Uniform exponential decay of the eigenfunctions of A ε

The phenomenon of exponential decay of eigenfunctions of various differential operators corresponding to the eigenvalues in the gaps of essential spectra has been extensively investigated for the few last decades, see e.g. [6, 22, 23, 39].

For example, in [6] a Schr¨odinger operator H with random perturbation is con-sidered. It is shown that the rate of the exponential decay is proportional to p∆₊(E)∆₋(E) , where E is an eigenvalue in a gap, ∆₊(E) and ∆₋(E) are distances from E to the right and left edges of the gap respectively. Roughly speaking, the exponent obtained in [6] is proportional to the distance from the essential spectrum near the centre of the gap and to the square root from the dis-tance near the both edges of the gap. This estimate is the best of the sort known at present. Another result, obtained in [23], can be straightforwardly applied to the operator A_ε when ε is fixed. It follows from [23] that an eigenfunction u^ε decays exponentially with an exponent proportional to dist(λ_ε, σ_ess(A_ε)) . Never-theless, this is not sufficient for our purposes. To gain compactness of a sequence

2The error bound (1.23) employs the results of [29] requiring, as stated, higher regularity of

∂Q0. The rest of the statement of the theorem applies potentially to less regular boundaries.

u^ε we need uniform decay, i.e. exponential decay with exponent independent of ε , which previous results do not guarantee. In this section we prove ε -uniform exponential decay of sequence of u^ε corresponding to converging in the gap λ_ε. Remark 1.2.1. We would like to draw attention to the qualitative difference between the previously obtained estimates for the rate of exponential decay and the one we prove in this work. As we mentioned above, known results give the rate of decay proportional to the distance to gap edges or to the square root of the distance. Our estimate (1.24) is entirely different. For small enough ε the rate of decay is O dist(λ_ε, σ_ess(A_ε))^1/2

at the right end of each interval β(λ) < 0 and proportional to (dist(λ_ε, σ_ess(A_ε))^−1/2 at the left end, cf. Figure 1-2 and (1.14).

We formulate the main result of this section (and also one of the principal results of the first part) in the following

Theorem 1.2.2. Let λ_εk and u^εk be sequences of eigenvalues of the operator A_ε and corresponding eigenfunctions normalised in L²(Rⁿ) , where ε_k is some positive sequence converging to zero as k → ∞ . Let λ₀ be such that β(λ₀) is negative and λ₀ is not an eigenvalue of the operator T given by (1.11). Suppose that λ_εk converges to λ₀. Then for small enough ε_k eigenfunctions u^εk decay uniformly exponentially at infinity, namely, for

0 < α < p

−β(λ₀)/a₁ (1.24)

the following holds:

ke^α|x|u^εkk_L²_(Rⁿ₎ ≤ C,

uniformly in ε_k, i.e. for any 0 < ε_k < ε(α) , with C = C(α) independent of ε . Proof. We drop the index k in ε_k for the sake of simplification of notation. So, when we say, for instance, ‘sequence λ_ε’ we actually mean ‘subsequence λ_εk’.

The plan of the proof is the following. We first derive ‘elementary’ a priori estimates for the eigenfunction u^ε outside the set of inclusions Ω^ε₀ ∪ eΩ^ε₀. Next we study the structure of the eigenfunction at the small scale and deduce some vital inequalities for ε∇u^ε inside the inclusions. As a central technical step, we then employ in the integral identity (1.6) a test function with exponentially growing weight g²(|x|) , see (1.37)–(1.38) below, and perform some delicate two-scale uniform estimates to achieve the result. Introducing a test function with exponentially growing weight we use the idea of Agmon, [1]. This on its own does

not lead to a straightforward conclusion. We have to develop some delicate two-scale analysis, studying the properties of eigenfunctions of A_ε both at large and small scales at the same time. The main auxiliary technical results are proven in Lemma 1.2.4 and Proposition 1.3.1.

Step 1. Due to the nature of the operator Aε (coefficient a(x, ε) is very small on the inclusion phase) one can expect that the eigenfunctions (more precise, their gradients) oscillate wildly on the inclusion phase. Nevertheless it is possible to control u^ε outside the inclusions. Setting w = u^ε in (1.6) we have

ε²a₀k∇u^εk²_L2(Ω^ε₀)+ a₁k∇u^εk²_L2(Ω^ε₁)+ a₂k∇u^εk²_L2(Ω^ε₂)+ +kea^1/2₀ (x, ε) ∇u^εk²_L2(eΩε

0)= λ_εku^εk²_L2(Rⁿ) = λ_ε.

(1.25)

Therefore

ku^εk_H1(Rⁿ\(Ω^ε₀S eΩ^ε₀))≤ C (1.26) uniformly in ε . From now on C denotes a generic constant whose precise value is insignificant and can change from line to line.

Step 2. Now we will represent u^ε as a sum of two functions, one of them has ε -uniformly bonded norm in H¹, another preserves the ‘uncontrollable’ os-cillations of the gradient of u^ε. Let us consider u^ε in a cell εQ corresponding to such ξ = ξ(ε) ∈ Zⁿ, see 1.1, that the respective ‘inclusion’ εQ₀ has a nonempty intersection with Ω₁. There exists an extension eu^ε of u^ε|_εQ1 to the whole cell εQ such that

keu^εk_L²_(εQ0) ≤ Cku^εk_L²_(εQ1), k∇eu^εk_L²_(εQ0) ≤ Ck∇u^εk_L²_(εQ1), (1.27) where C does not depend on ε or ξ , see e. g. [35, Ch. 3, §4, Th. 1], which is a version of the so-called ‘extension lemma’, see also e.g. [27, §3.1, L. 3.2]. In particular, we can choose the following extension:

e

u^ε ≡ u^ε, x ∈ Ω^ε₁∪ Ω^ε₂,

−∇ · a(x,ε)∇eu^ε(x)

= 0, x ∈ Ω^ε₀∪ eΩ^ε₀,

which minimises ka^1/2(x, ε)∇eu^εk_L²_(εQ0) subject to the prescribed boundary con-ditions, with (1.4) and (1.5) ensuring that (1.27) still holds. From (1.26) and (1.27) we conclude that

keu^εk_H¹_(Rⁿ₎≤ C. (1.28)

We represent u^ε in the form

u^ε(x) = eu^ε(x) + v^ε(x) (1.29) and consider the function v^ε∈ H₀¹(Ω^ε₀∪ eΩ^ε₀) . We assume that v^ε is extended by zero to the whole Rⁿ. In each inclusion εQ₀ ⊂ Ω^ε₀ ∪ eΩ^ε₀ we have the following boundary value problem for v^ε(x) :

−∇ · (a(x, ε)∇v^ε)−λ_εv^ε= λ_εeu^ε, x ∈ εQ₀,

v^ε(x) =0, x ∈ ∂(εQ₀). (1.30) When a(x, ε) = a₀ε², i.e. everywhere in Ω^ε₀ and also in eΩ^ε₀ in the case ea₀(x, ε) = a₀ε², after changing the variables x → y = x/ε we obtain

−a₀∆_yv^ε(εy) − λ_εv^ε(εy) = λ_εue^ε(εy), y ∈ Q₀,

v^ε(εy) = 0, y ∈ ∂Q₀. (1.31) Since λ₀ 6= λ_j by the assumptions of the theorem, λ_ε is separated uniformly from the spectrum of the operator T , (1.11), for small enough ε . Hence the resolvent of T at λ_ε is bounded uniformly in ε and (1.31) implies

kv^ε(εy)k_H¹_(Q0)≤ Ckeu^ε(εy)k_L²_(Q0). (1.32) In the case when eA₀ε^2−θ ≤ ea₀(x, ε) ≤ σ₀ε^2−θ, θ ∈ (0, 2] , we multiply equa-tion (1.30) by v^ε and integrate by parts to obtain after rescaling

ε⁻² Z

Q0

ea₀(εy, ε)|∇_yv^ε(εy)|²dy − λ_ε Z

Q0

v^ε(εy)
₂

dy = λ_ε Z

Q0

e

u^ε(εy)v^ε(εy) dy. (1.33)

Notice that ε⁻²ea₀(εy, ε) ≥ eA₀ε^−θ → ∞ as ε → 0 . Then using Poincar´e inequal-ity for functions from H¹(Q₀)

Z

Q0

f²dy ≤ C Z

Q0

|∇_yf |²dy,

and H¨older inequality



 Z

Q0

f g dy





2

≤ Z

Q0

f²dy Z

Q0

g²dy,

we derive from (1.33) that

(Cε^−θ− λ_ε)kv^ε(εy)k²_L2(Q0) ≤ λ_εkv^ε(εy)k_L²_(Q0)keu^ε(εy)k_L²_(Q0). The latter immediately implies that

kv^ε(εy)k_L²_(Q0) ≤ Ckeu^ε(εy)k_L²_(Q0). (1.34) In fact an even stronger relation is valid, kv^ε(εy)k_L²_(Q0) = o keu^ε(εy)k_L²_(Q0)

. From (1.33) and (1.34) one directly obtains

ε⁻²kea^1/2₀ ∇_yv^ε(εy)k²_L2(Q0) ≤ Ckeu^ε(εy)k²_L2(Q0), (1.35) for small enough ε . Returning in (1.32) and in (1.34), (1.35) to the variable x we arrive at the following inequality that describes the behaviour of v^ε and its gradient in Ω^ε₀∪ eΩ^ε₀,

ka^1/2∇v^ε(x)k²_L²_(εQ0)+ kv^ε(x)k²_L²_(εQ0) ≤ Ckeu^ε(x)k²_L²_(εQ0), (1.36) with an ε -independent constant C .

Step 3. In order to get the uniform exponential decay of the eigenfunctions we next substitute in (1.6) a test function of a special form:

w = g²(|x|)eu^ε(x). (1.37) Here we define function g as follows

g(t) =

( e^αt, t ∈ [0, R],

e^αR, t ∈ (R, +∞), (1.38)

where R is some arbitrary positive number. The exponent α will be chosen later. This method was employed e.g. by Agmon, see [1], but in the present case its realisation is not straightforward. Namely, to obtain the desired estimates we

have to implement the approach of [1] in the context of the two-scale analysis.

We will show that g(|x|)eu^ε(x) , and consequently g(|x|)u^ε(x) , are bounded in L²(Rⁿ) uniformly with respect to R and ε . Then we will show via passing to the limit as R → ∞ that we can replace g(|x|) by e^α|x|.

Remark 1.2.3. We cannot use e^2α|x|eu^ε(x) as a test function directly, since it is not known at this stage that functions e^α|x|ue^ε(x) and e^α|x|u^ε(x) are square integrable.

The following identity holds by direct inspection

∇eu^ε∇(g²eu^ε) = |∇(geu^ε)|²− |∇g|²(eu^ε)². (1.39) Notice that the Euclidian norm of ∇g is bounded by g with constant α (uni-formly with respect to R ):

∇g(|x|)

 ≤ αg(|x|). (1.40)

After the substitution of (1.37) into (1.6) we have, via (1.29) and (1.39), ε²a₀ Notice that the right hand side is bounded by some constant C independent of ε and R due to (1.26), (1.28), (1.36) and the boundedness of the domains of integration.

The rough idea of the remaining part of the proof is the following. We argue that the second term on the left hand side of the latter is small and the first one is relatively small (compared to the other terms of the identity). One can notice that in equation (1.31) the right hand side is ‘almost’ a constant for every fixed Q0. Then one can expect that the solution of (1.31) is ‘approximately equal’ to

the solution of (1.12) corresponding to λ = λ_ε multiplied by λ_εeu^ε(εy) , v^ε(εy) ∼ λ_εb(y)eu^ε(εy).

Rearranging the integrated entities in the last two terms on the left hand side of (1.41) one obtain ‘approximately’

− λ_εg²(eu^ε)²− λ_εg²v^εue^ε ∼ − [λ_ε(1 + λ_εb(y))] g²(eu^ε(εy))². (1.42) The expression in the square brackets resembles the definition of β(λ_ε) , see (1.13). Rescaling back to variable x and integrating the right hand side of (1.42) one can obtain

− β(λ_ε)kgeu^εk²_L2(Ω^ε₀∪Ω^ε₁). (1.43) Notice that as λ_ε → λ₀, β(λ_ε) → β(λ₀) < 0 . Hence we obtain kgeu^εk²_L2(Ω^ε₀∪Ω^ε₁)

multiplied by a uniformly positive coefficient, end we need only to choose appro-priate exponent α , see (1.38), to ensure that the fourth term on the left hand side of (1.41),

Let us continue the proof. Consider the second term on the left hand side of (1.41). Since the coefficient ea₀(x, ε) is bounded uniformly in ε and the sequence of domains eΩ^ε₀ is also bounded (so g²|_Ωeε

0, ∇g²|_Ωeε

0 ≤ C uniformly) we derive that 

Then from (1.28), (1.36) follows that 

The right hand side of the latter converges to zero. Indeed, let us take an arbitrary

subsequence eu^ε. Since keu^εk_H¹_(Rⁿ₎ is bounded uniformly in ε , see (1.28), the set of functions eu^ε is weakly compact in H¹(B_R) , hence strongly compact in L²(B_R) for any R ; we take R large enough so that Ω₂ b B_R. Then there exists a further subsequence eu^ε that converges to some function u0 strongly in L²(BR) . Then

keu^εk_L2(eΩ^ε₀) ≤ ku₀k_L2(eΩ^ε₀)+ keu^ε− u₀k_L2(eΩ^ε₀) → 0 (1.44) as the Lebesgue measure of the set eΩ^ε₀ tends to zero. Since we have chosen in the beginning an arbitrary subsequence eu^ε, (1.44) holds for any sequence of ε .

Hence 

From (1.36) and (1.44) we also obtain

kv^εk_L2(eΩ^ε₀) → 0. (1.46) Step 4. The following Lemma approximates the last two terms and bounds the first term (both, in a sense, of a ‘two-scale’ nature) on the left hand side of (1.41).

Lemma 1.2.4. There exists ε₀ > 0 such that for all positive ε < ε₀ the following estimates are valid

The proof of this lemma is quite technical and we give it in the next section.

We make use of Lemma 1.2.4 and convergence (1.45) to transform identity (1.41)

into the following inequality, valid for small enough ε :

where C is independent of ε and R . Notice that β(λ_ε) is negative and uniformly bounded away from zero as λ_ε → λ₀. Applying (1.40) to the second term on the left hand side we arrive at

(a₁− 2Cε)k∇(geu^ε)k²_L2(Ω^ε₁)+ −β(λ_ε) − α²a₁− 2Cε

kgeu^εk²_L2(Ω^ε₀∪Ω^ε₁)≤ C. (1.49) Hence we should choose α such that −β(λ₀) − α²a₁ is positive, i.e.

α <p

−β(λ0)/a1.

Since g(|x|) coincides with e^α|x| on the ball B_R, taking ε small enough and restricting the L²-norms to BR we arrive at

e^α|x|eu^ε

Step 5. Despite the fact that the sequence of ∇v^ε is unbounded in L²-norm, the function v^ε itself is controlled by eu^ε, see (1.36). Therefore we can get the estimate for the function u^ε analogous to (1.50).

e^α|x|u^ε

In each cell we use inequality (1.36) and sup

and hence, finally,

e^α|x|u^ε

L²(Rⁿ)≤ C uniformly in ε .

Remark 1.2.5. It is easy to see that ∇eu^ε (unlike ∇u^ε) decays exponentially uniformly in ε with the same rate as u^ε. Indeed, from (1.51) and (1.27) it follows that

kg∇eu^εk²_L2(Ω^ε₀∪eΩ^ε₀) ≤ X

εQ0⊂Ω^ε₀∪eΩ^ε₀

sup

x⁰∈εQ

gk∇eu^εk²_L²_(εQ0)≤ Ckg∇eu^εk²_L²_(Π^ε₎,

where Π^ε := nS

εQ₁|εQ₁ is such that correcponding εQ₀ ⊂ Ω^ε₀∪ eΩ^ε₀ o

. Since k∇eu^εk²_L2(Ω^ε₂) and hence kg∇eu^εk²_L2(Ω^ε₂) are bounded uniformly, we have

kg∇eu^εk²_L2(Rⁿ) ≤ C + Ckg∇eu^εk²_L2(Ω^ε₁) = C + Ck∇(geu^ε) − ∇geu^εk²_L2(Ω^ε₁)≤

≤ C + Ck∇(geu^ε)k_L²_(Ω^ε₁₎+ Cαkgeu^εk²_L2(Ω^ε₁).

The latter is bounded uniformly in ε and R due to (1.49) and (1.50). Hence, passing to the limit as R → ∞ , we finally arrive at

e^α|x|∇eu^ε

L²(Rⁿ)≤ C (1.52)

uniformly in ε .

Remark 1.2.6. Estimate (1.24) is sharp in a sense. As we will show later, u^ε strongly two-scale converges to u₀, for which p

−β(λ₀)/a₁ is the optimal estimate for its decay exponent, cf. (2.51).