Spectral properties of Schr\"odinger operators with locally $H^{-1}$ potentials

We study half-line Schr\"odinger operators with locally $H^{-1}$ potentials. In the first part, we focus on a general spectral theoretic framework for such operators, including a Last--Simon-type description of the absolutely continuous spectrum and sufficient conditions for different spectral types. In the second part, we focus on potentials which are decaying in a local $H^{-1}$ sense; we establish a spectral transition between short-range and long-range potentials and an $\ell^2$ spectral transition for sparse singular potentials. The regularization procedure used to handle distributional potentials is also well suited for controlling rapid oscillations in the potential; thus, even within the class of smooth potentials, our results apply in situations which would not classically be considered decaying or even bounded.


Schrödinger operators in one dimension
dx 2 + V are often considered in the setting of locally L 2 or locally L 1 potentials; however, there are several reasons to investigate more general potentials.One is the ubiquity of non-integrable singularities such as Coulomb-or δtype potentials in models from mathematical physics; another is the Lax pair representation of the KdV equation, where H −1 (R) and H −1 (T) is the optimal regularity for well-posedness [32,35].Non-integrable singularities are often studied by specialized methods such as those for the Kronig-Penney model, and inverse scattering arguments in the distributional setting are considered in ways that circumvent the underlying Schrödinger operators.One of the goals of this paper is to extend some robust techniques in spectral theory to the greater generality of locally H −1 potentials, defined precisely below.
Schrödinger and Sturm-Liouville operators with distributional coefficients are often treated via the regularization method introduced in the pioneering work of Savchuk, Shkalikov [52].This approach has materialized into the main tool in the spectral theory of ordinary differential operators with measure and distributional coefficients.Indeed, it was employed, for example, by Eckhardt, Teschl [20] in the setting of measure coefficients; by Eckhardt, Gesztesy, Roger, Teschl [16,17,18] for L 1 loc ((a, b)) four coefficient Sturm-Liouville operators; by Eckhardt, Kostenko, Malamud, Teschl [19] for δ ′ potentials supported on Cantor sets; by Hryniv, Mykytyuk [28,29] for periodic singular potentials H −1 per (R) = H −1 (T); and by many other authors, see [18] for an extensive reference list.Most of the papers in this direction address foundational questions such as self-adjointness, Weyl-Titchmarsh theory, spectral decomposition, as well as some inverse spectral problems.We emphasize that the study of spectral types such as in the current paper, and of the associated dynamics for operators with singular coefficients, have received much less attention and have been mostly restricted to periodic [2, III.2.3] and some ergodic [8,9,10,13] Hamiltonians modeling point interactions.
In particular, Hryniv-Mykytyuk [28,29] introduced a class of uniformly locally H −1 potentials on R by the condition with the help of compactly supported H 1 multipliers and showed that real distributions in this class are precisely those with a representation where σ, τ are real-valued functions on R such that sup Note that this class includes the potentials V ∈ H −1 (R) and V ∈ H −1 (T) (when viewed as periodic distributions on R).In particular, the study of Schrödinger operators with locally H −1 potentials helps to bridge spectral theory with scattering arguments.This decomposition is related to the Miura transformation and the Riccati representation [37,31] for periodic V , in which every V ∈ H −1 (T) with zero average is represented uniquely in the form V = σ ′ + σ 2 − T σ 2 (t) dt.In the non-periodic case, in the construction of [28], τ takes the role of a local average, so the decomposition really requires two functions.Several classes of singular potentials are modeled by a suitable choice of σ, τ .For example, a Coulomb-type term |x − x 0 | −1 , x 0 ∈ (0, ∞) is realized by setting σ(x) = log |x − x 0 |, τ (x) = 0, and the point interaction δ(x − x 0 ) is realized by the characteristic function σ(x) = χ [x 0 ,∞) and τ (x) = 0. Remark 1.1.Of course, the decomposition V = σ ′ + τ is not unique; the procedure in [28] provides σ, τ such that with some universal constant C (the second inequality is general; the first is a consequence of the choice of σ, τ starting from V ).Accordingly, the quantity σχ [x,x+1) 2 + τ χ [x,x+1) 1 is interpreted as the local size of the potential.
Using the quasi-derivative, the eigenfunction equation can be written as a first-order system for u [1]  u .This is encoded by a family of transfer matrices T (z, x) which is locally absolutely continuous in x and solves the initial value problem There is a corresponding Weyl function m α and a canonical spectral measure µ α .We will provide all definitions in Section 2; for the purpose of this introduction, it suffices to know that µ α is a maximal spectral measure for H α , and we are using it to make precise statements about the spectral type of H α .We will use the Lebesgue decomposition µ α = µ α ac + µ α sc + µ α pp .One of the goals of this paper is to establish sufficient conditions for different spectral types, including a criterion for a.c.spectrum which extends the results of Last-Simon [39] for locally integrable V .One is a description of an essential support for the a.c.spectrum in terms of Cesarò-boundedness of the transfer matrices: Theorem 1.3.Assume Hypothesis 1.2.Then, for arbitrary α ∈ [0, π), the set is an essential support for the a.c.spectrum of H α in the sense that µ α ac is mutually absolutely continuous with the measure χ Σac (E) dE.In particular, Spec ac (H α ) = Σ ac ess .
Above we denoted the essential support of a Borel set S by A closely related result gives a sufficient criterion for absence of a.c.spectrum: Theorem 1.4.Assume Hypothesis 1.2 and fix arbitrary α ∈ [0, π).Let F ⊂ R be a measurable set and suppose there exist sequences Then, µ α ac (F) = 0.In the other direction one has: Theorem 1.5.Assume Hypothesis 1.2 and fix α ∈ [0, π).Suppose that for some p > 2, Then, H α has purely absolutely continuous spectrum on (E 1 , E 2 ).Theorems 1.3, 1.4, 1.5 generalize results of Last-Simon [39].The proofs are given in Section 2, which also includes a Carmona-type formula, subordinacy, and a Simon-Stolz criterion for absence of eigenvalues.
An important ingredient are new pointwise eigenfunction estimates which are stated and derived in Section 2. These relate the pointwise behavior of a formal eigenfunction and its derivative to its local L 2 behavior.For V ∈ L 2 loc they follow from Sobolev embedding theorems, but for V / ∈ L 2 loc , the local domain becomes V -dependent and different arguments are needed; estimates of this form were previously considered for locally L 1 potentials [56,41].The pointwise estimates are given in Lemma 2.7; here we point out one corollary of these estimates: Theorem 1.6.Assume Hypothesis 1.2 and let w : (0, ∞) → (0, ∞) obey For any E ∈ R there exists a positive constant C such that for any l > 1 and any solution u ∈ D, ℓu = Eu, one has In particular, if In this paper, we will only use the case w = 1; however, polynomial weights w(x) = (x+1) c and exponential weights w(x) = e cx for c ∈ R are also relevant for various criteria about the spectrum, spectral type, and dynamical properties which we expect to have a generalization to the current setting.
Remark 1.1 indicates that decay at ∞ should be quantified by the local L 2 -norm on σ and local L 1 -norm on τ .Thus, the following result generalizes the Blumenthal-Weyl criterion for preservation of essential spectrum under decaying perturbations: Lemma 1.7.Assume Hypothesis 1.2 and suppose that (1.6) We note in particular that Lemma 1.7 gives a more robust criterion even for locally L 1potentials.Any locally uniformly L 1 potential V can be decomposed as σ = 0, τ = V , but choosing a different decomposition can give better results.For instance Lemma 1.7 implies: ) is real-valued and the limit is convergent, then the operator − d 2 dx 2 + V is limit point at ∞ and its arbitrary self-adjoint realization This corollary applies to oscillatory potentials such as which was considered in [22] by a more specialized argument, and to potentials which aren't even locally uniformly integrable if α > 0. Similar growing oscillatory potentials were considered in [61,11].The description of the essential spectrum is the starting point in the theory of Schrödinger operators with decaying potentials, which are a classical subject and have been extensively studied over the past 30 years [1,3,13,14,12,15,34,36,38,47,49,58,60,61].Their spectral properties show a subtle competition between the rate of decay (with faster decay leading to absolutely continuous spectrum) and the disorder and oscillation in the potential (which promote more singular spectrum).Spectral transitions dependent on the rate of the decay have been studied by many authors, in particular: Pearson [47] in deterministic setting; Kiselev, Last, Simon [36], central to this paper; Delyon, Simon, Souillard [13] for discrete Schrödinger operators and Kronig-Penney models with decaying random potentials; and Kotani, Ushiroya [38] for continuous Schrödinger operators with decaying random potentials.This collection of papers gave rise to a number of subsequent investigations many of which are referenced in the review paper by Denisov, Kiselev [14].
We first prove that short-range perturbations preserve pure a.c.spectrum.In situations where different exponents are used to control local integrability and decay, the spaces of functions [6,50,51].The classical result about short-range perturbations is that V ∈ L 1 (R + ) implies purely a.c.spectrum on (0, ∞).The distributional analog of this criterion, informally speaking, is ℓ 1 (H −1 ); following Remark 1.1, we find the correct formulation.
Since such an example obeys σ ∈ L 2 (R + ), it shows that the condition σ ∈ ℓ 1 (L 2 ) cannot be relaxed in Theorem 1.9 and that the condition σ 2 − τ ∈ L 1 (R + ) cannot be relaxed in Theorem 1.10.
In the second part of the paper, we specialize to decaying sparse potentials and prove the following theorem.
Another new feature of our result is that the profile W n may vary with n.Note that this allows examples such as the locally integrable potential where χ denote characteristic functions.Since nχ [0,1/n] → δ 0 in H −1 (R), by Theorem 1.12, the spectrum is purely a.c. on (0, ∞) if the decaying sequence (d n ) ∞ n=1 is square-summable and purely s.c. on (0, ∞) otherwise.
Although stated in terms of H −1 (R), the starting point in our analysis is a decomposition W n = S ′ n + T n and the proof must treat these contributions to σ and τ separately.As in the classical case [36] our proof is based on the analysis of Prüfer variables.However, in the present case this analysis is more intricate due to the appearance of new terms in the differential equations obeyed by Prüfer variables.Namely, in the setting of H −1 potential V = σ ′ + τ , as shown in Proposition 2.13, one has whereas in the classical case V ∈ L 1 loc (R + ), as discussed in [36], An important ingredient in the proof of Theorem 1.12 is given by the fact that T (k 2 , x) is comparable to R(x), see Proposition 2.13.Hence, in order to establish growth or boundedness of eigensolutions and, respectively, the absence or existence of purely absolutely continuous spectrum on [E 1 , E 2 ], it suffices to study the asymptotics for R(x).In Sections 3.3 and 3.4, we describe the asymptotic behavior of R(x) depending on whether or not {d n } ∞ n=1 ∈ ℓ 2 (N).

Spectral analysis of Schrödinger operators with distributional potentials
In this section, we consider Schrödinger operators in the setting of Hypothesis 1.2.
2.1.Self-adjointness and form bounds. Associated with the differential expression ℓ are three linear, densely defined, unbounded operators H 0 , H min , H max acting on L 2 (R + ) defined as follows: and H min := H 0 , the closure of H 0 in L 2 (R + ).Then, upon setting q = τ, p = 1, r = 1, s = σ in [18, Section 3], we infer In the following Theorem, we discuss self-adjoint extensions of H min , prove that ℓ is limit point at ∞ and limit circle at 0, and obtain auxiliary resolvent estimates.
Proof.By Hypothesis 1.2, differential expression ℓ is regular at 0; thus, by [18,Lemma 3.1], all solutions of ℓu − zu = 0 can be extended by continuity to 0 so that u, u [1] are absolutely continuous in a neighborhood of 0. Hence, all such solutions are square integrable near and ℓ is limit circle near 0. In this setting, the Wronskian is defined for u, v ∈ D by and in order to show that ℓ is limit point at infinity, it suffices to check that cf. [18,Lemma 4.4].To that end, we will first prove that every f ∈ dom(H max ), f [1] lies in L 2 (R + ).Since f [1] ∈ AC loc (R + ), it suffices to analyze it near infinity.Let σ, τ be extensions of σ, τ by zero to the whole line R and consider the operator H acting on L 2 (R) given by This operator is self-adjoint, bounded from below and its form domain is given by H 1 (R), see [28,29]; in particular, dom( H) ⊂ H 1 (R).Fix u ∈ dom(H max ) and let u be the extension of u by zero to the whole line R. Pick any φ ∈ C ∞ (R) with supp(φ) ⊂ (1/2, ∞) and φ(x) = for x ≥ 1.Then, uφ ∈ dom( H) and, as dom( H) ⊂ H 1 (R), one also has uφ ∈ H 1 (R).Since ( uφ) ′ (x) = u ′ (x), x ≥ 1, we infer that u ′ lies in L 2 near infinity.Next, we show that σu lies in L 2 and u lies in H 1 near infinity.Recall from [28, Lemma 3.1] that for arbitrary interval I ⊂ R + of length 1, ε ∈ (0, 1), and ψ ∈ H 1 (I), ) In particular, inspired by the proof of [28,Theorem 3.4], we get That is, σu ∈ L 2 (R + ).Consequently, u [1] = u ′ − σu ∈ L 2 (R + ), and by Cauchy-Schwarz, the Wronskian W (u, v) lies in L 1 (R + ).Moreover, since W (u, v)(x) has a limit at infinity, see [18,Lemma 3.2], it must converge to zero as asserted in (2.3).In conclusion, ℓ is limit point at infinity.The fact that all self-adjoint extensions of H min are determined by the boundary conditions (2.1) follows from [18,Theorem 6.2] (where one should pick BC 1 0 (u) := u(0), BC 2 0 (u) := u [1] (0)).
Let us now switch to quadratic form h α .Our first objective is to show that it is relatively bounded with respect to the quadratic form of the Dirichlet or Neumann free Laplacian on R + , depending on the value of α.Note that for arbitrary ε > 0, employing (2.4), (2.5) as in the proof of [28,Lemma 3.2], for arbitrary u ∈ H 1 (R + ); moreover, by (2.4), , where dom(h D ) := H 1 0 (R + ) and dom(h N ) := H 1 (R + ).We will proceed with assuming α ∈ (0, π), the second case α = 0 can be handled similarly.For any a ∈ (0, 1), the inequalities (2.6), (2.7) yield b ∈ R such that That is, the lower order terms and the boundary term in the definition of h α , considered as quadratic form on H 1 (R + ), are relatively bounded with respect to Neumann form h N , with relative bound less than one, see [33,Section VI.3.3] or [48,Chapter X].Thus, by [48, Theorem X.17], h α is closed bounded from below quadratic form and there is a unique self-adjoint operator T α acting in L 2 (R + ) which satisfies We claim that H α ⊂ T α .Assume this claim, we note that both operators are self-adjoint and therefore must coincide.This implies that h α is the quadratic form of the operator H α which is consequently bounded from below.Returning to where in the second step, we used the boundary condition u [1] (0) = − cot(α)u(0).
In order to prove (2.2) (again we focus on the case α ∈ (0, π)), we invoke (2.6), (2.7) to obtain some Noting that the left-hand side above is the quadratic form of H α and the right-hand side is the quadratic form of C(−∆ N + λ), the assertion (2.2) follows from [33,Theorem VI 2.21], where it is shown that the ordering of quadratic forms implies the ordering of resolvents.
We say that the pair (σ + θ, τ − θ ′ ) is a gauge change of (σ, τ ).The domain dom(H max ) is gauge change invariant since for u ∈ AC loc (R + ) one has and a direct calculation shows that the action of the maximal operator H max is also gauge change invariant.The gauge change affects the definition of the quasi-derivative u 2 − θu.Therefore the self-adjoint boundary conditions u(0) cos α j + u [1] j (0) sin α j = 0 are relabelled by the formula cot This is due to the fact that the deficiency indices of H min are (2, 2) and the abstract Krein's resolvent formula [7,Theorem A.1].
We can now prove our version of the Blumenthal-Weyl criterion: Proof of Lemma 1.7.By Remark 2.3 and [21, Theorem 2.4], it suffices to prove the statement for α = 0. Let H D and h D denote respectively the Dirichlet Laplacian and its quadratic form on R + ; i.e., using the notation of Theorem 2.1 with α = 0, σ = τ = 0, write Our goal is to show that for σ, τ as in (1.6), the quadratic form h 0 is a relative compact perturbation of h D (see e.g., [45, Definition 2.12], [21,Section IV.4]).This assertion together with [45, Theorem 2.13] yields Spec ess (H 0 ) = Spec ess (H D ) and, when combined with Spec ess (H D ) = [0, ∞), proves the statement.Consider the quadratic form: In order to show that h 0 is a relative compact perturbation of h D , it suffices to verify cf. [45,Theorem 2.14].The first inequality (2.8) follows from (2.6), so it suffices to prove (2.9).First, let χ [a,b] denote the characteristic function of [a, b] and note that where in the last inequality, we used (1.6) and sup j u j H 1 (R + ) ≤ 1. Next, for t defined above, note that sup j χ [0,t] u j H 1 (R + ) ≤ 1 and, due to compactness of the embedding H 1 ((0, t)) ֒→ L 2 ((0, t)), there exists a subsequence {u j k } ∞ k=1 which is Cauchy in L 2 (R + ).For such a subsequence and arbitrary ε > 0, there exists where we used the Cauchy-Schwartz inequality and sup j χ [0,t] u j H 1 (R + ) ≤ 1.It follows from (2.10) with u := u jm − u jn , (2.11) and (2.12) that which yields (2.9) as required.
Proof of Corollary 1.8.Let σ(x) := Hence, for some C, c > 0 and sufficiently large x we have Remark 2.4.(i) The invariance of the essential spectrum under small at infinity perturbations of the coefficients has been investigated by many authors in various settings, see e.g., [24,27,59] and especially [40], which contains many relevant references.The central fact in the classical treatment of this problem via Weyl-type sequences, see [27,Section 10], is that H α has a locally compact resolvent; i.e., This still holds in our case, as readily seen from the explicit form of Green's function.However, there is a major obstacle in using the classical approach since dom(H α ), as a subset of L 2 (R + ), depends on σ, τ .Notably, one does not even have the inclusion C ∞ 0 (R + ) ⊂ dom(H α ) in general; e.g., such an inclusion does not hold when V is not locally L 2 .The key feature of our proof of Lemma 1.7 is that the form domain h α does not depend on σ, τ .Interestingly, the latter does depend on α, though the invariance of essential spectrum under perturbation of the boundary condition is handled by Krein's formula for the difference of resolvents of two self-adjoint extensions of the minimal operator H min , as discussed in Remark 2.3.
(ii) Relevant to this discussion is [25, Theorem 3.2] (see also [43], [44]), where the fullline version of (1.6) is shown to be equivalent to compactness of the multiplier given by an H −1 (R) potential.

2.2.
Weyl-Titchmarsh theory.Let us fix z ∈ C, g ∈ L 1 loc (R + ) and consider the differential equation −(u [1] ) ′ − σu [1] Rewriting it as a first order system, we get Assuming Hypothesis 1.2, since the matrix coefficients in A(z, x) lie in L 1 loc (R; C 2×2 ), the corresponding initial value problem has a unique locally absolutely continuous solution.In particular, for g = 0, α ∈ [0, π), we consider the initial value problem ℓu −zu = 0 and denote by φ α,z , θ α,z its solutions satisfying the initial conditions The solutions are entire with respect to z.In the special case α = 0, we denote θ z := θ α,z , φ z := φ α,z .Note that any u ∈ D solving ℓu = zu satisfies .
We will often denote T (z; x) := T (z; x, 0).Next, we turn to the Weyl-Titchmarsh theory for ℓ.Assuming Hypothesis 1.2, since ℓ is limit point at infinity, for any z ∈ C \ R, there is a 1-dim set of solutions in L 2 (R + ) to ℓu = zu, where any such non-trivial solution is called a Weyl-Titchmarsh solution at infinity and denoted by ψ α,z .Fix any ψ α,z ; the Weyl-Titchmarsh m-function is given by , where note that m α (z) is independent of the choice of ψ α,z .Note that the boundary condition affects the Weyl function by a rotation matrix: denoting by ≃ the projective relation on Our next objective is to show that m α (z) can be obtained by the intersection of Weyl disks defined as U α : = cos(α)u [1] (0) − sin(α)u(0) sin(α)u [1] (0) + cos(α)u(0) . (

2.15)
To motivate this definition, let us reformulate it using the Möbius transformations.To that end, denote Ĉ = C ∪ {∞} and introduce the quotient map π : where This identity, together with the fact that Möbius transformations map generalized disks in Ĉ to generalized disks, yields that D α x (z) is a generalized disk in Ĉ.We will show below that, for z ∈ C + and x > 0, the disks D α x (z) ⊂ C + shrink to a point in a monotone fashion as x ↑ +∞.(2.17) In addition, if u = 0, then the function is real-valued and strictly increasing in x.
Then, integrating both sides of the above identity from 0 to x yields (2.17).Next, since If u is a non-trivial eigensolution and u(y) = 0 for some y, then u [1] (y) = 0 and thus u only has isolated zeros.In particular, iW (u, u) ′ = 2(Imz)uu > 0 away from a discrete set, and so iW (u, u) is strictly increasing.
To conclude this subsection, we recall from [18, Section 9] the spectral decomposition for the operator H α .The Herglotz function m α discussed in Proposition 2.6 (iv) gives rise to a Borel measure µ α via the Stieltjes-Livsic inversion formula Im m α (α + iε)dλ, for real numbers λ 1 < λ 2 .The operator H α is unitarily equivalent to the operator of multiplication by the independent variable in the space L 2 (R, µ α ) and the classical spectral description via boundary values of m α (z) holds, see [18,Section 9].For instance, as in the classical setting, if α − β / ∈ πZ, then a.c.parts of µ α , µ β are mutually a.c., and their singular parts are mutually singular.
We will return to a detailed analysis of the absolutely continuous part of spectral measure µ α in Section 2. 4, where we will rely on estimates for eigensolutions discussed next.

Eigensolution estimates.
In this section, we derive auxiliary estimates for solutions of ℓu = Eu, E ∈ R, u ∈ D. To describe the main assertions, let us fix λ > 0 and denote In the estimates that follow, we give bounds with explicit dependence on the parameter E; we do not optimize these estimates, but we will use explicit estimates in some of the proofs that follow.The operator norm bound implies that A(t, E) is uniformly locally integrable: on every interval The case y < x follows analogously.
(ii) We fix and assume that for some y 0 ∈ I, u(y 0 ) ≥ |u [1] (y Combining, we conclude that for all x, y ∈ I, Since C 1 e λ|E| S < 1, this implies |u [1] (y In particular, u [1] has no zeros on the interval I, so it has constant sign there.Thus, On the other hand, denoting the end points of I by j − < j + , one has (2.21) Since u is not identically zero on I, combining (2.20) and (2.21), we obtain and contradicts (2.19).
(iii) Impose C 3 ≤ 1 to ensure δ ≤ 1. Assume that the claim is false: then u(y) = 0 and by continuity there exist By considering ±u, without loss of generality we can assume u(s) > 0.
Moreover, let us assume u [1] (s) ≥ 0 and work on the interval [s, x 2 ]; the other case is analogous by working on [x 1 , s].
The first step is an upper bound for the quasiderivative.For x ∈ [s, x 2 ], denote x s σ(t) dt u [1] (x).
As a first application, we prove a Simon-Stolz type criterion for absence of pure point spectrum, cf.[55].
2.4.The absolutely continuous spectrum via Last-Simon approach.The main goal of this section is to develop the Last-Simon approach, cf.[39], to absolutely continuous spectrum via growth of transfer matrices.To do this, we first discuss the relation between the subordinacy theory and the growth of transfer matrices.We say that u ∈ D is a subordinate solution of ℓu − zu = 0 if for some solution ℓv − zv = 0, v ∈ D \ {0}, Note that if (2.27) holds for some eigensolution v, it holds for every eigensolution linearly independent with u.Moreover, taking v = u, we see that if a subordinate solution exists, it must be linearly dependent with its complex conjugate, so it must be a multiple of φ α,z for some α.
For µ-a.e.λ ∈ R, the normal boundary value lim ǫ↓0 m(λ + iǫ) exists in C + .Subordinacy theory relates this value to the existence of subordinate solutions [26,30]; this was recently understood to be a special case of bulk universality in a general Hamiltonian system setting [23].To explain this, incorporate the boundary condition into the transfer matrix by defining .
This transfer matrix T α (z; x) obeys the initial value problem This is a special case of a so-called Hamiltonian system, and can be written as The transfer matrices generate a matrix kernel By the Cauchy-Schwarz inequality, the solution φ α,z is subordinate if and only if Scaling limits of K l are related to the normal limits of m-function: by [23, Theorem 1.8], Using (2.14) to restate in terms of m 0 , we conclude: Lemma 2.9.Assume Hypothesis 1.2.For any E ∈ R, We also denote N(ℓ) := {E ∈ R : no solution of ℓu − Eu = 0 is subordinate}.
Taking the union over α in Lemma 2.9 and taking negations, for every E for which the normal limit exists, E ∈ N(ℓ) if and only if Recall that we denote by µ α ac the absolutely continuous part of the spectral measure µ α .Lemma 2.10.Assume Hypothesis 1.2.For arbitrary α ∈ [0, π), N(ℓ) is an essential support for the absolutely continuous spectrum of H α in the sense that µ α ac is mutually absolutely continuous with χ N (ℓ) (E) dE.In particular, Proof.Recall from [18,Corollary 9.4] that an essential support for µ α ac is the set Since m α has a normal boundary value in C + for Lebesgue-a.e.E (see e.g., [57, Theorem 3.27, Corollary 3.29]), the set is also an essential support for the a.c.spectrum.This set is independent of α by (2.14).By the observation proceeding the Lemma, the set N(ℓ) is another essential support for the a.c.spectrum of H α .
Proof of Theorem 1.3.Since the spectral type of the a.c.part is independent of α (Lemma 2.10), it suffices to prove the claim for α = 0. Assuming this value, we drop symbol α from subsequent notation.
Due to preservation of Wronskian we have φ α,E (x) θ α,E (x) ≥ 1; thus, (2.28) Then, one has where in the last step, we used φ α,E (x) ≤ T (E, x) .If the solution φ α,E is subordinate, taking the limit l → ∞ shows In other words, for the set Σ ac defined by (1.2), we conclude Σ ac ⊂ N(ℓ).Therefore, to complete the proof, it is enough to show that lim inf To that end, let us fix γ > 1 and introduce the measure .
Since dρ is equivalent to µ ac , in order to prove that Σ ac is an essential support for µ ac , it is enough to show To that end, we will prove the following auxiliary inequalities: there exists Υ > 0 such that for all x ∈ (2, ∞), We will prove the first parts of (2.30), (2.31), the second parts can be proved analogously.
Since supp µ 0 is bounded from below, for some Λ < min supp µ 0 , Then, using spectral representation of Green's function [18, Lemma 9.6] and the last part of Theorem 2.1, we obtain where λ(σ, τ ) is as in (2.2), G and G f ree denote respectively the Green's functions for H 0 and the free Dirichlet Laplacian on R + , i.e., for σ = τ = 0, and the constants α, β depend only on Λ, σ, τ .Integrating (2.32) yields the first inequality in (2.30).
Next, we switch to the first inequality in (2.31).By Lemma 2.7, where in the last step we used (2.32 T (E; t, s) 2 dt ds dρ(E) < ∞, which implies by Fatou's lemma that for ρ-a.e.E, By Lemma 2.7, for any E there exists C > 0 such that Theorem 1.4 will be our principal tool for showing the absence of absolutely continuous spectrum for a class of slowly decaying potentials, see Theorem 1.12(b).
2.5.Carmona formula and pure a.c.spectrum on intervals.In this section, we discuss a Carmona-type, cf.[5], approximation result for the spectral measure of H α and use it to derive a criterion for pure a.c.spectrum on an interval.This is our main tool for showing purely absolutely continuous spectrum for a class of slowly decaying potentials, see Theorem 1.12(a).Theorem 2.11.Assume Hypothesis 1.2.For any α ∈ [0, π), the measures converge vaguely to the spectral measure µ α of H α as x → ∞ in the sense that
It follows from above that the measure corresponding to m x,α (z) has the restriction to (0, ∞) given by (2.40), which concludes the proof.
2.6.Prüfer variables.We now introduce Prüfer variables associated with real eigensolutions of ℓ and relate their growth to that of the transfer matrices.In the locally integrable setting, Prüfer variables are a well-established tool for spectral analysis for decaying potentials; we will use them in the proof of Theorem 1.12.
Proof of Theorem 1.10.Consider Prüfer variables R(x, E) associated to the solution

Distributional sparse potentials. Investigation of spectral types
In this section we prove Theorem 1.12.
3.1.Decomposition of sparse potentials.The first step in the proof of Theorem 1.12 is to reformulate it in terms of the Hryniv-Mykytyuk decomposition in a way that is consistent with the sparse structure of the potential.If we applied their decomposition directly to V , the dependence on integers in [28] would complicate matters; instead, note that [28, Lemma 2.2] gives a decomposition of supported in the same interval (the authors use ∆ = 1 but this is merely a matter of rescaling).Moreover, this decomposition is continuous in H −1 (R)-norm.Thus, we obtain In addition, without loss of generality, we can assume that S = 0 and T = 0: this is because if one of S, T is identically equal to zero, we can pick arbitrary In summary, we will use the following setup throughout this section: Let β > 1 be so that x n ≥ Cβ n for a fixed constant C > 0. Let T, S, T n , S n be as in (3.1) and suppose, in addition, T ≡ 0, S ≡ 0. Furthermore, fix a sequence Let sparse coefficients τ, σ be given by Fix arbitrary α ∈ [0, π), and let H α be the corresponding Schrödinger operator as defined in Theorem 2.1.
The rest of this paper is dedicated to the proof of Theorem 1.12; parts (a), (b) are provided in Sections 3.3, see page 32, and 3.4, see page 43, respectively.

3.2.
Auxiliary estimates for Prüfer variables.We begin with a series of auxiliary results.The first one concerns estimates for Prüfer variables and their k-derivatives near x n for large n.
To streamline the exposition, in the remaining part of the paper, we will use C for positive constants that vary from one inequality to the other but always remain n-independent.Also, whenever an inequality involving n is mentioned without a specified range of admissible values of n, it is assumed that the range is n ≥ n 0 for some n 0 .

Lemma 3.2. Assume Hypothesis 3.1 and fix any compact interval
and sufficiently large n, Let h(x) := ∂θ ∂k (x).Differentiating (2.42) with respect to k, we have ∂h ∂x = f + gh with which, for [a, b] := [x n − ∆, x n + ∆] and sufficiently large n, satisfy Our objective is to prove that there exists C > 0 such that for sufficiently large n, for sufficiently large n.Since β > 1, d n → 0, and x n → ∞, there exists n 1 ∈ N such that for all n ≥ n 1 , For such n 1 , let D ≥ 2 be such that We claim that for all n ≥ n 1 , Indeed, using (3.2) together with (3.11) and (3.12), for n ≥ n 1 , y ∈ [−∆, ∆], we have Proof of (3.4)(ii).Let w := ∂h ∂k = ∂ 2 θ ∂k 2 and differentiate (2.43) twice with respect to k, then where Note that for [a, b] := [x n − ∆, x n + ∆] and sufficiently large n, where we used (3.13) in the first inequality.Then, using (3.Since β > 1 and d n → 0, for any C > 0, there is large enough n 2 such that for all n ≥ n 2 , To complete the proof of (3.4)(ii), integrate (3.14) over [x n − ∆, x n + ∆] and use (3.18); then, Hence, Combining this with (3.19) concludes the proof for (3.4).
Remark 3.3.Lemma 3.2 and its proof are similar to [36,Propositions 5.1,5.2],where the case of S n = 0 and T n = T ∈ L ∞ (R) was considered.We extend that proof to the case S n = 0, and T n ∈ L 1 (R) by using (3.6), which is an L 1 version of the key inequality [36, eq.(5.7)], and verifying new inequalities (3.8), (3.15).
To streamline the exposition, we introduce the following notation Note that due to (3.1), for a fixed interval [α, β] ⊂ (0, ∞), we have In the following Lemma, we provide the second order expansion of variable θ with respect to d n as n → ∞.This result will be used in Lemma 3.10 and the proof of Theorem 1.12(a).Lemma 3.4.Assume Hypothesis 3.1 and fix any compact interval [E 1 , E 2 ] ⊂ (0, ∞).Then, the following asymptotic expansion , where, recalling (3.20), θ (1)  n (y) :=  Sections 5,6].In our case, notice that when S n = 0, the integral on the right-hand side of (3.24) contains an additional term σ n (s) sin(2θ n (s)).This will become relevant in the proof of Theorem 1.12(b).
Corollary 3.6.Assume the setting of Lemma 3.4.Then, holds for sufficiently large n and all k ∈ Proof.Differentiating (3.23) with respect to k and using (3.4), we get where we used (3.2) in the last step.
To conclude this section, we show that (3.3) rules out point spectrum for H α .
Proof.Consider the Prüfer variables corresponding to a nontrivial real eigensolution u at E > 0, normalized so that R(0) = 1.By (3.5)(iii), This means at most exponential decay of the sequence R(x n + ∆) 2 , since the sequence d n is bounded.Due to the superexponential growth (3.2), this implies and, by Theorem 1.6, this implies u / ∈ L 2 (R + ).

3.3.
Purely absolutely continuous spectrum.In this section, we provide the proof of Theorem 1.12 part (a).
Proof of Theorem 1.12 (a).By Lemma 1.7 Spec ess (H α ) = [0, ∞).Then, by Theorem 2.11, it suffices to show that for every finite interval [E 1 , E 2 ] ⊂ (0, ∞), In fact, we will show that for any θ ∈ [0, 2π) and any non-negative g ∈ C ∞ 0 (0, ∞) (after possibly passing to a subsequence), The latter together with Proposition 2.13 yields (3.27).Explicitly, we will derive a recursive inequality for a sequence {ρ n } ∈ ℓ 1 (N), ρ n > 0, which is sufficient for (3.28).To that end, we integrate (2.43) over the interval [x n − ∆, where Then, where d n S n (y) cos(2θ (0) n (y)) dy .(3.32) and θ n is as in (3.23).Indeed, (3.31) follows readily from Returning back to (3.30), notice that (3.31) together with n .To obtain (3.29), multiply the above inequalities by g(k) and integrate over (0, ∞); then, Recalling (3.32) and exchanging the order of integration, we obtain where note that all terms above are of the form with Proof of Claim.Let v be either sin or cos so that one has u = v ′ , and rewrite E n as Next, we integrate by parts with respect to k to obtain three integrals, each corresponding to applying ∂ k to one of the three functions in . (3.38) Case 1: ∂ k lands on the first term in (3.38).Then, where β > 1 is such that x n ≥ Cβ n (Hypothesis 3.1) and in the first inequality, we used log(R(k; where in the first inequality, we used (3.39) and with (3.40) proved in Remark 3.8.Case 3: ∂ k lands on the last term in (3.38).In this case, we have where in the first inequality, we used (3.4)(ii) and (3.39); in the second inequality, we set with (3.41) proved in Remark 3.8.
Since all three terms on the right-hand side of (3.34) are of the type E n , Combining this with (3.33), for sufficiently large n, ) and thus (3.28) holds.Remark 3.8.In the setting of Theorem 1.12(a), the numerical series introduced in (3.40) and (3.41) are convergent due to [36,Lemma 5.3]; we expand the concise proof provided therein.For a numerical sequence d = {d n } n∈N , consider the convolution operator By Young's inequality, T γ is a bounded linear operator on ℓ 2 (N).Let γ > 1 be such that and (3.41) follows from where β is as in Hypothesis 3.1 in the second inequality.
3.4.Purely singular continuous spectrum.In this section, we provide the proof of Theorem 1.12 part (b).Since Proposition 3.7 rules out the presence of positive eigenvalues, to demonstrate the absence of absolutely continuous spectrum, the strategy is to verify the conditions of Theorem 1.4 via (2.44) and lim We begin with a set of auxiliary results concerning the Fourier transform of the potential.We will use the notation Lemma 3.9.Assume Hypothesis 3.1.Then (i) For j = 0, 1, 2 one has uniformly for z in compact intervals I ⊂ (0, ∞) that contain no roots of T .In particular, for such I one has Identical assertions hold with T replaced by S.
(ii) Let Φ(z) := (2z) −1 T (z) − i S(z) and suppose that a compact interval J ⊂ (0, ∞) contains no roots of Φ.Then one has Proof.(i) Denote for simplicity f n (z) := T n (z), f (z) := T (z).Clearly, f is entire function which is not identically zero and f n converges to f uniformly on compacts.We claim that there exists n 0 ∈ N such that: (1) for all n ≥ n 0 and z ∈ I, f n (z) = 0, (2) for j = 0, 1, 2, arg f n → arg f (j) uniformly on I and in particular, sup To prove these two basics facts from complex analysis, first, recall that if f n → f uniformly on some compact K, then for any compact K ′ ⊂ int K, f ′ n → f ′ uniformly on K ′ ; this holds by Cauchy's differentiation formula On the set I d/2 , f n converge uniformly to f , so there exists n 0 such that for all n ≥ n 0 and z ∈ I d/2 , f n (z) = 0.By the above argument, f ′ n → f ′ uniformly on I d/3 .Thus, (log (ii) The proof follows directly from complex analytic facts (1), ( 2) stated above with Assuming Hypothesis 3.1, we say that a compact interval J ⊂ R + := (0, ∞) is (S, T )−admissible if J avoids zeros of S(z), T (z), and (2z) −1 T (z) − i S(z), that is, In the following Lemma, we derive a third order expansion for the increment of log R(x n + ∆, k) with respect to d n .For {z n } n≥0 ⊂ C we denote δz is (S, T )−admissible, and define Y n (k) := log R(x n + ∆, k).Then the following asymptotic expansion holds uniformly for k ∈ where the oscillatory terms X n , X n and are given by Next, by Theorem 1.4, Spec ac (H α )∩[E 1 , E 2 ] = ∅ and, since the union of all (S, T )−admissible intervals gives R + up ot a discrete set, we conclude Spec ac (H α ) = ∅.Therefore, the spectrum of H α is purely singular continuous on (0, ∞).
At this point let us prove the assertion made in Example 1.11.Proof of Example 1.11.The Wigner-von Neumann potential V , explicitly defined in [53, Section 3, Part B], admits a real-valued nontrivial eigenfuction u ∈ L 2 (R + ) corresponding to eigenvalue 1.In particular, for the choice of boundary condition at 0 corresponding to u, the Schrödinger operator − d 2 dx 2 + V does not have purely absolutely continuous spectrum on (0, ∞).We set σ(x) := − ∞ x V (t)dt and τ = 0. Then V = σ ′ + τ so this is a gauge change of the Wigner-von Neumann potential; in particular, spectral type is unchanged.To prove σ(x) = x→∞ O(1/x), we recall the asymptotic formula

. 5 )
(iv); in the second inequality, we used d m = o(1), (3.2), and uniformly on I d/3 .Thus, (log f n ) ′′ → (log f ) ′′ uniformly on I d/4 .Taking imaginary parts we conclude arg f ′ n → arg f ′ and arg f ′′ n → arg f ′′ uniformly on I. Choosing branches so that log f n (min I) → log f (min I) and taking limits of log f n (x) = log f n (min I) + x min I f ′ n (y) f n (y) dy and taking imaginary parts shows uniform convergence of arg f n to arg f on I.
1.1) holds, hence, by Theorem 2.1 H V is limit point at infinity.In addition one has (1.6), thus, by Lemma 1.7, Spec ess .36) Claim.For B n and E n defined in (3.28) and (3.35) respectively, and β > 1 as in Hypothesis 3.1, there is a sequence {s n