Semiclassical study of shape resonances in the Stark effect

Semiclassical behavior of Stark resonances is studied. The complex distortion outside a cone is introduced to study resonances in any energy region for the Stark Hamiltonians with non-globally analytic potentials. The non-trapping resolvent estimate is proved by the escape function method. The Weyl law and the resonance expansion of the propagator are proved in the shape resonance model. To prove the resonance expansion theorem, the functional pseudodifferential calculus in the Stark effect is established, which is also useful in the study of the spectral shift function.


Introduction
In this paper, we study the semiclassical behavior of the resonances for the Stark Hamiltonian: where V (x) ∈ C ∞ (R n ; R) is a non-globally analytic potential and β > 0. Throughout this paper, the constant β > 0 is fixed.We set the cone C(K, ρ) = {x ∈ R n ||x ′ | ≤ K(x 1 + ρ)}, where x ′ = (x 2 , . . ., x n ), and denote its complement by C(K, ρ) c .We denote the set of all bounded smooth functions with bounded derivatives by C ∞ b .Our assumption on the potential V is as follows: Assumption 1.The potential V (x) ∈ C ∞ b (R n ; R) has an analytic continuation to the region {x ∈ C n |Rex ∈ C(K 0 , ρ 0 ) c , |Imx| < δ 0 } for some ρ 0 ∈ R, K 0 > 0 and δ 0 > 0, and ∂V (x) goes to zero when Rex → ∞ in this region.
We introduce the complex distortion outside a cone to study semiclassical Stark resonances.This reduces the study of resonances to that of eigenvalues of a non-self-adjoint operator P θ .We take any K > K 0 and sufficiently large ρ > 0 (such that Lemma 2.1 holds) and deform P ( ) in C(K, ρ) c .Take a convex set C(K, ρ) which has a smooth boundary such that C(K, ρ) is rotationally symmetric with respect to x ′ and C(K, ρ) = C(K, ρ) in x 1 > −ρ + 1.We define We next set Φ θ (x) = x + θv(x).This is a diffeomorphism for real θ with small |θ|.We set U θ f (x) = (detΦ ′ θ (x)) ), which is unitary on L 2 (R n ).We define the distorted operator P θ ( ) = U θ P ( )U −1  θ .The P θ ( ) is an analytic family of closed operators for θ with |Imθ| < δ 0 (1 + K −2 ) − 1 2 and |Reθ| small (Proposition 2.1).Moreover P θ ( ) with Imθ < 0 has discrete spectrum in {Imz > βImθ} (Proposition 2.2).We note that we exclude the condition that |θ| is small by repeated applications of the Kato-Rellich theorem.We also note that we do not require that is small.We set L p cone = {f ∈ L p |suppf ⊂ C(K, ρ) for some K, ρ} (in the following, we can replace L p cone by L p comp ).We also set R + (z, ) = (z − P ) −1 for Imz > 0. Then we define the (outgoing) resonances of P by meromorphic continuations of cutoff resolvents: Theorem 1. Suppose that Assumption 1 holds.Fix any > 0. Then for any χ 1 , χ 2 ∈ L ∞ cone (R n ) such that χ j = 0 on some open sets, the cutoff resolvent χ 1 R + (z)χ 2 (Imz > 0) has a meromorphic continuation to Imz > −βδ 0 with finite rank poles.The pole z is called a resonance and the multiplicity is defined by The set of resonances is independent of the choices of χ 1 and χ 2 including multiplicities and denoted by Res(P ).Moreover, Res(P ) = σ d (P θ ) including multiplicities in {Imz > βImθ} if 0 > Imθ > −δ 0 (1 + K −2 ) − 1 2 and |Reθ| is small.
We emphasize that there is no restriction on Rez in Theorem 1.The resonances are also described including multiplicities in terms of meromorphic continuations of the matrix elements of the resolvent (f, R + (z)g) for f, g ∈ L ∞ cone (Proposition 2.3) or f, g ∈ A = {u ∈ L 2 |suppû is compact} (Proposition 2.4).The latter formalism based on analytic vectors for 1  i ∂ shows that our definition of resonances coincides with that based on the global analytic translation when the potential is globally analytic (Corollary 2.2).
The resonances for the Stark Hamiltonians have been investigated by many authors.Avron-Herbst [1] defined the Stark resonances by the translation analyticity.Herbst [11] defined the Stark resonances by the dilation analyticity.Herbst [12] discussed the exponential decay of matrix elements of Stark propagator and its relation with Stark resonances.
Dimassi-Petkov [7] studied resonances of − 2 ∆ + V (x) + x 1 and its relation with the spectral shift function in the semiclassical limit ( → 0).In [7], reso-nances are defined and studied in the region Rez < R by the complex distortion in the region x 1 < R.
We next state the non-trapping resolvent estimate in our setting.We denote the trapped set for the classical flow in the energy interval [a, b] by is the set of all (x 0 , ξ 0 ) ∈ T * R n such that a ≤ p(x 0 , ξ 0 ) ≤ b and sup t∈R |x(t)| < ∞, where (x(t), ξ(t)) is the solution of the Hamilton equation for p(x, ξ) = |ξ| 2 + βx 1 + V (x) with the initial value (x 0 , ξ 0 ).
Wang [27] proved the non-trapping limiting absorbtion principle bound for the Stark Hamiltonians, that is, the O( −1 ) bound of R + (z, ) for Imz > 0 with suitable weights (see also ).The following bound implies the bound for the analytically continued cutoff resolvent χR Theorem 2. Suppose that Assumption 1 holds and K [a,b] = ∅.Then for any 0 < M ≪ M there exists C > 0, which also depends on the construction of P θ , such that for small > 0 and z ∈ The proof of Theorem 2 is based on the escape function method as in [18], [24], where the same result is proved for decaying potentials.Theorem 2 implies the non-trapping time decay estimate (Corollary 3.1) as in [19].
Our principal motivation comes from the shape resonance model.Denote the full potential by where G int is compact and non-empty, G ext is closed, and Our first main theorem is the Weyl-type asymptotics for the Stark shape resonances: Theorem 3.Under Assumption 1 and 2, there exists S > 0 such that Our second main theorem is the resonance expansion theorem for Stark propagators (in this paper, the symbol O for some operator means O L 2 →L 2 unless otherwise stated).
Theorem 4. Suppose that Assumption 1 and 2 hold.Then for any ψ ∈ where In the decaying potential case, Helffer-Sjöstrand [10] and Stefanov [25] [26] proved Theorem 3. Nakamura-Stefanov-Zworski [19] provided a simplified proof of Theorem 3 and proved Theorem 4 after the work of Burq-Zworski [3].We follow the general line of [19] with a minor simplification given by direct resolvent estimates (Proposition 4.1), which does not depend on the maximal principle technique (see   [5] for related resolvent estimates).Note that Theorem 4 is the resonance expansion in the limit → 0 while the resonance expansion in Herbst [12] is valid in the limit t → ∞.
To prove the resonance expansion theorem, we study the pseudodifferential property of ψ(P ).The symbol class is defined by The Weyl quantization is defined by We set σ(x, ξ; y, η) = ξ, y − η, x .The composition of Weyl symbols is which makes sense also for the formal power series.We denote OpS(m) = {a W (x, D; )|a ∈ S(m)} and S(m ).In the case where β = 0, the usual functional pseudodifferential calculus implies f (P ) ∈ OpS( ξ −∞ ) with the principal symbol f (|ξ| [8, section 8]).In the case where β > 0, this does not hold since P is not elliptic in the semiclassical sense.In fact, f (|ξ| ) and |ξ| can be arbitrary large on the support of f (|ξ| 2 + βx 1 + V (x)) when x 1 → −∞.Thus f (P ) ∈ OpS(m) for any tempered m.
Nevertheless, we can treat the weighted function f (P )χ and the difference of functions f (P 2 ) − f (P 1 ).We set m = |ξ| 2 + x 1 , where x = (1 + |x| 2 ) For the weighted function f (P )χ, we prove the following.Suppose , which is the composition of the formal asymptotic expansion of the symbol of f (P ) and χ.
We note that Theorem 5 holds true for χ W f (P ) since it is the adjoint of c (R) with g = 1 near suppf .This is used in subsection 4.3.
For the difference of functions f (P 2 )−f (P 1 ), we prove the following.Suppose , where j = 1, 2.
, which is the difference of the formal asymptotic expansion of the symbols of f (P 2 ) and f (P 1 ).
Then the derivative of the spectral shift function ξ ′ defined by ξ ′ , f = tr(f (P 2 )− We can also discuss the spectral shift function by the formula ( [21]) tr(f (P )− f (P 0 )) = −tr((∂ x1 V )f (P )) and Theorem 5, where P 0 = − 2 ∆ + βx 1 .Dimassi-Petkov [7] and Dimassi-Fujiié [6] proved many properties of the spectral shift function by constructing an elliptic operator P such that −tr(( Remark 1.2.The trace class property and finite terms in the asymptotic expansion can be discussed even if we only assume This paper is organized as follows.In section 2, we define the Stark resonances in various manners and in particular prove Theorem 1.In section 3, we prove the non-trapping resolvent estimate for the Stark Hamiltonian (Theorem 2).In section 4, we study the shape resonance model in the Stark effect and prove the Weyl-type asymptotics (Theorem 3) and the resonance expansion (Theorem 4).In section 5, we prove the functional pseudodifferential calculus in the Stark effect (Theorem 5, 6).In the Appendix, we justify the commutator calculations of the Stark resolvent in section 5.

Definition of resonances
Throughout this section, we assume Assumption 1.

Complex distortion
We prove Theorem 1 in this subsection.Recall from section 1 that , and P θ ( ) = U θ P ( )U −1 θ .We first note that F ∈ C ∞ (R n ; R) is concave since C(K, ρ) is convex and the convolution with a positive function preserves convexity.We have v 1 (x) ≥ 1 on C(K, ρ + 1) c by the coefficient (1+K −2 ) 1 2 in the definition of F .Moreover (x 1 ) − ∂ α v j is bounded for |α| ≥ 1.This follows from the replacement of C(K, ρ) by C(K, ρ) for |α| = 1 and from the mollification for |α| ≥ 2. We also note that Φ ′ θ = I + θ∂ 2 F is symmetric.A calculation (using the invariance of Laplace-Beltrami operator) shows that where ( θ )).This expression defines P θ ( ) as a differential operator for complex θ with small |Reθ| and |Imθ| < (1 + K −2 ) − 1 2 δ 0 .We denote the semiclassical principal symbol of P θ ( ) by An advantage of our definition of P θ ( ) is as follows: , where y ranges over a small complex neighborhood of x.Thus for large ρ, |ImV and the closure is also denoted by P θ ( ).We first prove the analyticity of P θ with respect to θ. Proposition 2.1.For 0 < ≤ 1, P θ is an analytic family of type (A) with respect to θ with |Imθ| < δ 0 (1 + K −2 ) − 1 2 and |Reθ| small.That is, D(P θ ) = D(P ) and P θ u is analytic with respect to θ for any u ∈ D(P ) = D(P θ ).Thus, , where C is independent of θ with |Reθ| small.We only have to estimate The first term can be estimated as follows.We take χ(x 1 ) such that χ(x 1 ) = 0 for x 1 ≤ 1 and χ(x , where the last inequality follows from the standard elliptic estimate.Repeated applications of Kato-Rellich theorem (see [20, section X.2]) to 0 P θ P θ 0 show that P θ is closed on D(P θ ) = D(P ) and P θ = P * θ .This is valid for small |Reθ| and |Imθ| < c , an approximation argument shows that P θ u is analytic with respect to θ for u ∈ D(P ).This implies that (P θ − z) −1 is analytic with respect to θ by the general theory (see [17, section 7.1, section 7.2]).
We next prove the discreteness of the spectrum of P θ in {Imz > βImθ}.

Proof. Set
We next prove ( P θ − z)χ 1 u ≥ c χ 1 u for large M > R. We take small ε > 0 and set χ j,M = τ j (G(x)/M ), where where φ is as above.Then Denote the seminorms in We have Thus we have proved ).Thus we see that (q −1 ) W : H k → H k+2 is uniformly bounded with respect to M > 1.We also see that lim M→∞ q 1 = 0 in S(1) if q −1 ♯q = 1 + q 1 since ∂ x,ξ q is bounded in S( ξ 2 ) with respect to M and lim M→∞ ∂ x,ξ q −1 = 0 in S( ξ −2 ).Thus (1 + q We also have ( Remark 2.2.The proof will be simplified if we assume that 0 < ≪ 1. Proof of Theorem 1.Take any 0 < δ 1 < δ 0 .Take χ 1 , χ 2 ∈ L ∞ cone (R n ) such that χ j = 0 on some open sets.Construct P θ outside suppχ j and C(K, ρ) with for real θ and Imz > 0. The right hand side has an analytic continuation with respect to θ with |Imθ| < δ 1 and |Reθ| small by Proposition 2.1.This in turn implies that the left hand side has a meromorphic continuation to Imz > −βδ 1 by Proposition 2.2.If z ∈ σ d (P θ ), this is analytic near z.Suppose that z ∈ σ d (P θ ).Then the multiplicity of the pole z of χ 1 R + (z)χ 2 is given by rank is the generalized eigenprojection of P θ at z.We have (P θ − z) k Π θ z = 0 for some k by the general theory of closed operators.Then the repeated applications of the unique continuation theorem for second order elliptic operators imply that rankχ 1 Π θ z = rankΠ θ z .Since (Π θ z ) * = Π θ z , the same argument for the adjoint implies that rankχ 1 Π θ z = rankχ 1 Π θ z χ 2 .This proves that the definition of resonances is independent of χ 1 , χ 2 and the multiplicity is given by m z = rankΠ θ z .

Meromorphic continuations of matrix elements
The resonances are also described by meromorphic continuations of the matrix elements of the resolvent.
Proposition 2.3.The matrix element of the resolvent (f, R + (z)g) has a meromorphic continuation to Imz > −βδ 0 for any f, g ∈ L 2 cone .For z with Imz > −βδ 0 , z is a resonance of P if and only if z is a pole of (f, R + (z)g) for some f, g ∈ L 2 cone and the multiplicity m z is given by the maximal number k such that there exist Moreover, for any nonempty open bounded U ⊂ R n and an orthonormal basis The Proposition easily follows from this.Corollary 2.1.Res(P ) ∩ R = σ pp (P ).
The resonances are also described based on analytic vectors.Set A = {u ∈ L 2 (R n )|suppû is compact}, which consists of analytic vectors for the generators of the translations ( 1 i ∂ 1 , . . ., 1 i ∂ n ).Proposition 2.4.The matrix element of the resolvent (f, R + (z)g) has a meromorphic continuation to Imz > −βδ 0 for any f, g ∈ A. For z with Imz > −βδ 0 , z is a resonance of P if and only if z is a pole of (f, R + (z)g) for some f, g ∈ A and the multiplicity is given by the maximal number k such that there exist We first note that U θ f (f ∈ A) has an analytic continuation for small |Reθ| by the definition of A. Take f, g ∈ A.
for real θ and Imz > 0. The right hand side is analytic with respect to θ by Proposition 2.1.This in turn implies that the left hand side has a meromorphic continuation to Imz > −βδ 1 by Proposition 2.2.Then we have We note that if we replace φ(x) by ε n φ(εx) in the definition of F (x), v(x) and P θ , the Lipschitz constant of v(x) is bounded by Cε for some C > 0. Thus taking ε > 0 sufficiently small and arguing as in [16, Theorem 3], we see that These prove the Proposition.
Corollary 2.2.In addition to Assumption 1, suppose that V has an analytic continuation to |Imz| < δ 0 and is bounded in this region.Then for −δ 0 < Imθ < 0, the resonances of P in Imz > βImθ coincide with the eigenvalues of Proof.Arguing as above, the eigenvalues of P ′ θ, are described by the meromorphic continuation of (f, R + (z)g) for f, g ∈ A and thus coincide with Res(P ) by Proposition 2.4.

Non-trapping estimates
Proof of Theorem 2. We only sketch the proof since it is similar to that of [24,Theorem 1].The non-trapping assumption enables us to construct an escape function ) ∩ {|x| < R} for some a < a < b < b, where R > 0 is large.We set P θ,ε = e −εG W / P θ e εG W / , where M 1 ≤ ε ≪ |Imθ| and M 1 ≫ 1.We consider z with a ≤ Rez ≤ b and (Imz) − ≪ ε.

Shape resonance model
In this section, we discuss the shape resonances for the Stark Hamiltonian generated by potential wells.Recall that p(x, ξ) Fix a cutoff function χ 0 near G int such that supp∂χ 0 ⋐ {x ∈ R n |V (x) > b + 2δ}.Complex distorted operators in this section are constructed outside suppχ 0 .Let V ext (x) be a potential obtained by filling up the wells: V ext = V β near supp(1 − χ 0 ) and V ext > b + 2δ near G int , and P ext = − 2 ∆ + V ext with the corresponding distorted operator P ext θ .Let V int (x) be a potential flattened outside the wells: V int (x) = V β near suppχ 0 and V int (x) = b + 2δ outside a small neighborhood of suppχ 0 , and In the following we set α( ) = C and γ( The basic estimate in this section is the following Agmon estimate which is valid in more general settings (see [30, section 7.1]).
This is also valid for P θ if U is away from the region of deformation in the definition of P θ .In the following we fix S 0 such that Lemma 4.1 holds true where U is a small neighborhood of supp∂χ 0 , and moreover Lemma 4.1 with P replaced by P int holds true where U is a small neighborhood of supp(1 − χ 0 ).

Resolvent estimate
In [19] the resolvent estimate is obtained by the abstract method based on the maximum principle technique.In the shape resonance model, we give more direct resolvent estimate based on the commutator calculation and the Agmon estimate.
Proposition 4.1.For small > 0, Proof.We have The third inequality follows from the Agmon estimate.The last inequality follows if we subtract Cα( from both sides for small > 0. We also have The third inequality follows from the Agmon estimate.The last inequality follows if we subtract C dist(z, σ(P int )) −1 e −S0/ χ 0 (P θ − z) −1 ≤ C χ 0 (P θ − z) −1 from both sides for small > 0. Substituting the left hand side of each inequality for the right hand side of the other inequality and subtracting the small remainder from both sides, we obtain the desired results.Remark 4.2.This proposition shows the dichotomy for resonances: As in [26] and [19], we decompose resonances into clusters.Lemma 4.2.For small > 0, there exist a j ( ) < b j ( ) < a j+1 ( ) such that where where
Remark 4.3.In the decaying potential case, we immediately have for z ∈ ∂ Ω j by the Agmon estimate for P int and dist(z, σ(P int )) ≥ e −S0/ since P θ − P int has bounded coefficients.This and Lemma 4.2 imply ).
Since P θ − P int has an unbounded coefficient in our case, we need additional arguments.
Proof of Proposition 4.2.Since z − P int is elliptic near supp(P θ − P int ), where the last two inequalities follow from the Agmon estimate for P int and dist(z, σ(P int )) ≥ e −S0/ (note that [P int , P θ − P int ] has bounded coefficients).This and Lemma 4.2 imply ) by Lemma 4.2, and ) by the Agmon estimate.
Proof of Theorem 3. Proposition 4.2 implies that rankΠ θ j = rankΠ int j for small > 0. Thus the Weyl law for discrete eigenvalues of P int completes the proof.

Resonance expansion
We prove Theorem 4 in this subsection.Theorem 5 and Theorem 6 are used in this subsection.In the following, we take We first prove Theorem 4 after large time t > −n+1−ε (see ).
Proof.This is proved by Stone's formula, the almost analytic extension technique and Green's formula.If we employ Proposition 4.1 as the resolvent estimate, the claimed result follows.Since the argument of the proof is the same as [3], we omit the details.We note that calculations involving the energy cutoff ψ(P ) are justified by Theorem 5.
Remark 4.4.If we employ Remark 2.3 as the resolvent estimate, the result of Burq-Zworski [3] is obtained for the Stark Hamiltonian case.Namely, Proposition 4.3 remains true under Assumption 1 for t > −L for some choices of Ω( ) and L > 0.
We move to the proof of Theorem 4 up to large time C ≤ t ≤ e S/2 .We first prepare the Agmon estimate for continuous spectrum ([19, Lemma 4.3]): Proof.This follows from the Agmon estimate, the almost analytic extension technique and Green's formula.Since the proof is the same as [19, Lemma 4.3], we omit the details.
We next compare the different quantum dynamics [19,Lemma 4.4].

Functional pseudodifferential calculus in the Stark effect
In this section, we prove Theorem 5 and Theorem 6.In subsection 5.1 and subsection 5.2, we set P ( ) = − 2 ∆ + βx 1 + V (x), where V ∈ C ∞ b (R n ; R).The commutator calculations below are justified by Corollary A.1 in the Appendix.

Weighted resolvent estimates
We estimate the weighted resolvents in this subsection.Take w ∈ C ∞ (R n ; R ≥1 ) depending only on x 1 and w = |x 1 | for x 1 ≤ −2 and w = 1 for x 1 ≥ −1.
since |z| 1.Since P ( ) − z is elliptic near the support of χ, we have We next assume that Lemma 5.1 is true for k − 1.The case where k = 0 implies We have The first term can be estimated by |Imz| −1 ( /|Imz|) by the case where k = 0.
The second term can be estimated by 2k for |z| 1 and 0 < ≤ 1.

Proofs
Proof of Theorem 5. Applying x ′ s ′ from the right, we may assume that s ′ = 0. We take an almost analytic extension , where We set We easily see that (a j − a j )♯χ ∈ ∞ S(w −∞ m −1 ) and a j ∈ S(w −∞ m −∞ ).Thus we have in fact a ∼ j a j ♯χ in S(w −∞ m −1 ).We set f k (t) = (t − i) k f (t).Then f k (P )χ W has an asymptotic expansion in S(w −∞ m −1 ) by the above argument.Proposition 5.2 with z = i implies that f (P )χ W = (P − i) −k f k (P )χ W has an asymptotic expansion in S(w −∞ m −k−1 ), which coincides with the formal one j a j ♯χ.Since k is arbitrary, f (P )χ W has an asymptotic expansion in Proof of Theorem 6.The Helffer-Sjöstrand formula and the resolvent equation show that ) −1 has an asymptotic expansion which is uniform with respect to z with |Imz| > h δ in −2δ S δ (w −∞ m −2 x ′ −s ′ ).Thus the similar calculation as in the proof of Theorem 5 based on the partial fraction expansion shows that f (P 2 ) − f (P 1 ) has an asymptotic expansion in OpS(w −∞ m −2 x ′ −s ′ ).We next prove that f (P 2 ) − f (P 1 ) has an asymptotic expansion in OpS(w −∞ m −N x ′ −s ′ ) for any N .Suppose that this is true for N .Applying this to g(t) = (t + i)f (t), we see that (P 2 + i)f (P 2 ) − (P 1 + i)f (P 1 ) has an asymptotic expansion in OpS(w −∞ m −N x ′ −s ′ ).Proposition 5.2 shows that f (P 2 ) − (P 2 + i) −1 (P 1 + i)f (P 1 ) has an asymptotic expansion in OpS(w −∞ m −N −1 x ′ −s ′ ).We observe that Theorem 5 and Proposition 5.2 show that the second term also has an asymptotic expansion in OpS(w . Finally, we calculate the asymptotic expansion of f (P 2 ) − f (P 1 ), whose existence has been proved now.Take χ ∈ C ∞ c (R n ) which is equal to 1 on a large ball.We see from Theorem 5 that (f (P 2 ) − f (P 1 ))χ has an asymptotic expansion in OpS(m −∞ x ′ −s ′ ) which coincides with the formal calculation.Since χ is arbitrary, we conclude that the asymptotic expansion of f (P 2 ) − f (P 1 ) coincides with the formal one.

A Commutator calculation
In this Appendix, we assume that V ∈ C ∞ b (R n ; R) and set P = −∆+βx 1 +V (x).We denote Schwartz space and its dual by S and S ′ .To justify the commutator calculations in section 5, we prove the following; Proposition A.1.For Imz = 0, (P − z) −1 is continuous from S to S .Thus, there is a unique continuous extension (P − z) −1 : S ′ → S ′ and this is the inverse of P − z : S ′ → S ′ .In particular, Ker(P − z) = {0} on S ′ .This enables us to compute the commutator with the resolvent.To apply the perturbation argument, we introduce the Banach space Y N = k+s≤N H k,s , where H k,s is the weighted Sobolev space We only consider k, s ∈ Z ≥0 .The following proposition implies the Proposition A.1 since S = k,s≥0 H k,s including the topology.if the above calculation is justified.We next give a rigorous proof.We first assume that V = 0. We set P 0 = −∆ + βx 1 .Then we have an explicit diagonalization F x ′ exp(− i 3β D 3 1 )P 0 exp( i 3β D 3 1 )F −1 x ′ = |ξ ′ | 2 + βx 1 , where F x ′ is the Fourier transform with respect to x ′ .Since F x ′ exp(− i 3β D 3 1 ) and (|ξ ′ | 2 + βx 1 − z) −1 preserve S , we conclude that (P 0 − z) −1 preserves S .Thus Proposition A.1 and Corollary A.1 are true for V = 0. Then the above calculation is justified and the estimate (A.1) is true for P 0 − z.
Remark A.2.All the results in this Appendix are true for β = 0.The free diagonalization is of course the Fourier transform.If we replace |Imz| by dist(z, σ(P )) in the proof, the results in this case are also true for any z in the resolvent set C \ σ(P ).

Lemma 4 . 1 .
For any open set U with U