Canonical systems whose Weyl coefficients have dominating real part

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on the half-line $(0,\infty)$ whose Hamiltonian $H$ is a.e. positive semi-definite, denote by $q_H$ its Weyl coefficient. De Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. The main result of the paper is a criterion when the singular integral of the spectral measure, i.e. Re $q_H(iy)$, dominates its Poisson integral Im $q_H(iy)$ for $y\to+\infty$. Two equivalent conditions characterising this situation are provided. The first one is analytic in nature, very simple, and explicit in terms of the primitive $M$ of $H$. It merely depends on the relative size of the off-diagonal entries of $M$ compared with the diagonal entries. The second condition is of geometric nature and technically more complicated, but explicit in terms of $H$ itself. It involves the relative size of the off-diagonal entries of $H$, a measurement for oscillations of the diagonal of $H$, and a condition on the speed and smoothness of the rotation of $H$.


Introduction
We investigate the spectral theory of two-dimensional canonical systems where −∞ < a < b ≤ ∞, z ∈ C is the spectral parameter, J is the symplectic matrix J := 0 −1 1 0 , and H is the Hamiltonian of the system.We deal with systems whose Hamiltonian satisfies ⊲ H(t) ∈ R 2×2 and H(t) ≥ 0 a.e.; ⊲ for all c ∈ (a, b) we have c a tr H(s) ds < ∞; ⊲ H(t) = 0 a.e.
We further assume that H is in the limit point case at the right endpoint b, i.e. (1.2) A central role in the theory of such equations is played by the Weyl coefficient q H associated with H.For Sturm-Liouville equations its construction goes back to H. Weyl [16].Let us recall the definition of q H for canonical systems.To this end, let W (t, z) be the (transpose of) the fundamental solution of the system (1.1), i.e. the unique 2 × 2-matrix-valued solution of the initial value problem W (a, z) = I. ‡ The second and third authors were supported by the project P 30715-N35 of the Austrian Science Fund (FWF).The third author was supported by the joint project I 4600 of the Austrian Science Fund (FWF) and the Russian foundation of basic research (RFBR).
Note that the transposes of the rows of W are solutions of (1.1), and let us write W (t, z) = w11(t,z) w12(t,z) w21(t,z) w22(t,z) .If (1.2) is satisfied, then the following limit exists and is independent of ζ in the closed upper half-plane C + ∪ R: q H (z) := lim t→b w 11 (t, z)ζ + w 12 (t, z) w 21 (t, z)ζ + w 22 (t, z) , z ∈ C \ R; the function q H is called Weyl coefficient associated with the Hamiltonian H.It is a Nevanlinna function or identically equal to ∞ (when h 2 (t) = 0 for a.e.t ∈ (a, b)); a Nevanlinna function 1is a function that is analytic in C \ R and satisfies q H (z) = q H (z) and Im q H (z) • Im z ≥ 0 for all z.The significance of the Weyl coefficient is that the measure µ in its Herglotz integral representation is a spectral measure for the differential operator constructed from the equation (1.1) (when β > 0, this differential operator is actually multi-valued and one can include a point mass at infinity with mass β).
A famous theorem by L. de Branges [3] says that the assignment H → q H establishes a bijective correspondence between the set of all suitably normalised Hamiltonians on the one hand, and the set of all Nevanlinna functions on the other hand.In view of de Branges' correspondence, it is a natural task to translate properties from H to q H (i.e.direct spectral relations) and vice versa from q H to H (i.e.inverse spectral relations).In the best case one can go both ways.For illustration, let us mention two examples of such theorems.It is possible to explicitly characterise those Hamiltonians H for which q H has an analytic continuation to C \ [0, ∞), see [17], or those Hamiltonians for which q H has a meromorphic continuation to all of C, see [15].The first result characterises that the differential operator associated with (1.1) is non-negative, the second one that it has discrete spectrum.
In the present paper we prove a direct and inverse spectral relation of a different kind.It belongs to a family of results which relate the behaviour of H locally at the left endpoint a with the behaviour of q H when z tends to +i∞; for physical reasons one also speaks of the high-energy behaviour of q H . Recall that the behaviour of Im q H (iy) at +∞ is related to the behaviour of the spectral measure at ±∞; see, e.g.[10,Section 4].Our main result is Theorem 1.1 stated further below, where we characterise those Hamiltonians H for which 2Im q H (iy) ≪ |q H (iy)|, y → +∞, (1.3) i.e. those Hamiltonians for which the singular integral Re q H (z) of the spectral measure strictly dominates the Poisson integral Im q H (z).
In our theorem, where (1.3) is listed as item (i), we give two different conditions on H, called (ii) and (iii), which are both equivalent to (1.3).Condition (ii) is analytic in nature, very simple, and explicit in terms of the primitive M (t) := t a H(s) ds of H, which is a nonnegative and nondecreasing matrix function.It says that, locally at a, the off-diagonal entries of M (t) should be as large as its diagonal entries.Condition (iii) is of geometric nature and somewhat more complicated.It involves the relative size of the offdiagonal entries of H compared with the diagonal entries, a measurement for oscillations of the diagonal of H, and a condition on the speed and smoothness of the "rotation" of H.
From a function-theoretic perspective, the behaviour exhibited by (1.3) is rather peculiar.For every Nevanlinna function q one has that for (in a measure-theoretic sense) most points on the boundary of the open upper half-plane (including +i∞) condition (1.3) fails; see [12] and recall that real and imaginary parts are comparable on approaching almost every point of the absolutely continuous spectrum.On the other hand, for a certain subclass of Nevanlinna functions it holds that for (in a topological sense) many boundary points (1.3) holds, cf.[4, Theorem 1] where one uses a curve that approaches the boundary tangentially.Neither of these statements has any implication for a single boundary point (in our case +i∞).The condition (iii) in Theorem 1.1 is a very strong restriction on H. Hence, one message of Theorem 1.1 is that (1.3), i.e. strict dominance of the singular integral at a specific boundary point, is a rather rare phenomenon.
Our interest in the class of Hamiltonians with (1.3) originates from the recent result [10, Theorem 1 .1].In this theorem we showed that, for every Hamiltonian H, the following estimates and hold, where L H (y) and A H (y) are certain functions defined explicitly in terms of the primitive M (t), and the constants in "≍" and " " are independent of H; we recall details in Section 2.6.
The question arises whether the lower bound L H (y) is sharp.The equivalence of (1.3) with Theorem 1.1 (ii) says that on a qualitative level the answer is affirmative: we have It is an open problem if there is a quantitative relation between Im q H (iy) and L H (iy) (assuming that Im q H (iy) ≪ |q H (iy)| and thinking up to universal multiplicative constants).This seems to be a rather involved question, and we expect that the equivalence of (1.3) with Theorem 1.1 (iii) will be of help to attack it.Let us give a brief overview of the contents of the paper.In the remainder of the Introduction we formulate the main theorem, Theorem 1.1, and a sequence variant, Theorem 1.4, and provide an illustrative example.In Section 2 we provide some preliminaries and set up notation.Section 3 contains the proof of the equivalence of (i) and (ii) in our main results.Section 4 contains preparations for the proof of the equivalence with (iii), which is then carried out in Section 5. Finally, in Section 6 we consider the situation when the diagonal entries of H, or their primitives, are regularly varying.

Formulation of the main theorem
We formulate our main theorem for Hamiltonians that satisfy ⊲ neither of the diagonal entries of H vanishes a.e. on some interval starting at the left endpoint 0.
Both assumptions are no loss in generality, and are only imposed for simplicity.The first one can always be achieved by a change of the independent variable in equation (1.1), and changes of variable do not alter the Weyl coefficient; see Section 2.2.The second condition excludes some exceptional cases where there is nothing to investigate: if it is not satisfied, then lim y→∞ Im qH (iy) |qH (iy)| = 1; we provide more details in Sections 2.2 and 2.3.Throughout the paper we write sometimes we write M (H, t) and m i (H, t) instead of M (t) and m i (t) respectively to indicate the dependence on H.Moreover, λ denotes the Lebesgue measure.Next, we have to introduce some notation which looks a bit technical on first sight, but actually is not.The intuition behind these quantities is discussed in Remark 1.3 below.The functions are well defined because h 3 (t) 2 ≤ h 1 (t)h 2 (t) for a.e.t > 0 and m 1 (t), m 2 (t) > 0 for all t > 0; the latter follows from the assumption that neither of the diagonal entries of H vanishes a.e. on an interval starting at 0. Set Let H be a Hamiltonian defined on the interval (0, ∞) such that (1.2) holds and neither h 1 nor h 2 vanishes a.e. on some neighbourhood of the left endpoint 0. Then the following statements are equivalent.
(ii) We have ), and all open intervals I, J ⊆ R \ {0} with I ∩ J = ∅ and at least one of I and J being bounded, the following limit relations hold: Under a certain additional assumption, the conditions in (iii) greatly simplify.This assumption is quite strong, and will, in many interesting cases, not be satisfied.Still, in order to understand the nature of (1.11) and (1.12) and the proof of their equivalence to (1.3), it is worth stating the following addition.
1.2 Addition to Theorem 1.1.Assume that, in addition to the assumptions of Theorem 1.1, the following conditions hold: Then the equivalent properties (i), (ii), (iii) in Theorem 1.1 are further equivalent to the following condition.
(iv) For all γ and I, J as in Theorem 1.1 (iii) we have where (1.17) Note that (1.13) implies that m 1 (t) + m 2 (t) = t.Hence, by [10, Theorem 1.1] (see also Proposition 2.9) we have lim inf (1.18) We come to the promised explanation of the conditions in (iii) (and (iv)).
The validity of this relation for all γ ∈ [0, 1) just says that the functions x → 1 − σ H (tx) 2 converge to 0 in measure as t → 0. Since they are non-negative and bounded by 1, this is also equivalent to the fact that their integrals converge to 0. Note that Hence the validity of (1.15) for all γ ∈ [0, 1) is (again by rescaling) equivalent to lim t→0 where the integrand is understood as equal to 1 at points where its denominator vanishes; this means that the Hamiltonian should be almost of zero determinant in the vicinity of the left endpoint 0 in a measure-theoretic sense.The role of π H is not so obvious.It is related to what one may think of as "rotation" of H.To see this, write H in the form where ⊙ denotes the Hadamard, i.e. entry-wise, product of the 2 × 2-matrices.The first factor takes the relative size of the off-diagonal entries into account; the second factor has zero determinant and corresponds to some kind of rotation.The factorisation in (1.19) is possible, for instance, with where Arccot is the branch with values in (0, π).Then We may say -descriptively -that ζ H is the rotation of H.
The statement (1.16) is equivalent to the following statement (see Section 5): there are no two separated arcs on the unit circle, such that, in the vicinity of the left endpoint 0, ζ H (t) often belongs to one arc and also often belongs to the other arc.In other words, the Hamiltonian should rotate so slowly that, on every interval close to 0, it looks -from a measure-theoretic viewpoint -as if its direction were constant; see also Example 1.7 The more complicated conditions (1.11) and (1.12) are weighted and rescaled variants of (1.15) and (1.16); see Section 2.7.The role of the function t s is to take care of heavy oscillations, and the purpose of the weight m2(s) m1(s) in the definition of π H,s is to level out the contributions of the two diagonal entries.Moreover, zooming into the vicinity of the left endpoint 0 is now achieved by sending the rescaling parameter s to 0.
Let us note that also the relation (1.11) can be rewritten in integral form, namely as To prove Theorem 1.1 we show the implications Interestingly, very different methods enter in the proofs of the various implications.
⊲ The implication "(i)⇒(ii)" is a direct consequence of [10,Theorem 1.1] in the form of Proposition 2.9 below.We recall that this theorem is proved by directly studying Weyl discs and estimating the power series coefficients of the fundamental solution of the canonical system.
⊲ The proof of "(ii)⇒(iii)" requires an elementary but elaborate analysis of the connection between H and its primitive M .In particular, estimates are proved where the constants are independent of the Hamiltonian.This is done in Section 4; see Propositions 4.1 and 4.2.
⊲ To show "(iii)⇒(i)" and "(ii)⇒(i)" we use cluster sets and compactness arguments for Hamiltonians endowed with the inverse limit topology of weak L 1 -topologies on finite intervals; see Section 2.1.Another necessary tool is provided in Section 2.5, and a crucial role is taken by a weighted variant of Y. Kasahara's rescaling trick [7], which relates the behaviour of q H towards i∞ with weighted rescalings of H; see Section 2.7.
The proof of "(ii)⇒(i)" was included in order to decouple the equivalences between (i) and (ii), and between (i) and (iii), respectively.This enables reading the proof of "(i)⇔(ii)" without having to go into the technical details of Section 4. We thank a referee for suggesting an argument which makes this possible.

A sequence variant of the theorem
We can also give a variant of Theorem 1.1 where limits are replaced by limits inferior.It reads as follows.
1.4 Theorem.Let H be a Hamiltonian defined on the interval (0, ∞) such that (1.2) holds and neither h 1 nor h 2 vanishes a.e. on some neighbourhood of the left endpoint 0. Then the following statements are equivalent.
(iii) For each T ∈ (0, ∞) there exists a sequence (s n ) n∈N with s n → 0, such that for all γ ∈ [0, 1), and all open intervals I, J ⊆ R \ {0} with I ∩ J = ∅ and at least one of I and J being bounded, the following limit relations hold: Also in this case, the analogous addition holds.(iv) There exists a sequence (t n ) n∈N with t n → 0, such that, for all γ and I, J as in Theorem 1.4 (iii), we have The conditions (1.23) and (1.24) can be rewritten in integral form in the very same way as before.Namely, (1.23) as and (1.24) as

Two examples
Let us illustrate Theorems 1.1 and 1.4 with two examples.The first one demonstrates a standard situation; it will be revisited in a more general form in Section 6 of the present paper, and in the forthcoming paper [11].The second example demonstrates a more peculiar situation, where and consider the Hamiltonian For this example a computation shows the following facts (this is elementary and we skip details): The situation that lim y→∞ Im qH (iy) |qH (iy)| = 0, equivalently that lim t→0 det M(t) m1(t)m2(t) = 0, appears only when q H (iy) grows very slowly.In fact, if 1.7 Example.Let (t n ) n∈N be a strictly decreasing sequence of positive numbers such that tn+1 tn → 0 (and hence t n → 0), set t 0 := ∞ and consider the partition (0, ∞) = I + ∪ I − where Further, let ϕ + , ϕ − ∈ (0, π) \ { π 2 } with ϕ + = ϕ − and define the Hamiltonian H by where Clearly, (1.13) and (1.14) are satisfied, so that we can apply the Additions to Theorems 1.1 and 1.4.Since σ H (t) = 1 for t > 0, the limit relation (1.15), and hence also (1.24), holds for every γ ∈ [0, 1 The limit relations (1.16) and (1.25) hold trivially whenever By symmetry, we only have to consider the case when c + ∈ I and c − ∈ J, which we assume in the following.For t > 0 we have Then as n → ∞ and, similarly, F (t 2n+1 ) → 0. This shows that (1.25) is satisfied and hence also (i) in Theorem 1.4.On the other hand, for n ∈ N such that t2n t2n−1 ≤ 1 2 , we have as n → ∞.This implies that (1.16) is not fulfilled and hence neither is (i) in Theorem 1.1.To summarise, Theorems We recall how H 1 can be topologised appropriately.This is already used in the work of L. de Branges.An explicit formulation is given in [14]; for a more structural approach see [13], which we use as our main reference in the following.For each T < ∞ the set H 1 T is a subset of L 1 ((0, T ), R 2×2 ), and hence naturally topologised with the 1 -topology or the weak L 1 -topology.It turns out that the latter is more suitable because the weak L 1 -topology on H 1 T is compact and metrisable; see [13,Lemma 2.3].Now consider the family (H 1 T ) T ∈(0,∞) with the restriction maps ρ T ′ T : The set H 1 can be naturally viewed as the inverse limit of this family: every function on (0, ∞) can be identified with the family of all its restrictions to finite intervals.Endowed with the inverse limit topology (see, e.g.[2, §I.4.4]),where we use the weak L 1 -topology on H 1 T , the set H 1 becomes a compact metrisable space; see [13,Lemma 2.9].The map that assigns to a Hamiltonian H its Weyl coefficient q H is continuous when the set of Nevanlinna functions is endowed with the topology of locally uniform convergence; see [13,Theorem 2.12].
Throughout the remainder of the paper we often deal with limit points of families of Hamiltonians.In general, for a net (x i ) i∈I in some topological space X, we denote by LP(x i ) i∈I the set of its limit points, i.e.

LP(x
If there is a need to specify the topology, we shall add an index.For example, if X is a normed space, we write LP (x i ) i∈I for limit points w.r.t. the norm topology, and LP w (x i ) i∈I for limit points w.r.t. the weak topology.2.2 Remark.In our context the space X is usually metrisable, and the index set I is N, (0, 1] or [1, ∞), each endowed with the natural order (or the reverse order in the case of (0, 1]).In these situations one can restrict attention to subsequences rather than subnets: Note that in the cases when I = (0, 1] or We need the following simple fact about constant singular limit points.It is proved using the compactness of H 1 , continuity of the restriction maps ρ T : H 1 → H 1 T , and the obvious fact that 2.3 Lemma.Let (H i ) i∈I be a net in H 1 .Then the following two equivalences hold.
T for all T > 0. (ii)"⇐": Assume that, for each T > 0, there exists Since H 1 is compact and ρ T is continuous, we find HT ∈ LP(H i ) i∈I such that ρ T ( HT ) = H T .Again by compactness, there exists a limit point H ∈ LP( HT ) T >0 , say H = lim n→∞ Htn with some sequence t n → ∞.Then H ∈ LP(H i ) i∈I , and for each T > 0 we have For the last inclusion recall that H cs T is 1 -compact as a homeomorphic image of R ∪ {∞}, see [13, §2.3], and hence also weakly closed.Again referring to (2.1) we obtain H ∈ H cs .
We also need the Weyl coefficients for constant Hamiltonians with zero determinant, which can be found by an elementary calculation; see [5, Example 2.2(1)]4 .
2.4 Lemma.Let H as in (1.5) be a constant Hamiltonian such that h 2 3 = h 1 h 2 .Then

Reparameterisation
Reparameterisation is the equivalence relation on the set of all Hamiltonians defined as follows.
If H and H are related as in (2.2) and y is a solution of (1.1), then y • ϕ is a solution of (2.2) with H replaced by H. Similarly, the fundamental solutions satisfy W (x, z) = W (ϕ(x), z) and hence q H (z) = q H (z). (2.3) Moreover, the following obvious transformation rules hold: Based on the transformation rule for the trace, we see that every equivalence class of Hamiltonians modulo reparameterisation contains exactly one element that is defined on the interval (0, ∞) and is trace-normalised, i.e. whose trace is equal to 1 a.e.In fact, given a Hamiltonian H defined on some interval (a, b), we set t(t) := t a tr H(x) dx and use ϕ := t −1 .This function is admissible to make a reparameterisation, since tr H(t) > 0 a.e., and hence t −1 is absolutely continuous.
Based on the transformation rule of the primitive M , we see that the quotient in (1.10) transforms correspondingly.Let

Hamiltonians starting with a vanishing diagonal entry
If a Hamiltonian starts with an interval where a diagonal entry vanishes, then its Weyl coefficient has a simple, and extremal, asymptotics towards +i∞.
Let H be a Hamiltonian defined on some interval (a, b).Recall the following classical facts; see, e.g.[6].
⊲ Denote by (a, â) the maximal interval starting at a such that h 2 (t) = 0 for t ∈ (a, â) a.e., and assume that â > a. Then The leading order term is the term that is linear in z: The case â = b is formally included and corresponds to q H ≡ ∞.
⊲ Denote by (a, ǎ) the maximal interval starting at a such that h 1 (t) = 0 for t ∈ (a, ǎ) a.e., and assume that ǎ > a. Then .
Again the linear term gives the leading order asymptotics: .
The case ǎ = b is formally included and corresponds to q H ≡ 0.
Translated to the spectral measure, â > a means that it should include a "point mass at infinity", and ǎ > a means that it has finite total mass.In particular, the above relations show that, if â > a or ǎ > a, then

Representation of Hamiltonians by scalar functions
We study the representation of a Hamiltonian H by means of the functions σ H and ζ H , defined in (1.6) and (1.22) respectively, a bit more systematically.Denote by T the unit circle in the complex plane and, for 0 < T ≤ ∞, set where f ∼ g means that f and g coincide almost everywhere.As usual, we suppress explicit notation of equivalence classes.Moreover, we write a function f ∈ L(T ) generically as a pair f = (σ, ζ) with σ : (0, T ) → [0, 1] and ζ : (0, T ) → T.
In particular, for T < ∞, we may consider L(T ) topologised with the 1 -topology or the weak L 1 -topology.
From this representation we see that Γ is a left-inverse of Ξ: given H ∈ H 1 T , the matrices H and Γ[σ H , ζ H ] both have trace 1, their quotients of diagonal entries coincide, and the relative size and sign of their off-diagonal entries coincide.Thus indeed Furthermore, observe the following continuity property, which holds since L(T ) is uniformly bounded: for each T < ∞ we find a constant C > 0 such that, for all (σ (2.7) In particular, for each T < ∞, the map Γ :

Nets with constant limit points
In the proof of the implication (iii)⇒(i) in Theorems 1.1 and 1.4 we need the following fact about sequences in L 1 -spaces which have only constant limit points.We do not know an explicit reference to the literature, and hence give a complete proof.In the formulation we tacitly identify C with the µ-a.e.constant functions in L 1 (µ).
2.7 Proposition.Let µ be a finite positive measure on a set Ω with µ = 0, and let (f n ) n∈N be a sequence in L ∞ (µ) with sup n∈N f n ∞ < ∞.We consider (f n ) n∈N as a sequence in L 1 (µ).Then the following two statements are equivalent: (2.8) (2.9) If the equivalent conditions (2.8) and (2.9) hold, then

Relation (2.11) implies that lim
and hence also the limit in (2.9) along the subsequence (n k(l) ) l∈N is zero.Since we started with an arbitrary sequence (n k ) k∈N , the limit relation (2.9) follows.Now let f ∈ LP w (f n ) n∈N and choose a subsequence (f n k ) k∈N with f n k w → f .Then we find a further subsequence (f n k(l) ) l∈N and a constant g such that f n k(l) and this implies (2.10).
We come to the converse implication "(2.9)⇒(2.8)".Assume from now on that (2.9) holds.Moreover, since (2.9) is inherited by subsequences, it is enough to prove (2.8) for the sequence (f n ) n∈N itself.Further, let us set M := sup n∈N f n ∞ .
There exist a subsequence (n k ) k∈N and a ∈ R such that (2.12) Let ε > 0 be arbitrary and consider the compact, disjoint sets by assumption, (2.9) holds with these sets.Suppose that there exist a subsequence (k(l)) l∈N such that lim l→∞ µ f −1 which is a contradiction to (2.12).Therefore (2.9) implies that lim k→∞ µ f −1 n k (A) = 0, which is equivalent to lim In a similar way one shows that −→ b with some b ∈ R.This proves (2.8).
In the context of Hamiltonians on a finite interval, Proposition 2.7 implies the following fact.
2.8 Corollary.Let T < ∞ and (H n ) n∈N be a sequence in H 1 T , and denote by λ the Lebesgue measure on (0, T ).Assume that and hence H = Γ(1, ζ) and H ∈ LP 1 (H n ) n∈N .

Estimates for imaginary part and modulus of the Weyl coefficient
In this subsection we recall lower and upper estimates for Im q H and |q H | on the positive imaginary axis.This result is a special instance of [10, Theorem 1.1] with q = 1 4 and ϑ = π 2 there and is used, in particular, in the proof of the implication (i)⇒(ii) in Theorem 1.1; the estimates for the modulus are also used in the proof of the implication (iii)⇒(i).
2.9 Proposition.Let H be a Hamiltonian defined on the interval (0, ∞) such that (1.2) holds and neither h 1 nor h 2 vanishes a.e. on some neighbourhood of the left endpoint 0, and let m i be as in (1.5) and d(H, t) as in (2.5).For r > 0, let t(r) ∈ (0, ∞) be the unique number that satisfies (2.14) the inequalities hold for all r > 0.
Note that the mapping t → (m 1 m 2 )(t) is a strictly increasing bijection from (0, ∞) onto itself, and therefore t(r) is uniquely defined via (2.14).The mapping r → t(r) is a strictly decreasing bijection from (0, ∞) onto itself.It is the inverse of the function r(t) := 1 8 (m 1 m 2 )(t) .

A weighted rescaling transformation
In order to study the behaviour of q H towards i∞, we use a weighted rescaling transformation on the set of Hamiltonians.This is a variant of Y. Kasahara's rescaling trick invented in [7] for Krein strings, and also used in slightly different forms in [5,8,11,13].The main idea of the rescaling is to zoom into a neighbourhood of the left endpoint 0 when s in the following definition tends to 0.
2.10 Definition.Let g 1 , g 2 : (0, ∞) → (0, ∞) be continuous such that g 1 (s), g 2 (s) → ∞ as s → 0. Further, let T ∈ (0, ∞] and set g 3 (s) := g 1 (s)g 2 (s).For every s > 0 define the map A s : In the following we shall use two special choices of g 1 , g 2 , namely Situation 1: If, in addition, (1.2) holds, then q H (rz). (2.21) In the following lemma we prove an a priori estimate for the modulus of the Weyl coefficient of A s H at a particular point, which is used in the proof of Theorems 1.1 and 1.4.This property follows from the choice of g 1 , g 2 in (2.16) in the general case or from the assumption (1.14) in the additions to the main theorems.
2.12 Lemma.Let H ∈ H such that (1.2) holds, let g 1 , g 2 , g 3 be as in Definition 2.10, and assume that (2.16) or (2.17) is satisfied.Then Proof.If g 1 , g 2 are as in (2.16), the assertion is clear from (2.21) and Proposition 2.9.Assume that (2.17) holds.Set x s := t( 1 8 ) where t(r) is the unique number that satisfies (2.14) for A s H instead of H. Then This is equivalent to (m 1 m 2 )(sx s ) = s 2 .The latter relation implies that sx s → 0 as s → 0. Assumptions (1.13) and (1.14) yield m i (t) ≍ t, i = 1, 2 and hence We obtain from Proposition 2.9 that In the proof of Theorems 1.1 and 1.4 in Section 5 we also need the trace of the primitive of the rescaled Hamiltonian.Let g 1 , g 2 be as in Definition 2.10 and H ∈ H.For s > 0 set , the function τ s is strictly increasing.If, in addition, H is in the limit point case at ∞, then τ s is a bijection from (0, ∞) onto itself.Note that for the choice (2.16) we have τ s = t s .
3 Proof of "(i) ⇔ (ii)" in Theorems 1.1 and 1.4 We use the following fact which also plays a role later.
Proof of " (i)⇔(ii)" in Theorems 1.1 and 1.4.Let H be as in the formulation of the theorems.
➀ The implications "(i)⇒(ii)" in Theorems 1.1 and 1.4 are a direct consequence of [10, Theorem 1.1] in the form of Proposition 2.9 since this result implies for every r > 0. It remains to recall that t, defined via (2.14), is a strictly decreasing bijection from (0, ∞) onto itself.
➁ In this step we show that Let r n → ∞.Then we have the equivalences The first one holds because of (2.21) and Lemma 2.12, and the second by the maximum principle and compactness of H. Remembering that t is a decreasing bijection we obtain (3.7) and (3.8).
➂ To prove the implication "(ii)⇒(i)" in Theorem 1.1, assume that lim t→0 d(H, t) = 0.By Lemma 3.1 we have lim s→0 d(H s , T ) = 0 for all T > 0. Since tr H s = 1 a.e., it follows that also lim s→0 det M s (T ) = 0 for all T > 0 where M s is the primitive of H s .
➃ For "(ii)⇒(i)" in Theorem 1.4 assume that lim inf t→0 d(H, t) = 0. Then for each T > 0 we have lim inf s→0 d(H s , T ) = 0 and, arguing as above, obtain a limit point H T ∈ LP(A s H) s∈(0,1] for which the interval (0, T ) is indivisible.Let φ T ∈ [0, π) be the type of this indivisible interval.Choose a sequence (T n ) n∈N such that (φ Tn ) n∈N converges, say, φ Tn → φ.Then (H Tn ) n∈N converges to the Hamiltonian for which (0, ∞) is indivisible of type φ.Since LP(A s H) s∈(0,1] is closed, we can refer to (3.8) to finish the proof.

Bounds for the off-diagonal entries and the rotation
In this section we show that the relative size, σ H (t), of the off-diagonal entries of a Hamiltonian and its rotation, ζ H (t), can be estimated from above by d(H, t); recall that the latter is defined in (2.5).These estimates are used in the proof of the implication (ii)⇒(iii) in Theorems 1.1 and 1.4.
We start with an estimate for the off-diagonal entry.As usual, λ denotes the Lebesgue measure.

Proposition.
Let H ∈ H 1 , and assume that neither h 1 nor h 2 vanishes a.e. on some neighbourhood of the left endpoint 0. For each pair of closed, disjoint subsets A, B ⊆ T there exists a constant c(A, B) > 0, which is independent of H, such that Heading towards the proof of this proposition, we present two lemmata.The first one is an easy observation, which shows how information about the Hamiltonian H on an interval I ⊆ (0, ∞) can be used to estimate d(H, t).In these two lemmata we use the following notation, which extends the notation of the primitive to functions that may vanish on sets of positive measure: for a Hamiltonian H, I ⊆ (0, ∞) and t > 0, set H(s) ds.
Together with m i (H, t) ≤ t, which is a consequence of tr H = 1 a.e., we obtain The second lemma contains the crucial estimates.For α, β ∈ R with α ≤ β we denote the corresponding arc on T by 4.4 Lemma.The following estimates hold.
(i) Let φ 0 , ψ 0 satisfy 0 ≤ φ 0 < ψ 0 ≤ π and set Then, for all H and t > 0, we have (ii) Let α, β ∈ (0, π] and set Then, for all H and t > 0, we have The same holds for Then, for all H and t > 0, we have The same holds for ➀ We start with a general calculation.Let K 1 , K 2 ⊆ [0, t] be disjoint and set We can use the inequality |h 3 | ≤ √ h 1 h 2 and the Cauchy-Schwarz inequality in the last step to obtain det M (H1 Using once more |h 3 | ≤ √ h 1 h 2 and the Cauchy-Schwarz inequality we arrive at a complete square: For the rest of the proof set K i := I i ∩ (0, t) for i = 1, 2, and K := K 1 ∪ K 2 .We consider the three cases in (i), (ii), (iii) separately.
➂ Let us first consider the situation in item (i).It follows from (4.6) that This, together with Lemma 4.3 and (4.5), implies which is the asserted statement in (i).
➃ Next, we consider the situation in item (ii).Here h 3 is non-negative on I 1 and non-positive on I 2 , or vice versa.Thus, Lemma 4.3 and inequality (4.4) yield The implication "(iii ′ )⇒(i)" is a consequence of Corollary 2.8.
Thus, for every open arc For the lower half-plane we proceed analogously.Consider the function which is a differentiable homeomorphism from (−∞, 0) onto T ∩ C − such that open intervals in (0, ∞) correspond to open arcs in T ∩ C − , and that, for an interval I ⊆ (−∞, 0), we have As above one shows that, for an arc Now we combine the mappings φ + and φ − ; let φ : R \ {0} → T \ R be defined by φ| (0,∞) = φ + and φ| (−∞,0) = φ − .The above considerations show that (iii ′ ) is equivalent to the following condition (iii ′′′ ).
⊲ Finally, assume that (1.13) and (1.14) in the addition to Theorem 1.1 hold.Let us choose g 1 , g 2 as in (2.17).Then τ s (t) = 1 s m 1 (st) + m 2 (st) = t by (1.13) and hence σ Hs (t) = σ H (st).For fixed T, s ∈ (0, ∞) and γ ∈ [0, 1) we have Hence, for fixed γ ∈ [0, 1), the following equivalences hold: In a similar way one shows that (5.12) is true for every T ∈ (0, ∞) if and only if (1.16) holds.This establishes the equivalence of (iii ′′′ ) and (iv) and finishes the proof of Theorems 1.1 and 1.4 and their additions.with ρ, β 1 , β 2 ∈ R, where higher iterates of logarithms can be added.In the theorem below we show that a Hamiltonian with regularly varying diagonal primitives is well behaved in the sense that d(H, t) 1 unless its diagonal entries are of the same size on the power scale, i.e. their indices coincide.This is closely related to our forthcoming paper [11], where we investigate Hamiltonians whose Weyl coefficients have regularly varying asymptotics towards +i∞.
6.1 Theorem.Let H be a Hamiltonian defined on the interval (0, ∞) and assume that neither h 1 nor h 2 vanishes a.e. on some neighbourhood of the left endpoint 0. Assume that m 1 and m 2 are regularly varying at 0 with positive indices ρ 1 and ρ 2 respectively.Then Proof.Let (A s H) s>0 be the family of rescaled Hamiltonians as in Definition 2.10 with g 1 , g 2 from (2.16), and let (H s ) s>0 be the corresponding trace-normalised family as in (3.1).
➀ In the first step of the proof we show that every accumulation point of (H s ) s>0 , for s → 0, is of a special form.It follows from (2.18) that t 0 where [C] ii denotes the ith entry on the diagonal of a matrix C, and hence where t s and τ s are defined in (1.8) and (2.22) respectively.Set t(t) := t ρ1 + t ρ2 for t ∈ (0, ∞).
The assumptions about m 1 and m 2 and the Uniform Convergence Theorem for regularly varying functions (see, e.g.[1, Theorem 1.5.2])imply that lim s→0 t s (t) = t(t) locally uniformly for t ∈ (0, ∞).The functions t s and t are continuous and increasing bijections from (0, ∞) onto itself, and it follows that also lim s→0 t −1 s (T ) = t −1 (T ) for all T ∈ (0, ∞).Let s n → 0 be a sequence such that the limit H := lim n→∞ H sn exists, and let H be the reparameterisation defined by H := ( H •t)•t ′ .Using (6.1) we find, for T ∈ (0, ∞) and i ∈ {1, 2}, where the off-diagonal entries are unknown.
➁ For Hamiltonians H of the form (6. from which we find that, for all t > 0, ➂ We make a limiting argument to complete the proof.Let (t n ) ∞ n=1 be a sequence of positive numbers with t n → 0. Fix T > 0 and let again u(s) be the function in (3.5).For large enough n, choose s n → 0 such that u(s n ) = t n , and extract a subsequence (s n(k) ) k∈N such that the limit H := lim k→∞ H s n(k) exists.Using (2.19), (2.6) and (6.Since the (t n ) was arbitrary, the claim follows.
As a consequence, if ρ 1 = ρ 2 in Theorem 6.1, then (ii) in Theorem 1.4 is not satisfied and hence neither is (i) (under the assumption that (1.2) holds), i.e. one has lim inf y→∞ Im qH (iy) |qH (iy)| > 0. If, on the other hand, the diagonal entries themselves (and not just their primitives) are regularly varying with the same index, then the situation is different.

1. 5
Addition to Theorem 1.4.Assume that, in addition to the assumptions of Theorem 1.4, relations (1.13) and (1.14) hold.Then the equivalent properties (i), (ii), (iii) in Theorem 1.4 are further equivalent to the following condition.

2. 5
Definition.Two Hamiltonians H and H, defined on respective intervals [a, b) and [â, b), are called reparameterisations of each other if there exists a function ϕ : [â, b) → [a, b) that is strictly increasing, bijective and absolutely continuous with absolutely continuous inverse such that H the class of constant, singular, trace-normalised Hamiltonians can be represented as follows: H cs T = {Γ(1, ζ) : ζ ∈ T} where we identify the constant (1, ζ) with the constant function in L(T ).

1
(s n t −1 sn (T )) m 1 (s n ) = t −1 (T ) ρ1 ,again by the Uniform Convergence Theorem.Hence H is of the form

ρ 1 ρ 2 1 2 (ρ 1 + ρ 2 ) 2 .
3) we obtaind(H, t n(k) ) = d H, u(s n(k) ) = d A s n(k) H, t −1 s n(k) (T ) = d(H s n(k) , T ) k→∞ −→ d( H, T ) ≥ 1 − √ 1.3 Remark.Let us first discuss the simpler conditions (1.15) and(1.16).The role of σ H is to quantify the relative size of the off-diagonal entries of H compared with the diagonal entries.Condition (1.15) can be written as lim t→0 We use the following notation for Hamiltonians on a finite or infinite interval.
2.1 Definition.Let T ∈ (0, ∞].(i) H T is the set of all measurable functions H : (0, T ) → R 2×2 (up to equality a.e.) such that H(t) ≥ 0 and tr H(t) > 0 a.e.; (ii) H 1 T is the set of all H ∈ H T such that tr H(t) = 1 a.e.; (iii) H cs T is the set of all H ∈ H 1 T that are constant and satisfy det H(t) = 0 a.e.If T = ∞, we often drop T from the notation and just write H, H 1 and H cs instead of H ∞ , H 1 ∞ and H cs ∞ respectively.
together with (2.13), implies that Re f n k → a in measure.Since Re f n k is uniformly in- The condition (i) says that σ Hn → 1 in measure.Since |σ Hn (t)| ≤ 1 for a.e.t, σ Hn tends to 1 also w.r.t.Hn k ) k∈N .This provides us with a constant ζ ∈ T and a further subsequence (ζ Hn k(l) ) l∈N such that ζ Hn k(l) → ζ w.r.t.
1 .Consider a subsequence (H n k ) k∈N of (H n ) n∈N that converges weakly to some H ∈ H 1 T .By (ii), we can apply Proposition 2.7 to the sequence (ζ Let g 1 , g 2 be as in Definition 2.10 and H ∈ H. Then 16) or Situation 2: g 1 (s) = g 2 (s) = 1 s and H satisfies (1.13) and (1.14); (2.17) in both cases g 1 , g 2 satisfy the assumptions in Definition 2.10.The functions in (2.16) are used in the proof of Theorems 1.1 and 1.4; the functions in (2.17) are used in the proof of the additions of these theorems.In the following lemma we collect how the quantities defined in (1.6)-(1.8),(1.17) and (2.5) are transformed.2.11 Lemma.