Perturbation theory for almost-periodic potentials I. One-dimensional case

We consider the family of operators $H^{(\epsilon)}:=-\frac{d^2}{dx^2}+\epsilon V$ in ${\mathbb R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(\epsilon)};\lambda)$ when $\epsilon\to 0$ and $\lambda$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $\lambda$ the IDS has a complete asymptotic expansion in powers of $\epsilon$; these powers are either integer, or in some special cases half-integer. These results are new even for periodic $V$. We also prove that when the potential is neither periodic nor quasi-periodic, there is an exceptional set $\mathcal S$ of energies (which we call $\hbox{the super-resonance set}$) such that for any $\sqrt\lambda\not\in\mathcal S$ there is a complete power asymptotic expansion of IDS, and when $\sqrt\lambda\in\mathcal S$, then even two-terms power asymptotic expansion does not exist. We also show that the super-resonant set $\mathcal S$ is uncountable, but has measure zero. Finally, we prove that the length of any spectral gap of $H^{(\epsilon)}$ has a complete asymptotic expansion in natural powers of $\epsilon$ when $\epsilon\to 0$.


Introduction
We consider the operator where ε > 0 is a small parameter and V is a real-valued almost-periodic potential. We are interested in various quantitative and qualitative spectral properties of H as ε → 0, and this paper is the first one in a series of articles devoted to the study of these properties of H under various assumptions. In this paper we assume that the dimension d = 1, so that (1.2) H = H (ε) := − d 2 dx 2 + εV. The quantities we will be interested in are: the length of the spectral gaps, and the behaviour of the integrated density of states (IDS) N (λ; H (ε) ) when the spectral variable λ is fixed (and ε → 0). It has been noticed by Arnold, [1] that if H is a Hill operator (1.2) with V being a finite trigonometric periodic polynomial 3) is infinite, then the size of all gaps is (generically) proportional to ε. Our first result is the extension of this observation to the case of almost-periodic potentials; moreover, we prove that the length of each spectral gap has a complete asymptotic expansion in natural powers of ε. We also prove similar expansions for the upper and lower ends of each spectral gap. The leading power in each expansion will depend on whether the potential is a finite or infinite linear combination of trigonometric functions (we call such operators quasi-periodic and almost-periodic respectfully). In the quasi-periodic case the leading power of the length of the gap opened around the square of each frequency θ will increase together with the order of θ (see the next section for the precise definitions and formulation of the results), whereas in the almost-periodic setting when no Fourier coefficients vanish, all expansions begin with the first power of ε. These expansions are formally uniform, but effectively they are not, because the higher the order of a frequency θ is, the smaller ε we need to choose to 'see' the expansion of the length of the gap generated by θ (i.e. if we choose ε not very small, then the remainder in the expansion will be larger than the asymptotic terms). Somewhat similar problems were considered in [2] and, in the discrete setting, in [3] (see also [4] and references there). However, there is a significant difference between these papers and our results. In these papers the authors have fixed ε and studied the behaviour of the gap length as a function of the 'natural label' of the gap (corresponding, roughly, to what we call the order of a frequency, see below for details). So, they were able to obtain information about all gaps simultaneously, but this information was either bounds (upper and lower), or one asymptotic term, whereas we obtain more detailed information (complete asymptotic expansion) about smaller number of gaps.
The second problem we consider is as follows. Let λ ∈ R be a fixed number and consider the behaviour of the IDS of H (ε) at λ when ε → 0. Questions of this nature (how the value of IDS at a fixed energy depends on the value of a small coupling constant) have arisen in our study of perturbations of Landau Hamiltonians by almost-periodic potentials. Despite the slightly esoteric feel of this type of questions, we believe they are more natural than it may seem at the first sight, especially given that the answers are quite surprising. Let us briefly describe the effects happening in one dimension; we are going to devote the second paper in this series to discuss the multidimensional case, where the results are even more unexpected. Suppose first that V is quasi-periodic. Then, whenever λ is not a square of a frequency, there is a complete asymptotic expansion of N (λ; H (ε) ) in integer powers of ε. Suppose, λ = θ 2 = 0, where θ is a frequency. Then the type of the expansion will depend on the relationship between τ (the constant Fourier coefficient of V ) and ν (the Fourier coefficient at e i2θx ). First we notice that, as we will show in this paper, there is a spectral gap of H (ε) around θ 2 of length ∼ 2νε. Therefore, if |τ | < |ν|, then the point λ + τ ε stays inside this gap and, as a result, the IDS does not depend on ε when ε is small. If, on the other hand, |τ | > |ν|, then the shift by τ ε pushes our point λ well outside the spectral gap, and we obtain the standard asymptotic expansion in integer powers of ε. The most interesting case is |τ | = |ν|, when the point λ + τ ε is approximately at the edge of the spectral gap. In this case generically the answer will depend on the sign of τ . For one value of this sign the point λ + τ ε is still located in the gap and so the IDS is constant. However, for the opposite value of the sign of τ the point λ + τ ε will be pushed just outside the gap and, as a result, the IDS will have a complete expansion in half-integer powers of ε (where we define half-integers as (Z/2) \ Z). Similar situation happens when we look at the point λ = 0: we have expansion in half-integers whenever τ < 0; otherwise, the expansion is in integers. The bottom line is, if V is quasi-periodic, then for all λ we have a complete asymptotic expansion of N (λ; H (ε) ) as ε → 0, which contains either integer, or half-integer powers of ε.
An interesting phenomenon occurs when we look at this problem in the almost-periodic setting, for example, when all the Fourier coefficients are non-zero. Namely, in this case there is a substantial set S such that for λ 1/2 ∈ S there is no asymptotic expansion of N (λ; H (ε) ) at all; in fact, there are uncountably many values of λ for which the remainder N (λ; H (ε) ) − N (λ; H (0) ) is not even asymptotically equivalent to any power of ε. This set (which we call the super-resonance set) is uncountable, but has measure zero; the interesting feature of this set is that it is present no matter how quickly the Fourier coefficients of V go to zero -whether V is smooth, or analytic, the super-resonant set without the asymptotic expansion of IDS is always uncountable (but perhaps its dimension may depend on the smoothness of V ).
The method we use for obtaining these results is a version of the gauge transform method used in [7] and [6]. The only difference is that in [7] and [6] we had fixed coupling constant and assumed that the energy λ was large (so that the small parameter was λ −1 ), whereas in the present paper the small parameter is the coupling constant ε. This difference is not essential, so the construction of the gauge transform can be performed almost word-to-word as it is done in [7] and [6]. This method allows us to find two operators, H 1 and H 2 so that H 1 is unitarily equivalent to H, H 2 is close to H 1 in norm, and H 2 is almost diagonal (in the sense that most of the off-diagonal matrix coefficients of H 2 vanish). For the sake of completeness, we have written the details of the gauge transform construction relevant to our setting in the Appendix; in the main body of the paper we will give a brief description of the method and use the relevant properties of H 1 and H 2 without proof.
The structure of the rest of the paper is as follows: in the next section we will give all the necessary definitions and formulate the main results. In Section 3 we will discuss the quasi-periodic operators, and in Section 4 the almost-periodic operators. Finally, in the Appendix we will describe the method of the gauge transform. Here, V θ are complex numbers (called the Fourier coefficients of V ; since V is real, we havê V −θ =V θ ), and Θ = Θ(V ) ⊂ R d is a finite set, called the set of frequencies (or rather half-frequencies; the factor 2 is used purely for convenience) of V . We assume without loss of generality that Θ is symmetric about the origin and contains it. Denote by l the number of independent elements in Θ (so that |Θ| = 2l + 1). For each natural L we denote Θ L := Θ + Θ + · · · + Θ (the algebraic sum of L copies of Θ) and put Θ ∞ := ∪ L Θ L . When θ ∈ Θ ∞ , we denote by Z(θ) the smallest number L for which θ ∈ Θ L and call this number the order of the frequency θ. We put A simple combinatorial estimate shows that We also put τ :=V 0 , Θ := Θ \ {0} and V := V − τ , so that The second type of potentials we are going to consider are smooth almost-periodic, by which we mean that Θ is still a finite set, but we have for θ ∈ T m and arbitrary positive P . We also assume that Θ satisfies the diophantine condition, i.e. for θ ∈ Θ m we have |θ| m −P 0 , where P 0 > 0 is fixed. In either of these two cases (quasi-or almost-periodic potentials) we also assume (as we can do without loss of generality) that Our first main result concerns the spectral gaps.
Theorem 2.1. Suppose, V is either quasi-periodic, or infinitely smooth almost-periodic and satisfies all the above assumptions. Suppose, θ ∈ Θ ∞ . Then for sufficiently small ε operator H has a (possibly trivial) spectral gap around |θ| 2 , the length of which, as well as its upper and lower ends, have complete asymptotic expansions in natural powers of ε. IfV θ = 0 then the asymptotic expansion for the upper (lower) end of the gap starts with |θ| 2 ± |V θ |ε + O(ε 2 ).
Remark 2.2. IfV θ = 0, we cannot guarantee that an expansion for the gap-length is always non-trivial, i.e. it could happen, in principle, that the length of the gap is O(ε +∞ ).
The next result involves two quantities, s 2 (0) and g 2 (0) which will be defined in the next section (in formula (3.24)). Throughout the paper we use the convention that each time we use letters a j (or a j (λ)) for coefficients in asymptotic expansions, the exact values of these coefficients could be different. The same refers to the use of C which can mean a different positive constant each time we use it. Theorem 2.3. Suppose, V is quasi-periodic. Then for sufficiently small ε > 0 the following holds: (i) For λ < 0 we have N (λ; H) = 0.
Theorem 2.4. Suppose, V is infinitely smooth almost-periodic, but not periodic, and V θ = 0 for any θ ∈ Θ ∞ . Then there exists a set S (which we call a super-resonance set) such that a complete power asymptotic expansion of N (λ; H) exists if and only if λ ∈ S. The set S is uncountable and has measure zero.
Remark 2.5. As we will see in the proof, there are uncountably many values of λ for which the difference N (λ; H (ε) ) − N (λ; H (0) ) properly oscillates between C 1 ε j and C 2 ε j , where C 1 = C 2 and j equals 1 or 2.
We will think of a point ξ ∈ R as the exponential function e ξ (x) := e iξx lying in the Besikovich space B 2 (R) (the collection of all formal countable linear combinations of {e ξ } with square-summable coefficients). Then for arbitrary pseudo-differential operator W with symbol (in a left quantisation) w = w(ξ, x) being quasi-periodic in x, Thus, we can think of the Fourier coefficientsŵ(θ, ξ) of the symbol as the matrix element of W joining ξ and ξ + 2θ: (2.22)ŵ(ξ, θ) = W e ξ , e ξ+2θ B 2 (R) .
In our paper [6] it is explained that instead of working with operators acting in L 2 (R), we can consider operators with the same symbol acting in B 2 (R) and work with them. This will not change the spectral properties we are studying in our paper (for example, the spectrum as a set is the same whether our operator acts in L 2 (R) or B 2 (R)).

Quasi-periodic potential
In this section we assume that the potential V is quasi-periodic, i.e. that (2.1) holds.
3.1. Gauge transform: general description. First of all, we give a brief outline of the construction of the gauge transform of our operator. The details of this construction are similar to those in [6]; for the sake of completeness, we present them in the Appendix. Let us fix a natural number N . All the constructions are going to depend on the choice of N , but we will often omit writing N as the variable. Applying the gauge transform leads to a pair of operators, H 1 = H and H 2 is almost diagonal in the sense that it can be decomposed into a direct integral with all fibres being finite dimensional (moreover, as we will see, the dimension of all fibres will be 1 or 2). Also, the frequencies of H 2 are inside the set Θ 3N . Here, the coefficient 3 technically appears in the gauge transform approach (see Appendix). It reflects the fact that one has to make slightly more than N steps to achieve the error of order ε N . Once we have constructed these operators, it turns out that we can study spectral characteristics of H by means of studying the corresponding spectral characteristics of H 2 . Indeed, the spectra of H and H 1 are the same, and so are the lengths of the spectral gaps. Also, the lengths of the spectral gaps of H 1 and H 2 differ by at most ε N . Concerning the IDS, it was proved in [6] that More precisely, we have shown in [6] that the immediate consequence of (3.1) is 2 ). We also define and notice the obvious property This trivial consideration is important for understanding of some of the effects described later. Now we choose a small positive number δ = δ(N ), to be specified later and for each non-zero frequency θ ∈ Θ (H 2 ) we put ).
Next, let ψ = ψ(ξ) be a standard smooth non-negative cut-off function satisfying supp ψ ⊂ [−1/2, 1/2] and ψ(ξ) = 1 for ξ ∈ [−1/4, 1/4], and let ϕ := 1 − ψ. We put Note that We also putχ The region R(θ) is called the resonance zone corresponding to θ. Since (for fixed N ) the number of resonance zones is finite and the length of them goes to zero, it implies that for sufficiently small δ these zones do not intersect. We also denote by the 'overall' resonant set corresponding to ε; we obviously have In what follows we always assume that δ(N ) is sufficiently small so that different resonance zones R(θ; δ) do not intersect for all θ ∈ Θ 9N ; we also take ε so small that ε ≤ δ 2 .
Remark 3.1. It is not difficult to see that in case when Θ satisfies Diophantine condition on frequencies, the parameter δ(N ) can be chosen to be c N with some constant c = c(Θ) with all constructions and statements of Section 3 being valid.
The important property of the operator H 2 established in the appendix is as follows: the Fourier coefficientsĥ 2 (ξ; θ) satisfy This property implies that if a point ξ lies outside all the resonance zones, then the onedimensional subspace spanned by the corresponding e ξ is invariant with respect to H 2 . If, on the other hand, for some (unique) θ we have ξ ∈ R(θ), then the two-dimensional subspace spanned by e ξ and e ξ+2θ is invariant with respect to H 2 .
The most important property of G is the following one: we have where we have denoted {ξ, G(ξ) ≤ λ} =: Ω λ = Ω λ (G). This property was proved in [6] and it immediately implies that the spectrum of H 2 is Equation (3.23) shows that in order to study the spectrum of H 2 , we need to look at the range of G. Our discussions above and Figure 1 imply the following statement: Later, we will obtain more precise information on the location and the length of the gaps.
The characteristic polynomial of the matrix M (ξ) − µ is (3.25) and the eigenvalues of M (ξ) are 3.3. Spectral gaps. Let us find the size of the spectral gap around It is easy to see and will be even clearer in what follows that all the objects we are interested in require detailed information only from the interior of the resonant zones. In particular, maximum value of σ − and minimal value of σ + are attained inside the interval [− δ 100|θ 0 | , δ 100|θ 0 | ] (assuming of course, ε is small enough). This allows us to ignore cut-off functions ϕ θ introduced above as they are equal to zero in the region of interest.
3.4. Integrated Density of States. Now let us discuss the IDS of H 2 . Formula (3.22) implies that in order to study the integrated density of states, we need to solve the equation In the unperturbed case (when G(ξ) = ξ 2 ) this equation has two solutions whenever λ > 0. After the perturbation, this equation may have no solutions (when λ is inside a spectral gap), or it may have one solution (when λ is exactly at the spectral edge of H 2 ). As we will see later, in other cases equation (3.31) has exactly two solutions. If λ is negative, the above constructions imply that N (λ; H) = 0 for sufficiently small ε. Suppose now that λ is positive and √ λ ∈ Θ 3N (in particular, λ = 0). Then, for sufficiently small δ, points (both of them) ξ with ξ 2 = λ do not belong to any resonance region; the same is true for points of the form λ − ετ . This, together with (3.19), implies that the equation G(ξ) = λ has two solutions (recall that we use convention of not distinguishing two solutions that are within distance O(ε N ) from each other), call them G −1 (λ) > 0 and −G −1 (λ). Monotonicity of G implies that (again for sufficiently small δ) the following holds: whenever 0 < η < G −1 (λ), we have G(η) < λ, and whenever η > G −1 (λ), we have G(η) > λ.
The last case we have to consider is λ = 0. The only points ξ where there is a chance that G(ξ) is negative are located in a (1 + |τ |) 1/2 ε 1/2 -neighbourhood of the origin and are not located in any resonance zone. Therefore, we have Now the simple use of the Implicit Function Theorem immediately gives the answer. If τ > 0, then N (0; H 2 ) = 0 for small ε. If τ < 0, then Finally, if τ = 0, then we have to note that formula (3.17) implies that for small ξ and non-trivial V we have f 2 (ξ; 0) < 0 and, therefore, All the asymptotic formulas for N (λ; H 2 ) obtained above together with equations (3.2) and (3.3) lead to Theorem 2.3. Again, it is easy to see that the corresponding expansions are independent of the particular choice of the cut-off function ϕ.

Almost-periodic potential
Let us discuss the situation when the potential is not quasi-periodic, but smooth almost-periodic, i.e. Θ is still a finite set, but we have We also assume that Θ satisfies the diophantine condition, i.e. for θ ∈ Θ m we have |θ| m −P 0 , where P 0 > 0 is fixed.
Remark 4.1. We can relax the diophantine properties of the frequencies if we assume a faster decay of the Fourier coefficients: the only condition that we effectively need is that the resonance zones do not intersect, see (4.13).
The way we perform the gauge transform is, essentially, the same as in the quasiperiodic case, with one important difference: we cannot afford to have infinitely many resonance zones, therefore, before transforming the operator H to H 1 and H 2 as above, we need to turn H to a quasi-periodic operator by truncating the potential V . The level of the truncation depends on the size of ε -the smaller ε, the more frequencies (and resonance zones) we need to keep. Thus, the number of resonance zones will be finite for each fixed ε, but, as opposed to the quasi-periodic situation, will increase as ε goes to zero. More specifically, let us assume first that 0 < ε < ε 0 , where ε 0 is a positive number, to be chosen later. We put ε n := 2 −n ε 0 and I n := [ εn 4 , ε n ]. The gauge transform construction will be performed separately for each I n and the asymptotic expansions we will obtain will hold only for ε ∈ I n . In order to 'glue' these expansions together at the end, we will use the following lemma:  Here, a j;n are some coefficients depending on j and n (and M ) satisfying ] + 1 such that for all ε, 0 < ε < ε 0 we have: This Lemma (in slightly different form) is proved in Section 3 of [6]; see also [5]. In order to apply it, we have to establish (4.4)-(4.5). Whenever we will be using this lemma, it will be rather straightforward to check estimates (4.5) for the coefficients from the constructions, so in what follows we will concentrate on establishing (4.4).

Remark 4.3.
Note that (4.4) is not a 'proper' asymptotic formula, since the coefficients a j;n are allowed to grow with n. Now, we will describe the construction in more detail. Let us fix a natural number N (which signifies that our errors are going to be O(ε N )) and suppose that ε ∈ I n . All the constructions below depend on the choice of (n, N ), but we will often omit writing n and N as the variables. Recall that for each θ ∈ Θ ∞ we define Z(θ) := m for θ ∈ T m . We also fix the smoothness P of the potential so that (4.7) |V θ | Z(θ) −P ; this (large) P depends on P 0 and N and will be chosen later. For each natural L we define the truncated potential Estimate (2.4) implies assuming of course that P is sufficiently large. Now, we chooseL =L(n, N ) so large that the norm of the operator of multiplication by V − V L is smaller than ε N n . The previous estimate shows that it is enough to take (4.10)L(n; N ) := ε − 2N P n to achieve this. Then we run 3N steps of the gauge transform as described in the appendix, but for the operatorĤL := H + εVL. The main difference with the gauge transform procedure for the previous section is that now the width of each resonant zone decreases as n increases. More precisely, we put Then the frequencies of the resulting operator H 2 will be inside the set (ΘL) 3N = Θ 3NL . Note that the resonant zones obtained at each step do not intersect. Indeed, suppose that θ 1 , θ 2 ∈ Θ 3NL , θ 1 = θ 2 . Then θ 2 − θ 1 ∈ Θ 6NL and, therefore, our diophantine condition implies (4.13) n for sufficiently small ε n , assuming that P is chosen so large that (4.14) 3N P 0 P < 1/8. At the same time the length of the resonant zone corresponding to θ ∈ Θ 3NL is bounded from above by n . Remark 4.4. Of course, condition (4.14) means that the bigger N is (i.e. the more asymptotic terms we want to obtain), the bigger P we should take (i.e. the smoother potentials we have to consider).
This construction leads to two operators, H 1 and H 2 with the same properties as described in the previous section. For each θ ∈ Θ 3NL we denote by R(θ) = R(θ; n) the resonant zone -the interval centred at −θ of length ε 1/2 n 2|θ| . We also denote (4.15) R(ε n ) = R n := ∪ θ∈Θ 3NL R(θ; n); this is the resonant zone corresponding to I n . The meaning of this set is that the symbol h 2 of H 2 is diagonal for ξ ∈ R n . This means that all Fourier coefficientsĥ 2 (ξ; θ) = 0 whenever θ = 0 and ξ ∈ R n ; our construction implies that even more is true:ĥ 2 (ξ; θ) = 0 unless ξ ∈ R(θ; n). The main difference between the almost-periodic and quasi-periodic cases is the following: in the quasi-periodic case the resonant set was fixed for any given N as δ(N ) and decreasing as N grows (see (3.13)), whereas in the almost-periodic case R(ε n ) is fixed only when ε ∈ I n , and in general it is no longer true that R n+1 ⊂ R n (since the smaller ε n leads to bigger n and biggerL(n) given by (4.10) and, thus, R n+1 consists of a bigger number of smaller zones than R n ). Estimate (2.4) implies that the number of elements in Θ 3NL can be estimated by which implies (4.17) meas(R n ) < ε 1/6 n if we choose P large enough.
Let us now discuss the behaviour of the gaps of H 2 (and, therefore, of H). This can be done using the arguments from the quasi-periodic case. When ε ∈ I n , the operator H 2 has gaps around points |θ| 2 , θ ∈ Θ 3NL(n) , and the length of each such gap has asymptotic expansion in natural powers of ε, according to Theorem 3.6. Now we notice that if θ ∈ Θ 3NL(n) , then θ ∈ Θ 3NL(m) for any m ≥ n and, therefore, there is a gap of H 2 around θ for any m ≥ n. The length of this gap has an asymptotic expansion given by Theorem 3.6 for ε ∈ I m , m ≥ n (Here we assume that ε 0 is chosen to be small enough, depending only on N ). These expansions may be different in general, but we can use Lemma 4.2 to deduce that we have a complete power asymptotic expansion of the length of a gap valid for all ε < ε 0 . Thus, we obtain Theorem 2.1 for smooth almost-periodic case. Now we discuss the asymptotic behaviour of the IDS. Recall that all our constructions are made for fixed N ; sometimes, we will be emphasising this and make N an argument of the objects we consider. First, we introduce the set of ξ > 0 such that ξ ∈ Θ ∞ and there is an infinite sequence n j → ∞ and θ j ∈ Θ L(n j ) satisfying ξ ∈ R(θ j ; n j ). We denote this set byS 1 (N ). Since we have ∞ n=p meas(R n ) → 0 as p → ∞, the measure of S 1 (N ) is zero. Also, it is easy to see that the set ∩ n R n (N ) is the Cantor-type set (i.e. a perfect set with empty interior) and is, thus, uncountable (unless V is periodic and Θ ∞ is therefore discrete). Since, obviously, ∩ n R n (N ) ⊂ (S 1 (N ) ∪ Θ ∞ ) and Θ ∞ is countable, this implies that the setS 1 (N ) is uncountable. We also haveS 1 (N ) ⊂S 1 (Ñ ) for N <Ñ . Finally, we introduce S 1 := ∪ NS1 (N ) -global uncountable set of Lebesgue measure zero.
Let us assume at the moment that τ = 0. For each fixed λ > 0 there are the following three possibilities: 1. Let √ λ ∈ Θ ∞ . Then √ λ = |θ| ∈ R(−|θ|; n) for all sufficiently large n and we therefore can repeat the procedure from the previous Section to obtain the resonance asymptotic 'expansion' (3.57) (see also Lemma 4.2). 2 Thus, for all sufficiently large n we have λ 1/2 ∈ R n . Then again we can repeat the (non-resonant) procedure from the previous Section which, together with Lemma 4.2, guarantees the existence of the complete asymptotic expansion (3.34).
3. Let √ λ ∈ S 1 . This is the most interesting case. As we will see below, in general there is a big part of S 1 where no power asymptotic expansion exists. Let us make a pause for a moment and summarize what we have done so far. We have proved the following statement: Theorem 4.6. Suppose, V is smooth almost-periodic with the constant Fourier coefficient τ = 0. Then there exists a set S 1 such that for λ 1/2 ∈ R + \ (S 1 ∪ Θ ∞ ), we have a complete expansion of the form (3.34), whereas when λ ∈ Θ ∞ , we have (3.57). The set S 1 is uncountable and has measure zero.
Suppose now τ = 0. Let us denote by R (θ; n) the interval centred at −θ, but of twice larger length than R(θ; n); obviously, R(θ; n) ⊂ R (θ; n). We also denote byS 2 (N ) the set of points ξ ∈ Θ ∞ for which there is an infinite sequence n j → ∞ and θ j ∈ Θ L(n j ) such that ξ ∈ R (θ j ; n j ). We put S 2 = S 2 (τ ) := ∪ NS2 (N ). Then S 1 ⊂ S 2 , meas(S 2 ) = 0, and for √ λ ∈ S 2 we still have the complete asymptotic expansion. Indeed, if ε ∈ I n j and √ λ ∈ R (θ j ; n j ), then √ λ + τ ε ∈ R(θ j ; n j ) for sufficiently large n. This proves the following statement: Theorem 4.7. The statements of the previous theorem hold for any τ = 0 with the set S 1 replaced by a different uncountable zero measure set S 2 = S 2 (τ ). Now we will prove the opposite -that there is a substantial set S such that for √ λ ∈ S there is no asymptotic expansion in powers of ε for N (λ; H). Obviously, the measure of S has to be zero, but we will show that it is uncountable. However, as we have seen in the previous Section, such a set must be empty in the quasi-periodic case. This means that we need to make a further assumption on the potential. Namely, we will assume that V is not periodic (i.e., Θ ∞ is dense) andV θ = 0 for any θ ∈ Θ ∞ . Remark 4.8. We can replace the last condition by requiring that there are infinitely many non-zero Fourier coefficients located in 'strategically important' places.
We again start with the case τ = 0. The strategy of the proof will be as follows. First, we will make a natural attempt to construct a set S such that for √ λ ∈ S there is no asymptotic expansion in powers of ε of N (λ; H). This attempt will almost work, but not quite. Then we will see what the problem with our first attempt is and will modify it correspondingly.
So, we define R (θ; n) = (−θ − δ n (θ), −θ + δ n (θ)) as the interval centred at −θ of halflength δ n and at our first attempt we define δ n (θ) = ε n |V θ |(100|θ|) −1 ; obviously, then R (θ; n) ⊂ R(θ; n) for large n. Note that our constructions guarantee that if ξ ∈ R (θ; n) and ε ∈ I n , then |ξ| 2 is well inside the spectral gap of H 2 (n) (this is the operator H 2 , when we want to emphasize that we have performed the gauge transform for ε ∈ I n ). Now we consider the setS 3 (N ) of all λ for which the following two conditions are satisfied: a. There is an infinite sequence n j → ∞ and θ j ∈ Θ 3NL(n j ) such that λ 1/2 ∈ R (θ j ; n j ), and b. There is an infinite sequence n j → ∞ such that λ 1/2 ∈ R(n j ). A simple argument based on the fact that Θ ∞ is dense in R implies thatS 3 (N ) is uncountable.
Suppose, √ λ ∈S 3 (N ). Then, if ε ∈ I n j , the point λ is in the spectral gap of H 2 (n j ) and, therefore, we have the (trivial) resonant asymptotic expansion (3.49). On the other hand, if ε ∈ I n j , we have the non-resonant asymptotic expansion (3.33). It is very tempting to stop the proof here by stating that these two expansions are different. However, we cannot quite guarantee this -it may well happen that all the coefficients in the non-resonant expansion (3.33) turn to zero. One way of overcoming this is to show that for generic set of Fourier coefficients of V these coefficients are bounded away from zero. We, however, will assume a different strategy and reduce the setS 3 (N ) even further (by choosing smaller values of the parameters δ n (θ)).
Before doing this, let us see what happens with the position of the point ξ ∈S 3 (N ) related to different resonant zones as n changes. When n = n j , our point ξ is inside the resonant zone R (θ j ; n j ) and, therefore, we have a trivial expansion for ε ∈ I n j . If we consider values n bigger than n j , then ξ may stay inside R(θ j ; n) for a while, but since ∩ n R(θ j ; n) = |θ j | = ξ, for sufficiently large n our point ξ will get outside of the resonant zone R(θ j ; n); let us denote byk j the index when this happens (i.e.k j is smallest value of n > n j for which we have ξ ∈ R(θ j ; n)). Similarly, let k j be the biggest value of n < n j for which we have ξ ∈ R(θ j ; n). Since the width of a resonance zone shrinks by a factor √ 2 at each step, Remark 4.5 implies that ξ cannot 'enter' a different resonance zone immediately after 'leaving' R(θ j ; n), i.e. ξ ∈ (R(k j ) ∪ R(k j )). Then by our construction we have N asymptotic terms of N (λ; H (ε) ) when ε ∈ I k j , and the coefficient in front of ε 2 is easily computable and equal to Similarly, we have N asymptotic terms of N (λ; H (ε) ) when ε ∈ Ik j , and the coefficient in front of ε 2 equals Notice that the sum in (4.19) contains more terms than (4.18); one of the extra terms corresponds to θ = θ j and its modulus is at least . The rest of the extra terms give a total contribution of O(ε N k j ). Therefore, we have (4.20) Now we will readjust the definition of the subset R of the resonant zone R by requiring that the jump (4.20) is at least one, which can be achieved by asking that ε 18|θ j | . Another way of formulation this is requesting that if n > n j satisfies then ξ ∈ R(θ j , n). Now, we define a modified setS 3 (N ) which satisfies properties a and b above, but with a modified parameter δ n defining the resonant zone R given by δ n (θ) = min{ εn|V θ | 100|θ| , |V θ | 2 72|θ| 2 }. The calculations just above show that if ξ ∈ R (θ j , n j ), then, assuming once again that ε 0 = ε 0 (N ) is small enough, we have: and, therefore, we cannot have both these coefficients small at the same time. This shows that, indeed, we cannot have a complete power asymptotic expansion (nor even an asymptotic expansion with the remainder o(ε 2 )) for any ξ ∈S 3 (N ) with N ≥ 3. If we put (4.23) then this is an uncountable set such that there is no complete power asymptotic expansion of N (λ, H) for √ λ ∈ S 3 . We have proved the following result: Theorem 4.9. Suppose, V is smooth almost-periodic, but not periodic, the constant Fourier coefficient τ = 0, andV θ = 0 for any θ ∈ Θ ∞ . Then there exists an uncountable set S 3 such that when λ 1/2 ∈ S 3 , there is no complete power asymptotic expansion of N (λ; H).
Suppose now that τ = 0. Consider the setS 3 (N ) of all λ for which the following two conditions are satisfied: a. There is an infinite sequence n j → ∞ and θ j ∈ Θ 3NL(n j ) such that (λ + τ ε n j ) 1/2 ∈ R (θ j ; n j ), and b. There is an infinite sequence n j → ∞ such that (λ + τ ε n j ) 1/2 ∈ R(n j ). A slightly more difficult than before (but still quite elementary) argument shows that S 3 (N ) is uncountable for each τ . Also, similar to the case τ = 0, if ε ∈ I n j , the point λ is in the spectral gap of H 2 (n j ) and, therefore, we have the (trivial) resonant asymptotic expansion (3.49). On the other hand, if ε ∈ I n j , we have the non-resonant asymptotic expansion (3.33), and the first order term in this expression equals − τ 2π √ λ , which means that these two expressions are different starting with ε, i.e. it is enough to take N ≥ 2. Putting S 3 := ∪ N ≥2S3 (N ), we will prove the analogue of Theorem 4.9 in the case τ = 0.
Putting all the results proved in this section together, we have proved the following: Theorem 4.10. Suppose, V is smooth almost-periodic, but not periodic, andV θ = 0 for any θ ∈ Θ ∞ . Then there exists a set S (which we call a super-resonance set) such that a complete power asymptotic expansion of N (λ; H) exists if and only if λ 1/2 ∈ S. The set S is uncountable and has measure zero.
Remark 4.12. We have called the set S the super-resonance set. An interesting question which we have not studied so far is what is the dimension of this set.

Preparation.
Our strategy will be to find a unitary operator which reduces H = H 0 + ε Op(V ), H 0 := −∆, to another PDO, whose symbol, essentially, depends only on ξ (notice that now we have started to distinguish between the potential V and the operator of multiplication by it Op(V )). More precisely, we want to find operators H 1 and H 2 with the properties discussed in Sections 3 and 4. The unitary operator will be constructed in the form U = e iΨ with a suitable bounded self-adjoint quasi-periodic PDO Ψ. This is why we sometimes call it a 'gauge transform'. It is useful to consider e iΨ as an element of the group We assume that the operator ad(H 0 , Ψ) is bounded, so that U (t)D(H 0 ) = D(H 0 ). This assumption will be justified later on. Let us express the operator A t := U (−t)HU (t) via its (weak) derivative with respect to t: A t = H + t 0 U (−t ) ad(H; Ψ)U (t )dt . The operator Ψ is sought in the form (5.2) Ψ =k j=1 Ψ j , Ψ j = Op(ψ j ), with some bounded operators Ψ j . Substitute this formula in (5.1) and rewrite, regrouping the terms: Next, we switch the summation signs and decrease l by one in the second summation: We emphasise that the operators B l and T l depend only on Ψ 1 , Ψ 2 , . . . , Ψ l−1 . Let us make one more rearrangement: Let ϕ θ (ξ, ε n ) be a smooth cut-off function of the set (5.7)
Similar notation is used for corresponding operator, i.e. B . Now we can specify our algorithm for finding Ψ j 's. The symbols ψ j will be found from the following system of commutator equations: ad(H 0 ; Ψ 1 ) + B 1 = 0, (5.9) ad(H 0 ; Ψ l ) + B l + T l = 0, l ≥ 2, (5.10) and hence (5.11) Below we denote by yk the symbol of the PDO Yk. Obviously, the operators B l , T l are bounded, and therefore, in view of (5.9), (5.10), so is the commutator ad(H 0 ; Ψ). This justifies the assumption made in the beginning of the formal calculations in this section.
It is also convenient to introduce the following norm in the class of symbols: We notice that Op(b) ≤ b .

5.3.
Computing the symbol of the operator after gauge transform. The following lemma provides us with more explicit form of the symbol yk.
Here C (p) s (θ) depend on s, p and all vectors θ, θ j , θ j , φ j , θ j , φ j . At the same time, coefficients C (p) s (θ) can be bounded uniformly by a constant which depends on s only. We apply the convention that 0/0 = 0.
The proof is identical to the proof of Lemma 9.3 from [6] and we omit it here. Explicit value of the coefficients for the second term (see (3.17) and (3.21)) can be found directly as the second order perturbation or following more carefully the first two steps of the construction for A 1 from (5.11).