints a and b are singular and in the limit point case with respect to τ 0 .Let j = 0, 1 and denote the maximal domains by where AC(a, b) stands for the space of absolutely continuous functions on (a, b).Note that τ j f ∈ L 2 ((a, b); |r j |) if and only if ℓ j f ∈ L 2 ((a, b); |r j |).The maximal operators associated to ℓ j and τ j are defined as for j = 0, 1. Observe that Hypothesis 2.2 ensures that the definite Sturm-Liouville operator L 0 is self-adjoint in the Hilbert space L 2 ((a, b); |r 0 |) and from K 0 = JL 0 it follows that the indefinite Sturm-Liouville operator K 0 is self-adjoint in the Krein space L 2 ((a, b); r 0 ); cf.Appendix A. Besides the natural maximal operators in (2.3) we shall also make use of Sturm-Liouville operators associated to ℓ j and τ j on the subintervals (a, α), (α, β), and (β, b) equipped with Dirichlet boundary conditions at the regular endpoints α and β.More precisely, for j = 0, 1 we define the operators where the maximal domains D j (a, α), D j (α, β), and Hypothesis 2.3.The function q 0 /r 0 is bounded near a and b.
Remark 2.4.Note that the inner product (2.1)(α, β) can be indefinite or definite, depending on the properties of the weight functions r j , j = 0, 1, on (α, β).To avoid confusion we will always view L 2 ((α, β); r j ) as a Krein space (which reduces to a Hilbert space in the special case r j > 0 a.e. on (α, β)).Furthermore, the case α = β in Hypothesis 2.1 is understood in the sense that the regular indefinite Sturm-Liouville operators K j,αβ and the Krein spaces L 2 ((α, β); r j ) are absent in the operator and space decompositions appearing later in the text.
The next result is the first main result in this paper.We provide criteria on the coefficients of ℓ 0 and ℓ 1 such that the essential spectra of the indefinite Sturm-Liouville operators K 0 and K 1 coincide.It also turns out that the assumptions on the coefficients imply that the resolvent sets of K 0 and K 1 are both nonempty.
We now turn to the essential spectra of K 0 and K 1 and verify the remaining assertion (2.5).For this we first consider the orthogonal sums H j , j = 0, 1, given by on their natural domains dom in the Krein spaces (1.3).Note that the operators K 0,αβ and K 1,αβ are both regular indefinite Sturm-Liouville operators that are self-adjoint in the Krein space L 2 ((α, β); r 0 ) and L 2 ((α, β); r 1 ), respectively, and that the spectra of these operators consist of real discrete eigenvalues accumulating to ∞ and −∞.In addition, there may appear at most finitely many pairs of nonreal discrete eigenvalues which are symmetric with respect to the real line.This is a consequence of the fact that K 0,αβ and K 1,αβ have finitely many negative squares; we refer the reader to [25] and Theorem A.3 in Appendix A. Therefore, we conclude that H 0 and H 1 are self-adjoint in the Krein spaces L 2 ((a, b); r 0 ) and L 2 ((a, b); r 1 ), respectively, and from (2.6) and (2.7) we obtain that the essential spectra of H 0 and H 1 coincide and (2.8) As ρ(K 0 ) and ρ(K 1 ) are both nonempty we see that for λ ∈ ρ(K 0 ) ∩ ρ(H 0 ) and µ ∈ ρ(K 1 ) ∩ ρ(H 1 ) the resolvent differences are rank two operators.In fact, this follows from the observation that and it is clear that this subspace of functions is a two-dimensional restriction of the maximal domains.Hence the essential spectra of K 0 and H 0 coincide and the essential spectra of K 1 and H 1 coincide, that is, This observation together with (2.8) leads to (2.5); note that the essential spectrum is real as H 0,± and H 1,± are self-adjoint in Hilbert spaces.
In the next corollary we use Theorem 2.5 to express the essential spectrum in a more explicit way for the case where the coefficients of ℓ 0 and ℓ 1 admit the same limits r ±∞ , p ±∞ , q ±∞ at ±∞. Corollary 2.6.Assume that (a, b) = R and that the coefficients r j , p j , q j , j = 0, 1, admit the limits where ±r ±∞ > 0, p ±∞ > 0, and q ±∞ ∈ R. Then both indefinite Sturm-Liouville operators K 0 and K 1 are self-adjoint in the Krein spaces L 2 (R; r 0 ) and L 2 (R; r 1 ), respectively, the resolvent sets ρ(K 0 ) and ρ(K 1 ) are nonempty, and In particular, there is a gap in the essential spectrum if −q −∞ /r −∞ < q +∞ /r +∞ .
Proof.We shall define a suitable comparison operator K with constant coefficients and apply Theorem 2.5 to the pairs {K, K 0 } and {K, K 1 }.Consider the piecewise constant coefficients r, p, q defined by and note that the corresponding definite Sturm-Liouville expression is in the limit point case at both singular endpoints ±∞, that is, Hypothesis 2.2 holds.From the choice of q and r, and the assumptions on r 0 , r 1 it is also clear that Hypothesis 2.1 and Hypothesis 2.3 are satisfied.Now we associate the operators K, L, H + , K αβ , H − to τ and its indefinite counterpart ℓ = sgn(r)τ in the same way as in (2.2), (2.3), and (2.4).Then the essential spectrum of the operators is given by (cf. [65, Example on p. 209]), and hence we conclude in the same way as in (2.7), (2.8), and (2.10) that It is easy to see that the conditions hold for j = 0, 1 and hence we conclude from Theorem 2.5 that the indefinite Sturm-Liouville operators K 0 and K 1 are self-adjoint in the Krein spaces L 2 (R; r 0 ) and L 2 (R; r 1 ), the resolvent sets ρ(K 0 ) and ρ(K 1 ) are nonempty, and (2.11) follows from (2.13) and (2.5).

Discrete spectrum
In this section we study the point spectrum of the indefinite Sturm-Liouville operator K 0 from the previous section.Here we concentrate on the nonreal point spectrum and accumulation points in gaps of the essential spectrum.Recall from the previous section that under Hypotheses 2.1, 2.2, and 2.3 the essential spectrum of K 0 is given by σ ess (K 0 ) = σ ess (H 0,+ ) ∪ σ ess (−H 0,− ) (3.1) and hence it is clear that the nonreal spectrum of K 0 consists of discrete eigenvalues that may only accumulate to points in (3.1).In the next theorem we observe that accumulation of nonreal eigenvalues is possible only towards certain subsets of (3.1).
(i) The nonreal spectrum of K 0 consists of discrete eigenvalues with geometric multiplicity one which are contained in a compact subset of C. (ii) The nonreal spectrum may only accumulate to points in Furthermore, the nonreal spectrum cannot accumulate to all points λ from the boundary (in R) with the following property (P): There exists a left sided neighbourhood in R of λ in ρ(H 0,+ ) (ρ(−H 0,− )) and a right sided neighbourhood in R of λ in ρ(−H 0,− ) (ρ(H 0,+ ), respectively).(iii) In particular, if the interior (in R) int (σ ess (H 0,+ ) ∩ σ ess (−H 0,− )) = ∅ and all λ in the set (3.2) satisfy property (P), then K 0 has at most finitely many nonreal eigenvalues.
Proof.We shall first show in Step 1 and Step 2 that the self-adjoint operator acting in the Krein space in the sense of Definition A.4 (see also [38,40] and [66]).Using the perturbation result in Theorem A.5 (i) we conclude in Step 3 that the same is true for K 0 , which implies the assertions (i)-(iii).
Step 1.We verify Definition A.4 (ii).By Hypothesis 2.1 L 2 ((a, α); r 0 ) is an anti-Hilbert space, L 2 ((β, b); r 0 ) is a Hilbert space, and L 2 ((α, β); r 0 ) is (in general) a Krein space.Since H 0,+ and −H 0,− are self-adjoint in Hilbert spaces their spectra are real.In the present situation σ(H 0,+ ) is of positive type in the sense of Definition A.1 and σ(−H 0,− ) is of negative type in the sense of Definition A.1.As mentioned in the proof of Theorem 2.5 the operator K 0,αβ is a regular indefinite Sturm-Liouville operator with finitely many negative squares (for the notion of negative squares we refer to the Appendix) and the spectrum consists only of discrete eigenvalues; the real eigenvalues accumulate to ∞ and −∞.Hence for every real λ not in σ ess (H 0,+ ) ∩ σ ess (−H 0,− ) and all points in (3.2) which satisfy property (P) item (ii) from Definition A.4 is satisfied for the operator H 0 in (3.3).We show next that item (ii) from Definition A.4 is also satisfied for λ = ∞.In fact, it follows from Hypothesis 2.3 that H 0,+ is semibounded from below and −H 0,− is semibounded from above and hence (−∞, −γ) ⊂ ρ(H 0,+ ) and (γ, ∞) ⊂ ρ(−H 0,− ) for some γ ≥ 0. As mentioned above the spectrum of H 0,+ is of positive type and the spectrum of −H 0,− is of negative type.Since K 0,αβ has finitely many negative squares Theorem A.3 (iii) implies that one can choose γ in such a way that (−∞, −γ) ∩ σ(K 0,αβ ) is of negative type and (γ, ∞) ∩ σ(K 0,αβ ) is of positive type.Now the claim follows for λ = ∞.
Step 2. Observe that the nonreal spectrum of H 0 coincides with the nonreal spectrum of K 0,αβ and hence it follows from Theorem A.3 (i) that the nonreal spectrum of H 0 consists of isolated points which are poles of the resolvent of H 0 and no point of R ∪ {∞} is an accumulation point of non-real spectrum of H 0 .It remains to verify Definition A.4 (i).In fact, the growth condition for the resolvent of H 0 is valid since the resolvents of H 0,+ and −H 0,− are bounded by |Im λ| −1 for nonreal λ and the operator K 0,αβ satisfies the growth condition for the resolvent by Theorem A.3 (iv).Hence H 0 is definitizable over the domain (3.4).
Step 3. By (2.9) the difference of the resolvents of H 0 and K 0 is a rank two operator and hence Theorem A.5 implies that the operator K 0 is also definitizable over the domain (3.4); note that ρ(K 0 ) = ∅ follows from Hypothesis 2.3 and [12,Theorem 4.5].This yields (ii) in Theorem 3.1.Observe that the nonreal spectrum of K 0 is discrete by (3.1) and Hypothesis 2.2 implies that all eigenvalues of K 0 have geometric multiplicity one.As ∞ is contained in the domain (3.4) it is also clear that ∞ is not an accumulation point of nonreal eigenvalues of K 0 .This shows (i) and assertion (iii) is an immediate consequence of (i) and (ii).
Next we illustrate the decisive role of points with property (P) from Theorem 3.1 (ii) with some examples.
where α = β = 0 in Hypothesis 2.1, see also [7,59] and [34].In this situation Corollary 2.6 and (2.12) show σ ess (H 0,+ ) = [0, ∞) and σ ess (−H 0,− ) = (−∞, 0] and hence (3.2) turns into Note that the operators ±H 0,± from (2.4) act in L 2 (R ± ) and the operator K 0,αβ is not present in the decomposition (1.2); cf.Remark 2.4.A Kneser type argument (see, e.g., [65,Corollary 9.43]) shows that the discrete spectrum in (−∞, 0) of the operator H 0,+ accumulates to zero from below and the discrete spectrum in (0, ∞) of the operator −H 0,− accumulates to zero from above.Hence property (P) is not fulfilled for the point zero and thus accumulation of nonreal eigenvalues of K 0 to zero may appear In fact, it was shown in [59] that for the shifted (indefinite) Coulomb potential the nonreal discrete eigenvalues indeed accumulate to zero.In general it is an open problem formulated in [5] whether the accumulation of discrete eigenvalues from below to the lower boundary of the essential spectrum of H 0,+ or the accumulation of discrete eigenvalues from above to the upper boundary of the essential spectrum of −H 0,− leads to nonreal accumulation of discrete eigenvalues to points from (3.2) not satisfying property (P).
We shall illustrate the statements in the Theorems 3.1 and 3.4 in the next example for a special case with q 0 chosen as a so-called finite-zone potential near ±∞; cf.[57,58].
According to Theorem 3.1 the nonreal spectrum of K 0 consists of discrete eigenvalues (with geometric multiplicity one) which are contained in a compact subset of C. Possible accumulation points of nonreal eigenvalues are contained in the intersection of the two sets of bands of essential spectra in (3.6).Note that in the present situation possible points µ + i = λ − j for some i, j or µ − l = λ + k for some l, k satisfy property (P) in Theorem 3.1 (ii) as ±H 0,± have at most finitely many discrete eigenvalues.Hence, if for all k, l, then K 0 has at most finitely many nonreal eigenvalues.Concerning the real discrete eigenvalues Theorem 3.4 implies the following: real discrete eigenvalues of K 0 may only accumulate (from the left) to points λ + k = λ − l or (from the right) to points µ + i = µ − j for some k, l or i, j.Therefore, under the assumptions λ for all k, l and i, j the operator K 0 has at most finitely many discrete real eigenvalues.Note that condition (3.7) implies (3.8) and hence it follows under the assumptions µ − n−+1 ≤ λ + 1 , µ − 1 ≤ λ + n++1 , and (3.7) that K 0 has at most finitely many discrete (real and nonreal) eigenvalues.
In the next theorem we discuss a more special situation that arises when, roughly speaking, the essential spectra of H 0,+ and −H 0,− are separated by a gap.More precisely, if η ± denote the lower bounds of the essential spectra of H 0,± , η + := min σ ess (H 0,+ ) and η − := min σ ess (H 0,− ), where η ± := ∞ if σ ess (H 0,± ) = ∅, then we shall assume that In the following we fix some η ∈ (−η − , η + ). (3.9) Let H 0 be the operator in (1.2) and define the hermitian form •, • on dom where f, g have the obvious decomposition with respect to (1.3) with j = 0, f = f − + f αβ + f + and g = g − + g αβ + g + .Then the operator H 0 − η has finitely many negative squares and the number κ of negative squares is given by the sum of the negative squares of the three entries on the diagonal, that is, where κ η is the number of negative squares of the operator K 0,αβ − η, κ + is the number of eigenvalues of the operator − H 0,− − η in (0, ∞), κ − is the number of eigenvalues of the operator H 0,+ − η in (−∞, 0).This is due to the fact that the weight r 0 is negative on (a, α) and positive on (β, b).
In the next theorem we discuss further properties of the eigenvalues of K 0 and their algebraic eigenspaces.
Then the operator K 0 − η has at most negative squares which implies the following statements for the eigenvalues of K 0 .
(i) There are at most κ 0 different real eigenvalues of K 0 with corresponding Jordan chains of length greater than one.The length of each of these chains is at most 2κ 0 + 1. (ii) The nonreal spectrum of K 0 consists of at most κ 0 pairs {µ i , µ i }, Im µ i > 0, of discrete eigenvalues with corresponding Jordan chains of length at most κ 0 .
Proof.The operator H 0 −η has κ squares with κ as in (3.10).By (2.9) the resolvent difference of the resolvents of the operators H 0 and K 0 is a rank two operator and, hence, by Theorem A.3 (v) the operator K 0 − η has at most κ + 2 negatives squares.Now the assertions in Theorem 3.7 follow from Theorem A.3.
Next we provide quantitative estimates on the number of eigenvalues of K 0 in the setting of Theorem 3.7.Recall that a finite rank perturbation of self-adjoint operators in Krein spaces may change the discrete spectrum dramatically, in particular, for general self-adjoint operators in (infinite dimensional) Krein spaces the number of eigenvalues in a gap of the essential spectrum may change arbitrarily under a rank one or rank two perturbation; cf.[13,Theorem 3.1].The situation is different if the operators have finitely many negative squares; in the next example we illustrate how the estimates for rank perturbations in [11] can be applied successively in the present situation to obtain upper bounds on the number of eigenvalues.
Example 3.8.Consider the case η = 0 (3.9) and in Theorem 3.7, and fix an interval I ⊂ R \ σ ess (H 0 ) such that I ⊂ (0, η + ) or I ⊂ (−η − , 0).In the following we denote the number of eigenvalues of a closed operator A in I by n A (I).
In the case α = β the above reasoning simplifies (see also Remark 2.4) and one can directly apply [11,Corollary 3.2] to the operators K 0 and H 0 , so that instead of (3.11)  Next we take a closer look at the discrete eigenvalues of K 0 in the situation of Corollary 2.6 and consider again the case (a, b) = R, where the coefficients r 0 , p 0 , q 0 admit the limits r ±∞ = lim x→±∞ r 0 (x), p ±∞ = lim x→±∞ p 0 (x), and q ±∞ = lim x→±∞ q 0 (x), (3.15) with ±r ±∞ > 0, p ±∞ > 0, q ±∞ ∈ R, and, in addition, we assume Then, by Corollary 2.6, the essential spectrum of K 0 is given by and by (3.16) there is a gap in (3.17), that is, the set σ(K 0 ) ∩ (−q −∞ /r −∞ , q ∞ /r ∞ ) consists of discrete eigenvalues.It is also clear from Theorem 3.7 that K 0 − η with −q −∞ /r −∞ < η < q ∞ /r ∞ has finitely many negative squares, and hence the nonreal spectrum of K 0 consists of at most finitely many discrete eigenvalues.We proceed by providing a Kneser type result for the indefinite Sturm-Liouville operator K 0 , i.e., we obtain criteria for the accumulation/non-accumulation of eigenvalues of K 0 in (−q −∞ /r −∞ , q ∞ /r ∞ ) to the edges of the essential spectrum (see, e.g., [65,Theorem 9.42 and Corollary 9.43] or [15,51] for such type of results in the definite case).We first recall some notation from [15,51]: the iterated logarithm log n is defined recursively via with the convention log(x) := log |x| for negative values of x.Then log n (x) will be continuous for x > e n−1 and positive for x > e n , where e −1 := −∞ and e n := e en−1 .Furthermore, abbreviate and and Here the convention −1 j=0 ≡ 0 is used, so that, P 0 (x) = Q 0 (x) = 0. Theorem 3.9.Let (a, b) = R and assume that the coefficients r 0 , p 0 , q 0 admit the limits in (3.15) with ±r ±∞ > 0, p ±∞ > 0, q ±∞ ∈ R such that (3.16) holds.For and do not accumulate at ±q ±∞ /r ±∞ if Proof.It follows from [15,Theorem 3.5] that σ ess (H In the same way one obtains that σ ess (−H 0,− ) = (∞, −q −∞ /r −∞ ) and the set The assumption (3.16) ensures that q ∞ /r ∞ ∈ σ ess (−H 0,− ) and −q −∞ /r −∞ ∈ σ ess (H 0,+ ) and hence the assertions follow directly from Theorem 3.4.
For the case n = 0 Theorem 3.9 reduces to the following statement.
Then the set and do not accumulate at ±q ±∞ /r ±∞ if We end this section with the study of the point spectrum of the perturbed indefinite Sturm-Liouville operator K 1 .Under the assumptions of Theorem 2.5 the indefinite Sturm-Liouville operator K 1 is self-adjoint in the Krein space L 2 ((a, b); r 1 ) with ρ(K 1 ) = ∅ and the essential spectrum of K 1 is given by It is clear that the above considerations and results in this section also hold for K 1 .In particular, the Kneser type results Theorem 3.9 and Corollary 3.11 can be formulated in the same way with the coefficients r 1 , p 1 , q 1 .However, it is also possible to use information of the coefficients r 0 , p 0 , q 0 of the unperturbed operator K 0 to investigate possible accumulation of discrete eigenvalues of the perturbed operator K 1 .Theorem 3.12.Let (a, b) = R and assume that the coefficients r 0 , p 0 , q 0 admit the limits (3.15) with ±r ±∞ > 0, p ±∞ > 0, q ±∞ ∈ R such that (3.16) holds.Let ∆ 0,± be as in Theorem 3.9 and assume that (3.18) holds for some n ∈ N, n ≥ 1.Then the following assertions hold.
(i) The essential spectrum of K 1 is given by the nonreal spectrum of K 1 consists of at most finitely many discrete eigenvalues, and and do not accumulate at ±q ±∞ /r ±∞ if Proof.Observe first that (3.18) implies and since lim x→∞ L n (x) = ∞, Now the statement on the essential spectrum in (i) of the indefinite Sturm-Liouville operator K 1 follows from Corollary 2.6.It follows from (3.20), (3.15), and [15, Theorem 3.2] that Hypotheses 2.1, 2.2, and 2.3 hold for the differential expression τ 1 with coefficients r 1 , p 1 , q 1 .Hence Theorem 3.1 implies that the nonreal spectrum of K 1 consists of at most finitely many discrete eigenvalues and from Theorem 3.7 applied to K 1 we obtain that the operator K 1 − η with −q −∞ /r −∞ < η < q ∞ /r ∞ has finitely many negative squares.We now turn to the proof of the accumulation properties of the discrete eigenvalues of K 1 in (ii).For this, define (in the same way as ∆ 0,± ) the functions for |x| > e n .It is easy to see that and from (3.19) we conclude Therefore, the limes superior and limes inferior of ∆ 1,± and ∆ 0,± at ±∞ coincide.Now assertion (ii) follows from Theorem 3.9 applied to K 1 .
Next we consider the case n = 0 which was excluded in the previous result, as one has to assume in this case, in addition, that p 1 and p 0 have the same limits at ±∞. Corollary 3.13.Let (a, b) = R and assume that the coefficients r 0 , p 0 , q 0 admit the limits (3.15) with ±r ±∞ > 0, p ±∞ > 0, q ±∞ ∈ R such that (3.16) holds.If and do not accumulate at ±q ±∞ /r ±∞ if

Periodic problems
In this section we study indefinite Sturm-Liouville operators with periodic coefficients near the singular endpoints ±∞ and we treat L 1 -perturbations in the spirit of [16], see also [9,28,50,61] for related considerations in the indefinite setting.More precisely, we shall assume that Hypothesis 2.1 holds and that the coefficients 1/p 0 , q 0 , r 0 are ω-periodic in (β, ∞) and θ-periodic in (−∞, α) for some ω, θ > 0; this also implies that ∞ and −∞ are in the limit point case, that is, Hypothesis 2.2 is automatically satisfied.In this situation the essential spectra of H 0,+ and H 0,− are purely absolutely continuous and consist of infinitely many closed intervals where the endpoints λ + k and µ + k , λ + k < µ + k , denote the k-th eigenvalues of the regular Sturm-Liouville operator in L 2 ((β, β +ω); r 0 ) (in nondecreasing order) with periodic and semiperiodic boundary conditions, respectively, and −µ − l and −λ − l , −µ − l < −λ − l , denote the l-th eigenvalues of the regular Sturm-Liouville operator in L 2 ((α−θ, α); r 0 ) (in nondecreasing order) with periodic and semiperiodic boundary conditions, respectively; cf.[22] or [68,Section 12].
Below we determine the essential spectra of indefinite Sturm operators K 0 and K 1 , where K 0 is periodic near the endpoints and K 1 is an L 1 -perturbation.Theorem 4.1.Assume that Hypothesis 2.1 holds and suppose that 1/p 0 , q 0 , r 0 are ω-periodic in (β, ∞) and θ-periodic in (−∞, α) for some ω, θ > 0. Then the following assertions hold.
(i) The indefinite Sturm-Liouville operator K 0 is self-adjoint in the Krein space L 2 (R; r 0 ), the resolvent set ρ(K 0 ) is nonempty, and the essential spectrum is given by then also the indefinite Sturm-Liouville operator K 1 is self-adjoint in the Krein space L 2 (R; r 1 ), the resolvent set ρ(K 1 ) is nonempty, and the essential spectrum is given by Proof.(i) The periodicity of the coefficients 1/p 0 , q 0 , r 0 implies the limit point condition at both singular endpoints ±∞.Hence the maximal operator L 0 in (2.3) is self-adjoint in the Hilbert space L 2 (R; |r 0 |).Define H 0,+ and H 0,− as in (2.4).Then H 0,+ and H 0,− are both semibounded from below; cf.[22] or [68].This implies the semiboundedness of L 0 .Hence we conclude from [12,Theorem 4.5] that the indefinite Sturm-Liouville operator K 0 is self-adjoint in the Krein space L 2 (R; r 0 ) and that ρ(K 0 ) is nonempty.Furthermore, taking into account (4.1) it follows that the essential spectrum of the block diagonal operator H 0 in (2.7) is given by Now the same perturbation argument as in the end of the proof of Theorem 2. ) and in the same way as in the end of the proof of Theorem 2.5 a perturbation argument leads to σ ess (K 0 ) = σ ess (K 1 ), which finally shows (4.4).
For the discrete spectrum of the operator K 0 we obtain the following corollary as an immediate consequence of Theorem 3.1, Theorem 3.4 (see also Remark 3.5), and Theorem 4.1 (i).Note that the periodic operators ±H 0,± have no discrete (real) eigenvalues and that in the present situation property (P) holds for points that satisfy λ + k = µ − l or µ + k = λ − l for some k, l = 1, . . ., ∞; these points are automatically in int(σ ess (K 0 )).Corollary 4.2.Assume that Hypothesis 2.1 holds and suppose that 1/p 0 , q 0 , r 0 are ω-periodic in (β, ∞) and θ-periodic in (−∞, α) for some ω, θ > 0. Then the following assertions hold.
(i) The nonreal spectrum of K 0 consists of discrete eigenvalues with geometric multiplicity one which are contained in a compact subset of C and may only accumulate to points in Furthermore, the nonreal eigenvalues do not accumulate to points that satisfy λ ) for all l = 1, . . ., ∞, then the real discrete eigenvalues of K 0 do not accumulate from the left at λ + k (from the right at µ + k , respectively).In particular, if µ − 1 < µ + i , then (µ + k , λ + k+1 )∩σ ess (K 0 ) = ∅ for all k ≥ i and there are at most finitely many discrete eigenvalues of ) for all k = 1, . . ., ∞, then the real discrete eigenvalues of K 0 do not accumulate from the left at λ − l (from the right at µ − l , respectively).In particular, if λ − j < λ + 1 , then (µ − l+1 , λ − l )∩σ ess (K 0 ) = ∅ for all l ≥ j and there are at most finitely many discrete eigenvalues of K 0 in (µ − l+1 , λ − l ).Note that the first statement in Corollary 4.2 (i) holds also for the perturbed operator K 1 in Theorem 4.1 (ii); this follows directly from Theorem 3.1, Remark 3.5, and Theorem 4.1 (ii).However, in order to exclude accumulation of eigenvalues of K 1 one has to impose a stronger "finite first moment" condition.Theorem 4.3.Assume that Hypothesis 2.1 holds and suppose that 1/p 0 , q 0 , r 0 are ω-periodic in (β, ∞) and θ-periodic in (−∞, α) for some ω, θ > 0. Assume, in addition, that Then the following assertions hold.
(i) The nonreal spectrum of K 1 consists of discrete eigenvalues with geometric multiplicity one which are contained in a compact subset of C and may only accumulate to points in Furthermore, the nonreal eigenvalues do not accumulate to points that satisfy λ ) for all l = 1, . . ., ∞, then the real discrete eigenvalues of K 1 do not accumulate from the left at λ + k (from the right at µ + k , respectively).In particular, if µ − 1 < µ + i , then (µ + k , λ + k+1 )∩σ ess (K 1 ) = ∅ for all k ≥ i and there are at most finitely many discrete eigenvalues of ) for all k = 1, . . ., ∞, then the real discrete eigenvalues of K 1 do not accumulate from the left at λ − l (from the right at µ − l , respectively).In particular, if λ − j < λ + 1 , then (µ − l+1 , λ − l )∩σ ess (K 1 ) = ∅ for all l ≥ j and there are at most finitely many discrete eigenvalues of K 1 in (µ − l+1 , λ − l ).Proof.Observe first that condition (4.5) implies (4.3) and hence the indefinite Sturm-Liouville operator K 1 is self-adjoint in the Krein space L 2 (R; r 1 ), the resolvent set ρ(K 1 ) is nonempty, and the essential spectrum is given by (4.4).It follows from the condition (4.5) and [16,Theorem 2.3] that every gap in σ ess (H 1,+ ) and every gap in σ ess (−H 1,− ) contains at most finitely many eigenvalues and hence the same is true for the operator H 1 .Hence property (P) in Theorem 3.1 (ii) holds for points that satisfy λ + k = µ − l or µ + k = λ − l for some k, l = 1, . . ., ∞ and now (i) follows from Theorem 3.1.Furthermore, the same arguments as in the proof of Theorem 3.4 show that the discrete eigenvalues of K 1 do not accumulate to boundary points λ + k or µ + k of the essential spectrum, where for all l = 1, . . ., ∞.This implies (ii); assertion (iii) follows in the same way (see also the discussion below Corollary 4.2).

Appendix A. Some facts on self-adjoint operators in Krein spaces
In this appendix we recall the concept of Krein spaces and some properties of certain classes of self-adjoint operators therein, which appear in this paper.We refer the interested reader to the monographs [1,21,32] for a thourough introduction to operator theory in Krein spaces.
In the following (K, (•, •)) denotes a Hilbert space and J is a bounded self-adjoint operator in K with the property J 2 = I.Define a new inner product by [x, y] := (Jx, y), x, y ∈ K.
Since σ(J) ⊂ {−1, 1} we have the fundamental decomposition K = K + ⊕ K − , where K ± = ker(J ∓ 1).Note that K + and K − are also orthogonal with respect to [•, •] and that the inner product •] takes positive and negative values: The space (K, [•, •]) is then called a Krein space and the operator J is called fundamental symmetry.We mention that often Krein spaces are introduced by starting with an indefinite inner product and a fundamental decomposition, see [1,21,32].Let (K, [•, •]) be a Krein space and let A be a bounded or unbounded (with respect to the Hilbert space norm) operator in K.The adjoint A + is defined via the indefinite inner product [•, •] and one has A + = JA * J, where * denotes the adjoint with respect to the Hilbert space scalar product (•, •).It follows that A is self-adjoint with respect to (•, •) if and only if JA is self-adjoint with respect to [•, •].It is important to note that the spectral properties of operators which are self-adjoint with respect to a Krein space inner product differ essentially from the ones of self-adjoint operators in Hilbert spaces, e.g., the spectrum is in general not real and may also coincide with C.However, the indefiniteness of the inner product [•, •] can be used to further classify eigenvalues of operators in Krein spaces, e.g. an isolated point λ 0 ∈ σ p (A) is called of positive (negative) type if all corresponding eigenvectors x satisfy [x, x] > 0 ([x, x] < 0, respectively).This notion is extended to all points from the approximate point spectrum σ ap (A) in the next definition.Recall that for a self-adjoint operator A in a Krein space all real spectral points belong to σ ap (A); cf.Spectral theory in Krein spaces is often focusing on sign-type spectrum and further properties in a neighbourhood of such spectral points.The following proposition is of this nature; for a proof see [56] and also [2,40].
Proposition A.2. Let A be a self-adjoint operator in the Krein space (K, [•, •]) and let λ 0 be a spectral point of positive type of A. Then λ 0 is real, the non-real spectrum of A cannot accumulate to λ 0 , and there exists an open neighbourhood U in C of λ 0 such that the following statements hold.An analogous statement holds for spectral points of negative type.
Roughly speaking, Theorem A.2 (iii) states that in a neighbourhood of a spectral point of positive type A behaves (locally) like a self-adjoint operator in a Hilbert space.In what follows we present two classes of operators with the property that they have intervals with spectrum of positive or negative type: operators with finitely many squares and locally definitizable operators.
We shall say that a self-adjoint operator A in a Krein space (K, [•, •]) has κ negative squares, κ ∈ N 0 , if the hermitian form •, • on dom A, defined by f, g := [Af, g], f, g ∈ dom A, has κ negative squares, that is, there exists a κ-dimensional subspace M in dom A such that v, v < 0 if v ∈ M, v = 0, but no κ + 1 dimensional subspace with this property.In the following theorem we recall some spectral properties of self-adjoint operators with finitely many negative squares.The statements are well known and are consequences of the general results in [25,54,55], see also [18,Theorem 3.1].
Theorem A.3.Let A be a self-adjoint operator in the Krein space (K, [•, •]) such that ρ(A) = ∅ and assume that A has κ negative squares.Then the following holds.
(i) The nonreal spectrum of A consists of at most κ pairs {µ i , µ i }, Im µ i > 0, of discrete eigenvalues.Denote for an eigenvalue λ of A the signature of similar statement is valid for the accumulation of real discrete eigenvalues.This is the content of the next theorem from [4] and item (ii) from [10].

( i )
All spectral points in U ∩ R are of positive type.(ii) There exists a number M > 0 such that(A − λ) −1 ≤ M |Im λ| for all λ ∈ U \ R.(iii) There exists a local spectral function of A of positive type: To each interval δ with δ ⊂ U there exists a self-adjoint projection E(δ) which commutes with A, the space (E(δ)K, [•, •]) is a Hilbert space and σ(A|E(δ)K) ⊂ σ(A)∩ δ.

Theorem A. 5 .
Let Ω be as in Definition A.4 and let A and B be self-adjoint operators in a Krein space (K, [•, •]) with ρ(A) ∩ ρ(B) ∩ Ω = ∅, and assume that for some λ 0 ∈ ρ(A) ∩ ρ(B) the difference(A − λ 0 ) −1 − (B − λ 0 ) −1is a finite rank operator.Then the operator A is definitizable over Ω if and only if B is definitizable over Ω; in this case the following holds.(i)IfI ⊂ Ω ∩ Ris an open interval with boundary point λ ∈ Ω ∩ R and the spectral points in I are of positive (negative) type with respect to A, then there exists an open interval I ′ , I ′ ⊂ I, with boundary point λ such that the spectral points in I ′ are of positive (negative, respectively) type with respect to B. (ii) Let J be an open interval such that J ⊂ Ω ∩ R. Then σ(A) ∩ J consists of finitely many discrete eigenvalues if and only if σ(B) ∩ J consists of finitely many discrete eigenvalues.
b) are defined in the same way as D j (a, b) in (2.2).It follows from Hypothesis 2.2 that the operators H j,− are self-adjoint in the Hilbert spaces L 2 ((a, α); |r j |) and the operators H j,+ are self-adjoint in the Hilbert spaces L 2 ((β, b); |r j |).The next hypothesis ensures that the operators L 0 and H 0,± are semibounded from below (see, e.g.[15, Proof of Theorem 3.2]).