aa r X i v : . [ m a t h - ph ] F e b RESONANCES AS VISCOSITY LIMITS FOR BLACK BOXPERTURBATIONS
HAOREN XIONG
Abstract.
We show that the complex absorbing potential (CAP) method for com-puting scattering resonances applies to an abstractly defined class of black box per-turbations of the Laplacian in R n which can be analytically extended from R n toa conic neighborhood in C n near infinity. The black box setting allows a unifyingtreatment of diverse problems ranging from obstacle scattering to scattering on finitevolume surfaces. Introduction and statement of results
The complex absorbing potential (CAP) method has been used as a computationaltool for finding scattering resonances – see Riss–Meyer [RiMe95] and Seideman–Miller[SeMi92] for an early treatment and Jagau et al [J*14] for some recent developments.Zworski [Zw18] showed that scattering resonances of − ∆ + V , V ∈ L ∞ comp , are limits ofeigenvalues of − ∆ + V − iεx as ε →
0+ . The situation is very different for potentialsof the Wigner–von Neumann type, in which case Kameoka and Nakamura [KaNa20]showed that the corresponding limits exist away from a discrete set of thresholds. Usingan approach closer to [KaNa20] than [Zw18], the author extended Zworski’s result topotentials which are exponentially decaying [Xi20]. In this paper we show that the CAPmethod is also valid for an abstractly defined class of black box perturbations of theLaplacian in R n which can be analytically extended from R n to a conic neighborhoodin C n near infinity.We formulate black box scattering using the abstract setting introduced by Sj¨ostrandand Zworski in [SjZw91] except that the operator P is not assumed to be equal to − ∆near infinity. For that we follow Sj¨ostrand [Sj97] and assume that P is a dilationanalytic perturbation of − ∆ near infinity. The black box formalism allows an abstracttreatment of diverse scattering problems without addressing the details of specificsituations – see Examples 1–3 later in this section. We recall the setup as follows:Let H be a complex separable Hilbert space with an orthogonal decomposition: H = H R ⊕ L ( R n \ B (0 , R )) , (1.1)where B ( x, R ) = { y ∈ R n : | x − y | < R } and R is fixed. The corresponding orthogonalprojections will be denoted by u u | B (0 ,R ) , and u u | R n \ B (0 ,R ) or simply by the characteristic function 1 L of the corresponding set L . We consider an unboundedself-adjoint operator P : H → H with domain D . (1.2)We assume that D| R n \ B (0 ,R ) ⊂ H ( R n \ B (0 , R )) , (1.3)and conversely, u ∈ D if u ∈ H ( R n \ B (0 , R )) and u vanishes near B (0 , R ); and that1 B (0 ,R ) ( P + i ) − is compact . (1.4)We also assume that,1 R n \ B (0 ,R ) P u = Q ( u | R n \ B (0 ,R ) ) , for all u ∈ D ,Q = − n X j,k =1 ∂ x j ( g jk ( x ) ∂ x k ) + c ( x ) , g jk , c ∈ C ∞ b ( R n ) . (1.5)Here C ∞ b denotes the space of C ∞ functions with all derivatives bounded. Note thatif ψ ∈ C ∞ b ( R n ) is constant near B (0 , R ), then there is a natural way to define themultiplication: H ∋ u ψu ∈ H , and we have ψu ∈ D if u ∈ D .It is further assumed that Q is formally self-adjoint, i.e. g jk , c are real-valued func-tions on R n satisfying | n X j,k =1 g jk ( x ) ξ j ξ k | ≥ C − | ξ | , n X j,k =1 g jk ( x ) ξ j ξ k + c ( x ) → ξ , | x | → ∞ . (1.6)We will use the method of complex scaling – see § P .For that we follow [Sj97] to make the following assumptions:There exist θ ∈ [0 , π/ , δ > , and R ≥ R , such thatthe coefficients g jk ( x ) , c ( x ) of Q extend analytically in x to { sω : ω ∈ C n , dist( ω, S n − ) < δ, s ∈ C , | s | > R, arg s ∈ ( − δ, θ + δ ) } and the second half of (1.6) remains valid in this larger set. (1.7)We can now define the resonances z j of P in { z ∈ C \ { } : arg z > − θ } as theeigenvalues of P on a suitable contour in C n , see [SjZw91] and § regularized operator, P ε := P − iε (1 − χ ( x )) x , ε > , (1.8)where χ ∈ C ∞ c ( R n ) is equal to 1 near B (0 , R ); x := x + · · · + x n . It follows from § P ε is an unbounded operator on H with a discrete spectrum. We have ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 3
Theorem 1.
Suppose that { z j ( ε ) } ∞ j =1 are the eigenvalues of P ε . Then, uniformly onany compact subset of the sector { z ∈ C \ { } : − θ < arg z < π/ θ } , z j ( ε ) → z j , ε → , where z j are the resonances of P . Remark:
We will prove a more precise version of this theorem in §
6: it involvesthe multiplicities of z j and z j ( ε ) defined in § § Example 1. Obstacle scattering.
Suppose that
O ⊂ B (0 , R ) is an open set suchthat ∂ O is a smooth hypersurface in R n . Let H = L ( R n \ O ), and P = − ∆ | R n \O on the exterior domain realized with any self-adjoint boundary conditions on ∂ O . Forinstance, the Dirichlet boundary condition D = { u ∈ H ( R n \ O ) : u | ∂ O = 0 } or the Neumann/Robin boundary condition D = { u ∈ H ( R n \ O ) : ∂ ν u + ηu | ∂ O = 0 } where ∂ ν is the normal derivative with respect to ∂ O and η is a real-valued smoothfunction on ∂ O . Theorem 1 shows that the eigenvalues of P − iεx converge to theresonances of P (the irrelevance of the missing iεχ ( x ) x term comes from continuityof resonances under compactly supported perturbations – see Stefanov [St94]). Example 2. Scattering on asymptotically Euclidean space.
Let M be a realanalytic manifold which is diffeomorphic to R n near infinity and equipped with a realanalytic metric g which is asymptotically Euclidean. More precisely, let g ij = δ ij + h ij be the metric tensor then we assume that h ij ( x ) extend analytically in x to { sω : ω ∈ C n , dist( ω, S n − ) < δ, s ∈ C , | s | > R, arg s ∈ ( − δ, θ + δ ) } HAOREN XIONG for some θ ∈ [0 , π/ δ > R ≥ R , and that h ij → P = − ∆ g , the Laplace–Beltrami operator with respect to the metric g , thenall the black box assumptions are satisfied. Suppose that χ ∈ C ∞ c ( M ; [0 , K and that M \ K is diffeomorphic to R n \ B (0 , R ).Then the operator − ∆ g − iε (1 − χ ( x )) x has a discrete spectrum for ε > − ∆ g uniformly on compact subsets of − θ < arg z < π/ θ . Example 3. Scattering on finite volume surfaces.
This example was alreadydiscussed in [Zw18] but this paper provides a complete proof via the black box setting.Consider the modular surface M = SL ( Z ) \ H (or any surfaces with cusps – see[DyZw19, § g and ∆ M ≤ M . We choose the fundamental domain of SL ( Z ) to be { x + iy ∈ H : | x | ≤ / , x + y ≥ } then ∆ M in the cusp y > y ( ∂ x + ∂ y ). Let r = log y , θ = 2 πx , then M in ( r, θ ) coordinates admits the following decomposition: M = M ∪ M , ( M , g | M ) = ([0 , ∞ ) r × S θ , dr + (2 π ) − e − r dθ ) , S = R / π Z . We recall the black box setup in this case from [DyZw19, § H = H ⊕ L ([0 , ∞ ) , dr ) , H = L ( M ) ⊕ H , where (with Z ∗ := Z \ { } ) H = ( { a n ( r ) } n ∈ Z ∗ : a n ∈ L ([0 , ∞ )) , X n ∈ Z ∗ Z ∞ | a n ( r ) | dr < ∞ ) . We can identify L ( M ) with H via the following isomorphism: ι : L ( M ) ∋ u (cid:0) u | M , { e − r/ u n ( r ) } n ∈ Z ∗ , e − r/ u ( r ) (cid:1) ∈ H ,u n ( r ) := 12 π Z S u ( r, θ ) e − inθ dθ, r > . Then P := − ∆ M − / H which equals − ∂ r on L ([0 , ∞ ) , dr ) – see [DyZw19, § x, y ) coordinates P ε = − ∆ M − / − iε (1 − χ ( y ))(log y ) Π , Π u ( x, y ) := Z / − / u ( x ′ , y ) dx ′ . where χ ∈ C ∞ c ([0 , ∞ )), χ ( y ) ≡ y < χ ( y ) ≡ y >
3. The eigenvaluesof P ε converge to the resonances of P uniformly on compact subsets of arg z > − π/ s ( ε ) ∈ Σ ε ⇔ s ( ε )(1 − s ( ε )) − / ∈ Spec( P ε ), then the limitpoints of Σ ε , ε → s < /
2, arg( s − / = 11 π/ ζ (2 s ) where ζ is the Riemann zeta function – see [Zw18, Example 2] and[DyZw19, § ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 5
The paper is organized as follows. In § P as the eigenvalues of the complex scaled operator P θ .In § P ε has a discrete spectrum in C \ e − iπ/ [0 , ∞ ), which is invariantunder complex scaling. Since our operator is an abstract perturbation of − ∆, in § P ε,θ , ε ≥
0. More precisely, we use a reference operator reviewed in § N ε,θ ( z ) associated with P ε,θ and an artificial smoothobstacle O . The artificial obstacle problem is needed to separate the abstract blackbox from the differential operator outside. The operator N ε,θ ( z ) is well-defined for all z except for a discrete set depending on the obstacle, and we show that the eigenvalues of P ε,θ can be identified with the poles of z
7→ N ε,θ ( z ) − , with agreement of multiplicities.In § N ε,θ ( z ) iswell-defined near the resonances z j . The proof of Theorem 1 is completed in § N ε,θ ( z ). Notation.
We use the following notation: f = O ℓ ( g ) H means that k f k H ≤ C ℓ g wherethe norm (or any seminorm) is in the space H , and the constant C ℓ depends on ℓ .When either ℓ or H are absent then the constant is universal or the estimate is scalar,respectively. When G = O ℓ ( g ) : H → H then the operator G : H → H has its normbounded by C ℓ g . Also when no confusion is likely to result, we denote the operator f gf where g is a function by g . Acknowledgments.
The author would like to thank Maciej Zworski for helpfuldiscussions. This project was supported in part by the National Science Foundationgrant DMS-1901462. 2.
Preliminaries
Review of Complex Scaling.
Complex scaling has been a standard techniquein resonance theory since the works of Aguilar–Combes [AgCo71], Balslev–Combes[BaCo71] and Simon [Si79]. Here we follow rather closely the presentation in [Sj97]since our assumptions on the operator P is weaker than [SjZw91].A smooth submanifold Γ ⊂ C n is said to be totally real if T x Γ ∩ iT x Γ = { } forevery x ∈ Γ, where we identify T x Γ with a real subspace of T x C n ≃ C n . We say thatΓ is maximally totally real if Γ is totally real and of maximal (real) dimension n , thenatural example is Γ = R n . Let Γ ⊂ C n be smooth and of real dimension n , thenlocally Γ can be represented using real coordinates: R n ∋ x f ( x ) ∈ Γ. Let ˜ f be analmost analytic extension of f so that ¯ ∂ ˜ f vanishes to infinite order on R n . Let x ∈ R n ,then since d ˜ f ( x ) is complex linear, iT f ( x ) Γ = d ˜ f ( x )( iT x R n ). Hence Γ is totally real ina neighborhood of f ( x ) if and only if d ˜ f ( x ) is injective, i.e. det df ( x ) = 0. HAOREN XIONG
Let Ω ⊂ C n be an open neighborhood of Γ such that Γ is closed in Ω, and let A ( z, D z ) = X | α |≤ m a α ( z ) D αz , D z j := 1 i ∂ z j , D αz = D α z · · · D α n z n , be a differential operator on Ω with holomorphic coefficients. Define A Γ : C ∞ (Γ) →C ∞ (Γ) by A Γ u = ( A ˜ u ) | Γ , (2.1)where ˜ u is an almost analytic extension of u , that is, a smooth extension of u to aneighborhood of Γ such that ¯ ∂ ˜ u vanishes to infinite order on Γ. A Γ is then a differentialoperator on Γ with smooth coefficients, and for the principal symbols we have a Γ = a | T ∗ Γ , where a is the principal symbol of A .We recall a deformation result from [SjZw91, Lemma 3.1]: Lemma 2.1.
Suppose that W ⊂ R n is open and that F : [0 , × W ∋ ( s, x ) F ( s, x ) ∈ C n , is a smooth proper map satisfying for all s ∈ [0 , ∂ x F ( s, x ) = 0 , and x F ( s, x ) is injective,and assume that x ∈ W \ K = ⇒ F ( s, x ) = F (0 , x ) for some compact K ⊂ W .Let A ( z, D z ) be a differential operator with holomorphic coefficients defined in aneighborhood of F ([0 , × W ) such that for ≤ s ≤ and Γ s := F ( { s } × W ) , A Γ s iselliptic.If u ∈ C ∞ (Γ ) and A Γ u extends to a holomorphic function in a neighborhood of F ([0 , × W ) , then the same holds for u . The lemma will be applied to a family of deformations of R n in C n . We aim torestrict the operators P ε , ε ≥
0, to the corresponding totally real submanifolds. Forgiven α > R > R , we can construct a smooth function[0 , θ ] × [0 , ∞ ) ∋ ( θ, t ) g θ ( t ) ∈ C , injective for every θ , with the following properties:(i) g θ ( t ) = t for 0 ≤ t ≤ R ,(ii) 0 ≤ arg g θ ( t ) ≤ θ, ∂ t g θ ( t ) = 0,(iii) arg g θ ( t ) ≤ arg ∂ t g θ ( t ) ≤ arg g θ ( t ) + α ,(iv) g θ ( t ) = e iθ t for t ≥ T , where T depends only on α and R .We now define the totally real submanifolds, Γ θ , as images of R n under the maps f θ : R n ∋ x = tω g θ ( t ) ω ∈ C n , t = | x | . ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 7
Then a dilated operator P θ can be defined as follows. Let H θ = H R ⊕ L (Γ θ \ B (0 , R )) , where B (0 , R ) denotes the real ball as before. If χ ∈ C ∞ c ( B (0 , R )) is equal to 1 near B (0 , R ), we put D θ = { u ∈ H θ : χu ∈ D , (1 − χ ) u ∈ H (Γ θ \ B (0 , R )) } . Let P θ be the unbounded operator H θ → H θ with domain D θ , given by P θ u := P ( χu ) + Q θ ((1 − χ ) u ) , Q θ := − n X j,k =1 ( ∂ z j ( g jk ( z ) ∂ z k ) + c ( z )) | Γ θ . These definitions do not depend on the choice of χ .We recall some properties of the dilated Laplacian from [SjZw91, § θ := (∆ z ) | Γ θ , x θ := z | Γ θ . Parametrizing Γ θ by [0 , ∞ ) × S n − ∋ ( t, ω ) g θ ( t ) ω , we obtain − ∆ θ = ( g ′ θ ( t ) − D t ) − i ( n − g θ ( t ) g ′ θ ( t )) − D t + g θ ( t ) − D ω , (2.2)where D t = − i∂ t and D ω = − ∆ S n − . If ω ∗ denotes the principal symbol of D ω andwe let τ be the dual variable of t , then the principal symbol of − ∆ θ is σ ( − ∆ θ ) = g ′ θ ( t ) − τ + g θ ( t ) − ω ∗ , so pointwise on Γ θ , − ∆ θ is elliptic and the principal symbol takes values in an angle ofsize ≤ α , while globally, σ ( − ∆ θ ) takes values in the sector − θ − α ≤ arg z ≤ − θ + δ < arg z < π − θ − δ, | z | > δ = ⇒ ( − ∆ θ − z ) − = O δ ( | z | j − ) : L (Γ θ ) → H j (Γ θ ) , j = 0 , , . (2.3)This follows from [SjZw91, Lemmas 3.2–3.5 and §
4] applied with P = − ∆. P θ , as a perturbation of − ∆ θ , is also elliptic – see [Sj97, § R large enough, it follows from the assumptions (1.6) and (1.7) thatIn Γ θ \ B (0 , R ), P θ is an elliptic differential operator whose principalsymbol pointwise on Γ θ takes its values in an angle of size ≤ α , and globally in a sector − θ − α ≤ arg z ≤ α . (2.4)The coefficients of P θ − e − iθ ( − ∆) tend to zero when Γ θ ∋ x → ∞ , where we identify Γ θ and R n , by means of f θ . (2.5)We recall some basic results about P θ from [Sj97, § HAOREN XIONG
Lemma 2.2. If z ∈ C \ { } , arg z = − θ , then P θ − z : D θ → H θ is a Fredholmoperator of index . In particular the spectrum of P θ in C \ e − iθ [0 , ∞ ) is discrete.Proof. We follow closely the proof of [SjZw91, Lemma 3.2] (see also [DyZw19, Theorem4.36]) except that P θ is more general here. We shall invert P θ − z modulo compactoperators. On the complex contour Γ θ we introduce a smooth partition of unity:1 = χ + χ + χ with supp χ ⊂ B (0 , R ), supp χ contained in the region whereΓ θ ∋ x θ = e iθ x, x ∈ R n , supp χ compact and disjoint from B (0 , R ). Let ˜ χ j have thesame properties as the χ j except that they do not form a partition of unity, satisfying˜ χ j = 1 near supp χ j . Now we put E ( z ) = ˜ χ ( P − z ) − χ + S ( z ) χ + ˜ χ e iθ ( − ∆ − e iθ z ) − χ , (2.6)where z ∈ C \ R and S ( z ) is a properly supported parametrix of the elliptic operator P θ − z in Γ θ \ B (0 , R ). Then we have( P θ − z ) E ( z ) = I + K ( z ) + K ( z ) , (2.7)where K ( z ) = ( z − z ) ˜ χ ( P − z ) − χ + [ P, ˜ χ ]( P − z ) − χ + (( P θ − z ) S ( z ) − I ) χ +[ − e − iθ ∆ , ˜ χ ] e iθ ( − ∆ − e iθ z ) − χ ,K ( z ) = ( P θ − ( − e − iθ ∆)) ˜ χ e iθ ( − ∆ − e iθ z ) − χ . Using (2.5) we may assume that supp χ ⊂ { z ∈ C n : | z | ≥ T } for T sufficiently largesuch that k K ( z ) k H θ →H θ ≤ /
2, thus I + K ( z ) is invertible and we get( P θ − z ) E ( z )( I + K ( z )) − = I + K ( z )( I + K ( z )) − . It follows from the assumptions (1.3) and (1.4) that K ( z ) is compact: H θ → H θ , thus E ( z )( I + K ( z )) − is an approximate right inverse. The construction of an approximateleft inverse is similar, we omit the details and refer to [SjZw91, Lemma 3.2].It remains to show that P θ − z is invertible for some z ∈ C \ e − iθ [0 , ∞ ). For z = iL, L ≥
1, we can replace (2.6) with E ( z ) = ˜ χ ( P − z ) − χ + (1 − χ )( − ∆ θ − z ) − (1 − χ ) , (2.8)where χ ∈ C ∞ c ( B (0 , R )) is equal to 1 near supp χ and χ = 1 on B (0 , R − δ ), forsome δ > K ( z ), K ( z ) given by K ( z ) = [ P, ˜ χ ]( P − z ) − χ + [∆ θ , χ ]( − ∆ θ − z ) − (1 − χ ) ,K ( z ) = ( P θ − ( − ∆ θ ))(1 − χ )( − ∆ θ − z ) − (1 − χ ) . Choosing R sufficiently large, we may assume by (2.3) and (2.5) that k K ( z ) k H θ →H θ ≤ /
2, for all z = iL , L ≥
1. Then we get( P θ − z ) E ( z )( I + K ( z )) − = I + K ( z )( I + K ( z )) − . ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 9
It follows from (2.3) that K ( iL ) = O ( L − / ) : H θ → H θ , thus P θ − z is inveritible for z = iL, L ≫ P θ − z ) − = E ( z )( I + K ( z )) − ( I + K ( z )( I + K ( z )) − ) − , (2.9)which completes the proof. (cid:3) Lemma 2.3.
Assume that ≤ θ < θ ≤ θ and let z ∈ C \ e − i [ θ ,θ ] [0 , ∞ ) . Then dim ker( P θ − z ) = dim ker( P θ − z ) . This is identical to [SjZw91, Lemma 3.4] and the proof is the same as there usingLemma 2.1.Lemma 2.3 shows that the spectrum in the sector − θ < arg z ≤ θ in the following sense: We say that z ∈ C \ { } , − θ < arg z ≤ P if and only if z ∈ Spec( P θ ) with − θ < arg z ≤ θ ∈ (0 , θ ]. For sucha resonance z ∈ e − i [0 ,θ ) (0 , ∞ ), the spectral projectionΠ θ ( z ) = 12 πi I z ( z − P θ ) − dz, (2.10)where the integral is over a positively oriented circle enclosing z and containing noresonances other than z , is of finite rank. The restriction of P θ − z to Ran Π θ ( z ) isnilpotent. If ˜ θ ∈ [0 , θ ] is a second number with z ∈ e − i [0 , ˜ θ ) (0 , ∞ ), then since Lemma2.3 can be extended to dim ker( P θ − z ) k = dim ker( P ˜ θ − z ) k for all k , Π θ ( z ) andΠ ˜ θ ( z ) have the same rank, which by definition is the multiplicity of the resonance z : m ( z ) := rank Π θ ( z ) , − θ < arg z ≤ . (2.11)2.2. A reference operator.
As explained in §
1, to separate the abstract black boxfrom the differential operator outside we introduce a reference operator P O associatedwith an open set O ⊂ R n containing B (0 , R ). We assume that ∂ O is a smoothhypersurface in R n . In the notation of (1.1), we put H O := H R ⊕ L ( O \ B (0 , R )) . (2.12)The corresponding orthogonal projections are denoted by u B (0 ,R ) u = u | B (0 ,R ) , u O\ B (0 ,R ) u = u | O\ B (0 ,R ) . If P is a black box Hamiltonian introduced in § D , then we define D O := { u ∈ H O : ψ ∈ C ∞ c ( O ) , ψ = 1 near B (0 , R ) ⇒ ψu ∈ D , (1 − ψ ) u ∈ H ( O ) ∩ H ( O ) } (2.13)and, for any ψ with the property (2.13), P O : D O → H O ,P O u := P ( ψu ) + Q ((1 − ψ ) u ) . (2.14) Assumptions (1.3), (1.5) show that this definition is independent of the choice of ψ .We recall some basic properties of the reference operator from [SjZw91, § Lemma 2.4.
Suppose that
O ⊂ R n is an open set containing B (0 , R ) such that ∂ O is a smooth hypersurface in R n . Let P O be the reference operator defined in (2.14) .Then, with H O given by (2.12) , P O : H O → H O , is a self-adjoint operator with domain D O defined in (2.13) . Furthermore, the resolvent ( P O + i ) − is compact and thus P O has discrete spectrum which is contained in R . For the proof we refer to Dyatlov–Zworski [DyZw19, Lemma 4.11] and we remarkthat the arguments there is still valid if we replace the assumption there: P = − ∆ in R n \ B (0 , R ), by the assumption (1.5).3. The regularized operator
In this section we show that the spectrum of P ε is invariant under complex scaling.Choosing R such that supp χ ⊂ B (0 , R ) when we construct the complex contoursΓ θ , the complex absorbing potential − iε (1 − χ ( x )) x can be analytically extended toΓ θ , thus it defines a multiplication on the following subspace of H θ : b H θ := H R ⊕ | x θ | − L (Γ θ \ B (0 , R )) , where x θ := f θ ( x ) denotes the parametrization of Γ θ . We now introduce the deforma-tion of P ε on Γ θ , θ ∈ [0 , θ ): P ε,θ := P θ − iε (1 − χ ( x θ )) x θ , with the domain b D θ := D θ ∩ b H θ . (3.1)It follows from (2.5) that P ε,θ near infinity is close to the operator H ε,θ := − e − iθ ∆ − iεe iθ x , (3.2)which was considered by Davies [Da99] as an interesting example of a non-normaldifferential operator. We recall the following basic result: Lemma 3.1.
For ε > , ≤ θ < π/ , H ε,θ is a closed densely defined operator on L ( R n ) equipped with the domain H ( R n ) ∩ h x i − L ( R n ) . The spectrum is given by Spec( H ε,θ ) = { e − iπ/ √ ε (2 | α | + n ) : α ∈ N n } , | α | := α + · · · + α n . (3.3) In addition for any δ > we have uniformly in ε > , ( H ε,θ − z ) − = O δ ( | z | j − ) : L ( R n ) → H j ( R n ) , j = 0 , , , for − θ + δ < arg z < π/ θ − δ, | z | > δ. (3.4) ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 11
Proof.
For every ε > ≤ θ ≤ θ , H ε,θ can be viewed as the quantization of thecomplex-valued quadratic form q : R nx × R nξ → C , ( x, ξ ) e − iθ ξ − iεe iθ x , whichshall be viewed as a closed densely defined operator on L ( R n ) equipped with the do-main D ( H ε,θ ) := { u ∈ L ( R n ) : H ε,θ u ∈ L ( R n ) } . For the analysis of the spectrum forgeneral quadratic operators see Hitrik–Sj¨ostrand–Viola [HSV13] and references giventhere, in particular we obtain (3.3). Noticing that the numerical range of q is the sector { z ∈ C : 3 π/ θ ≤ arg z ≤ π − θ } , H ε,θ − i is elliptic with respect to the orderfunction m = 1 + x + ξ in the sense that | q − i | ≥ Cm for some C = C ( ε ) >
0. Since H ε,θ − i is invertible by (3.3), we conclude that( H ε,θ − i ) − : L ( R n ) → m − ( x, D x ) L ( R n ) = H ( R n ) ∩ h x i − L ( R n ) . Hence u ∈ D ( H ε,θ ) ⇒ u = ( H ε,θ − i ) − ( H ε,θ u − iu ) ∈ H ( R n ) ∩ h x i − L ( R n ). Nowwe rescale y = √ εx , then H ε,θ is unitary equivalent to − e − iθ ε ∆ y − ie iθ y , thatis a semiclassical quadratic operator with h = √ ε . The bounds (3.4) follow fromsemiclassical ellipticity of − e − iθ ε ∆ y − ie iθ y − z for − θ + δ < arg z < π/ θ − δ , | z | > δ . (cid:3) Then we show that P ε,θ is a Fredholm operator for z / ∈ e − iπ/ [0 , ∞ ). Lemma 3.2. If z ∈ C \ { } , arg z = − π/ , then for each ε > and ≤ θ < θ , P ε,θ − z : b D θ → H θ is a Fredholm operator of index . In particular the spectrum of P ε,θ in C \ e − iπ/ [0 , ∞ ) is discrete.Proof. We choose χ j ∈ C ∞ c (Γ θ ), j = 0 , , ,
3, such that χ j = 1 near supp χ j − andthat χ ( g θ ( t ) ω ) = 1 for any t ≤ T , thus 1 − χ j are supported in the region whereΓ θ ∋ x θ = e iθ x, x ∈ R n . Lemma 3.1 then shows that if arg z = − π/ − χ )( H ε,θ − z ) − (1 − χ ) : H θ → b D θ . Now we fix z ∈ C \ { } with arg z = − π/
4. Using (2.5) we may assume that supp χ is large enough so that k ( P ε,θ − H ε,θ )(1 − χ )( H ε,θ − z ) − (1 − χ ) k H θ →H θ ≤ /
2. Thenwe choose z = iL , L ≫ k ε ( χ − χ ) x θ ( P θ − z ) − k H θ →H θ ≤ / P θ − iε ( χ − χ ) x θ − z ) − = ( P θ − z ) − ( I − iε ( χ − χ ) x θ ( P θ − z ) − ) − exists. We put E ( z ) = χ ( P θ − iε ( χ − χ ) x θ − z ) − χ + (1 − χ )( H ε,θ − z ) − (1 − χ ) . Then we get ( P ε,θ − z ) E ( z ) = I + K ( z ) + K ( z ) , where K ( z ) = (( z − z ) χ + [ P θ , χ ])( P θ − iε ( χ − χ ) x θ − z ) − χ +[ e − iθ ∆ , χ ]( H ε,θ − z ) − (1 − χ ) K ( z ) = ( P ε,θ − H ε,θ )(1 − χ )( H ε,θ − z ) − (1 − χ ) . Recalling that k K ( z ) k H θ →H θ ≤ /
2, we obtain that I + K ( z ) is invertible thus( P ε,θ − z ) E ( z )( I + K ( z )) − = I + K ( z )( I + K ( z )) − . Since ( P θ − z ) − : H θ → D θ , we conclude that K ( z ) is compact: H θ → H θ . Hence E ( z )( I + K ( z )) is an approximate right inverse of P ε,θ − z .As an approximate left inverse, we put F ( z ) = χ ( P θ − iε ( χ − χ ) x θ − z ) − χ + (1 − χ )( H ε,θ − z ) − (1 − χ ) . Then F ( z )( P ε,θ − z ) = I + L ( z ) + L ( z ) , where L ( z ) = χ ( P θ − iε ( χ − χ ) x θ − z ) − (( z − z ) χ − [ P θ , χ ]) − (1 − χ )( H ε,θ − z ) − [ e − iθ ∆ , χ ] L ( z ) = (1 − χ )( H ε,θ − z ) − (1 − χ )( P ε,θ − H ε,θ ) . We may assume again by (2.5) that k L ( z ) k b D θ → b D θ ≤ /
2, then( I + L ( z )) − F ( z )( P ε,θ − z ) = I + ( I + L ( z )) − L ( z ) . Using (1.3), we see that [ e − iθ ∆ , χ ] is compact: b D θ → H θ , thus L ( z ) is compact: b D θ → b D θ , ( I + L ( z )) − F ( z ) is an approximate left inverse.Since k K ( z ) k H θ →H θ and k L ( z ) k b D θ → b D θ can be controlled by the operator normsof ( P − z ) − , ( − ∆ θ − z ) − and ( H ε,θ − z ) − . It then follows from (2.3) and (3.4)that k K ( z k H θ →H θ , k L ( z ) k b D θ → b D θ ≪ z = iL, L ≫
1, thus P ε,θ − iL isinvertible for L sufficiently large, which implies that P ε,θ has a discrete spectrum in C \ e − iπ/ [0 , ∞ ). (cid:3) Lemma 3.3.
For each ≤ θ < θ and ε > , let ψ ∈ C ∞ c ( B (0 , R ); [0 , be equal to near B (0 , R ) so that ψ is a function on Γ θ and defines a multiplication on H θ . Thenwe have, meromorphically in the region − π/ < arg z < π/ , ψ ( P ε − z ) − ψ = ψ ( P ε,θ − z ) − ψ. (3.5) Proof.
We modify the proof of [Zw18, Lemma 2]. It is sufficient to show that for0 ≤ θ < θ < θ , | θ − θ | ≪ ψ ( P ε,θ − z ) − ψ = ψ ( P ε,θ − z ) − ψ. (3.6) ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 13
It is also enough to establish this for z ∈ e i ( − θ + π/ (1 , ∞ ) as then the result follows byanalytic continuation. For that we show that for f ∈ H R ⊕ L ( B (0 , R ) \ B (0 , R )) ⊂H θ j there exists U holomorphic in a neighborhood Ω θ ,θ of [ θ ≤ θ ≤ θ (Γ θ \ B (0 , R )) ⊂ C n such that U | Γ θj ( x ) = [( P ε,θ j − z ) − ψf ]( x ) , ∀ x ∈ Γ θ j \ B (0 , R ) . (3.7)To show the existence of U such that (3.7) holds we apply Lemma 2.1 to a modifiedfamily of deformations, which is obtained as follows. Let ρ ∈ C ∞ c ((1 , , , T ≥ g θ ,θ ,T ( t ) := g θ ( t ) + ρ ( t/T )( g θ ( t ) − g θ ( t )) , Γ θ ,θ ,T := { g θ ,θ ,T ( t ) ω : t ∈ [0 , ∞ ) , ω ∈ S n − } ⊂ C n . We can apply Lemma 2.1 to the family of totally real submanifolds interpolatingbetween Γ θ and Γ θ ,θ ,T , [0 , ∋ s Γ θ , (1 − s ) θ + sθ ,T . It follows that there exists aholomorphic function U T defined in a neighborhood of the union of these submanifoldswhich restricts to u := ( P ε,θ − z ) − ψf ∈ H θ . Varying T we obtain a family offunctions agreeing on the intersections of their domains and that gives a holomorphicfunction U defined in the neighborhood Ω θ ,θ .It remains to show that U restricts to u ∈ H θ (the equation ( P ε,θ − z ) u = ψf isautomatically satisfied). For T large we putΩ ( T ) = { z ∈ C n : T ≤ | z | ≤ T } ∩ Γ θ ,θ ,T ⊃ Γ θ ,θ ,T \ Γ θ , Ω ( T ) = { z ∈ C n : T / ≤ | z | ≤ T } ∩ Γ θ ,θ ,T , Ω ( T ) \ Ω ( T ) ⊂ e iθ R n , and choose χ T ∈ C ∞ (Ω ( T ); [0 , χ T = 1 on Ω ( T ) with derivative boundsindependent of T . We recall the following estimate from the proof of [Zw18, Lemma3]: for u ∈ C ∞ (Γ θ ,θ ,T ), τ > | h ( − ∆ | Γ θ ,θ ,T − iε ( x | Γ θ ,θ ,T ) − ie − iθ τ ) u, u i | ≥ ( k u k L + k Du k L ) /C, with C > τ, T , here h· , ·i is the L inner product on Γ θ ,θ ,T . Writing P ε,θ ,θ ,T := P | Γ θ ,θ ,T − iε ( x | Γ θ ,θ ,T ) , it then follows from (1.5) that h ( P ε,θ ,θ ,T − ( − ∆ | Γ θ ,θ ,T − iε ( x | Γ θ ,θ ,T ) )) u, u i = Z Γ θ ,θ ,T ( g jk − δ jk ) ∂ k u∂ j ¯ u + c | u | . In view of (1.6) and (1.7), we obtain that for T sufficiently large, | h ( P ε,θ ,θ ,T − ie − iθ τ ) χ T U, χ T U i | ≥ ( k χ T U k L + k D ( χ T U ) k L ) /C, thus k χ T U k L ≤ C k ( P ε,θ ,θ ,T − ie − iθ τ ) χ T U k L . We note that( P ε,θ ,θ ,T − ie − iθ τ ) U T = 0 = ⇒ ( P ε,θ ,θ ,T − ie − iθ τ ) χ T U = [ P ε,θ ,θ ,T , χ T ] U, which is supported on Ω ( T ) \ Ω ( T ) ⊂ Γ θ . Hence, k T ≤| z |≤ T u k L (Γ θ ) ≤ C k [ P ε,θ ,θ ,T , χ T ] U k L ≤ C k T/ ≤| z |≤ T u k H (Γ θ ) . We now take T = 2 j and sum over j , it follows that u ∈ H θ . (cid:3) Lemma 3.4.
For ≤ θ < θ , ε > , the spectrum of P ε,θ is independent of θ . Moreprecisely, for any z ∈ C \ e − iπ/ [0 , ∞ ) we have m ε,θ ( z ) := rank I z ( P ε,θ − z ) − dz = rank I z ( P ε − z ) − dz, (3.8) where the integral is over a positively oriented circle enclosing z and containing nopoles other than possibly z .Proof. Lemma 3.2 shows thatΠ ε,θ ( z ) := − πi I z ( P ε,θ − z ) − dz, (3.9)is a finite rank projection which maps H θ to the generalized eigenspace of P ε,θ at z .In view of Lemma 3.3, it suffices to show that for each 0 ≤ θ < θ ,rank Π ε,θ ( z ) = rank ψ Π ε,θ ( z ) ψ. First we show that rank Π ε,θ ( z ) = rank Π ε,θ ( z ) ψ , which is equivalent to show thatrank ψ Π ε,θ ( z ) ∗ = rank Π ε,θ ( z ) ∗ , since the adjoint of a finite rank operator is of finiterank with the same rank. For that we shall argue by contradiction. Suppose thatrank ψ Π ε,θ ( z ) ∗ < rank Π ε,θ ( z ) ∗ , there would exist 0 = ˜ v ∈ Ran Π ε,θ ( z ) ∗ satisfying ψ ˜ v = 0. Note that Π ε,θ ( z ) ∗ is also a projection of the form (3.9) except that P ∗ ε,θ and¯ z replace P ε,θ and z there, we may assume( P ∗ ε,θ − ¯ z ) k ˜ v = 0 , ˜ u := ( P ∗ ε,θ − ¯ z ) k − ˜ v = 0 , for some k ≥ . But that would mean that ˜ u can be identified with an element of H (Γ θ ) satisfying( Q ∗ ε,θ − ¯ z )˜ u = 0 , ˜ u | B (0 ,R ) ≡ , Q ε,θ := Q θ − iε (1 − χ ( x θ )) x θ . Since Q ∗ ε,θ is elliptic, unique continuation results for second order elliptic differentialequations – see H¨ormander [H¨oIII, Chapter 17] show that ˜ u ≡
0, thus a contradiction.It remains to show that rank ψ Π ε,θ ( z ) ψ = rank Π ε,θ ( z ) ψ . Otherwise there wouldexist solutions v ∈ b D θ to ( P ε,θ − z ) ℓ v = 0, u := ( P ε,θ − z ) ℓ − v = 0 with ψv = 0. Itfollows that u can be identified with an element of H (Γ θ ) satisfying( Q ε,θ − z ) u = 0 , u | B (0 ,R ) ≡ . ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 15
Again by the unique continuation results for second order elliptic differential equations,we obtain that u ≡
0, thus a contradiction. (cid:3)
The next lemma shows that the spectrum of P ε,θ must stay close to the spectrum of P θ when ε is sufficiently small: Lemma 3.5.
Suppose that ≤ θ < θ and that Ω ⋐ { z : − θ < arg z < π/ θ } isdisjoint with Spec( P θ ) , then there exist ε = ε (Ω) and C = C (Ω) such that, uniformlyin < ε < ε , Spec( P ε,θ ) ∩ Ω = ∅ and k ( P ε,θ − z ) − k H θ →D θ ≤ C, z ∈ Ω . Proof.
We follow closely the proof of [Zw18, Lemma 5] except that P θ replaces − ∆ θ there. Let χ j ∈ C ∞ c ([0 , ∞ ); [0 , , R ] and satisfy χ j = 1 nearsupp χ j − , j = 1 ,
2. Parametrizing Γ θ by f θ : [0 , ∞ ) × S n − ∋ ( t, ω ) g θ ( t ) ω ∈ Γ θ , wedefine functions χ hj ∈ C ∞ c (Γ θ ) by χ hj ( g θ ( t ) ω ) := χ j ( th ) , < h ≤ . For z ∈ Ω we put E hε,θ ( z ) := χ h ( P θ − z ) − χ h + (1 − χ h )( H ε,θ − z ) − (1 − χ h ) , so that ( P ε,θ − z ) E hε,θ ( z ) = I + K hε,θ ( z ), where K hε,θ ( z ) := − iε (1 − χ ) x θ χ h ( P θ − z ) − χ h + [ P θ , χ h ]( P θ − z ) − χ h + ( P ε,θ − H ε,θ )(1 − χ h )( H ε,θ − z ) − (1 − χ h ) − [ P θ , χ h ](1 − χ h )( H ε,θ − z ) − (1 − χ h ) . Using (2.5) and (3.4) we see that for h small enough, k ( P ε,θ − H ε,θ )(1 − χ h )( H ε,θ − z ) − (1 − χ h ) k L (Γ θ ) → L (Γ θ ) < / . Since [ Q θ , χ hj ] = O ( h ) : H (Γ θ ) → L (Γ θ ) and x θ χ h = O ( h − ) : L (Γ θ ) → L (Γ θ ),we can first choose h sufficiently small then there exists ε = ε ( h, Ω) such that forall ε < ε ( h, Ω) and z ∈ Ω, k K hε,θ ( z ) k H θ →H θ < /
2, thus I + K hε,θ ( z ) has a uniformlybounded inverse and ( P ε,θ − z ) − = E hε,θ ( z )( I + K hε,θ ( z )) − exists. It follows from (3.4)that there exists C = C (Ω) independent of ε such that for z ∈ Ω, k E hε,θ ( z ) k H θ →D θ ≤ C ,which completes the proof. (cid:3) The obstacle problem and the Dirichlet-to-Neumann operator
In the black box case we cannot use the strategy of [Zw18] which covers the case P = − ∆ + V , V ∈ L ∞ comp . Instead we introduce an artificial obstacle to separate theabstract black box from the differential operator outside. By an obstacle we mean anopen set O with smooth boundary as in § O contains B (0 , R ) and that χ in (1.8) be equal to 1 near O . Let ν ( x ) be the Euclidean normal vector of ∂ O at x pointing into O , we put ν g ( x ) := ( g jk ( x )) n × n · ν ( x ) , x ∈ ∂ O . (4.1)First we introduce the interior Dirichlet-to-Neumann operator of P : N in P ( z ) ϕ := ∂u∂ν g , where u solves the problem ( P − z ) u = 0 in O u = ϕ on ∂ O . (4.2) N in P ( z ) is well-defined once we establish the existence and uniqueness of the solution u to the boundary-value problem in (4.2). This can be done if z is not an eigenvalue of theoperator P O introduced in § E in : H / ( ∂ O ) → H ( O ) as a linearbounded extension operator such that E in ϕ | ∂ O = ϕ and supp E in ϕ ⊂ O \ B (0 , R ) forany ϕ . Then for z / ∈ Spec( P O ), u = E in ϕ − ( P O − z ) − ( Q − z ) E in ϕ is the uniquesolution to (4.2), we obtain that N in P ( z ) ϕ = ∂ ν g ( E in ϕ − ( P O − z ) − ( Q − z ) E in ϕ ) , (4.3)Hence z
7→ N in P ( z ) : H / ( ∂ O ) → H / ( ∂ O ) is a meromorphic family of operators on C with poles contained in Spec( P O ).Similarly, we can define the exterior Dirichlet-to-Neumann operator of P ε,θ for every0 ≤ θ < θ and ε ≥ N out ε,θ ( z ) ϕ := ∂u∂ν g , where u solves the problem ( Q ε,θ − z ) u = 0 in Γ θ \ O u = ϕ on ∂ O . (4.4)To show the well-definedness of N out ε,θ ( z ), we introduce the restriction of Q ε,θ to Γ θ \ O with Dirichlet boundary condition as follows: Q O θ : H (Γ θ \ O ) ∩ H (Γ θ \ O ) → L (Γ θ \ O ) , Q O θ u := Q θ u,Q O ε,θ := Q O θ − iε (1 − χ ) x θ with domain D ( Q O θ ) ∩ | x θ | − L (Γ θ \ O ) . (4.5)Since Q O θ and Q O ε,θ can also be viewed as black box perturbations of − ∆ θ and H ε,θ respectively, we conclude from Lemma 2.2 and Lemma 3.2 that Q O ε,θ − z, ε ≥ − θ < arg z < π/ θ . We claim that N out ε,θ ( z ) iswell defined if z / ∈ Spec( Q O ε,θ ). For that let E out : H / ( ∂ O ) → H (Γ θ \ O ) be a linearbounded extension operator with E out ϕ | ∂ O = ϕ and supp E out ϕ ⋐ Γ θ \ O , then N out ε,θ ( z ) ϕ = ∂ ν g ( E out ϕ − ( Q O ε,θ − z ) − ( Q ε,θ − z ) E out ϕ ) . (4.6)It follows that z
7→ N out ε,θ ( z ) : H / ( ∂ O ) → H / ( ∂ O ) is a meromorphic family ofoperators in the region − θ < arg z < π/ θ , with poles contained in Spec( Q O ε,θ ).Now we put N ε,θ ( z ) := N out ε,θ ( z ) − N in P ( z ) . (4.7) ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 17
Lemma 4.1.
Suppose that ≤ θ < θ , ε ≥ and that − θ < arg z < π/ θ with z / ∈ Spec( P O ) ∪ Spec( Q O ε,θ ) , then N ε,θ ( z ) : H / ( ∂ O ) → H / ( ∂ O ) is a Fredholmoperator of index .Proof. Let Q O in and N in Q ( z ) be the the reference operator and the interior Dirichlet-to-Neumann operator associated with Q , defined as in (2.14) and (4.2) respectivelyexcept that Q replaces P there. Choosing z / ∈ Spec( Q O in ) ∪ Spec( Q O ε,θ ) ∪ Spec( Q ε,θ ),we claim that N out ε,θ ( z ) − N in Q ( z ) : H / ( ∂ O ) → H / ( ∂ O ) is invertible. (4.8)To show injectivity, we argue by contradiction. Suppose that 0 = ϕ ∈ H / ( ∂ O )satisfies N out ε,θ ( z ) ϕ = N in Q ( z ) ϕ , it follows from (4.2) and (4.4) that there exist u ∈ H ( O ), u ∈ H (Γ θ \ O ) ( | x θ | u ∈ L (Γ θ \ O ) when ε >
0) such that u solves ( Q − z ) u = 0 in O u = ϕ on ∂ O , and u solves ( Q ε,θ − z ) u = 0 in Γ θ \ O u = ϕ on ∂ O , (4.9)and that ∂ ν g u = ∂ ν g u . Let u = 1 O u + 1 Γ θ \O u , we aim to show that u ∈ H (Γ θ ).For that it suffices to show the regularity of u near ∂ O . For any x ∈ ∂ O , we choose B x := B ( x , r ) ⊂ B (0 , R ) such that Q ε,θ = Q in B x and put v ∈ C ∞ c ( B x ). Then weintegrate by parts to obtain: Z B x n X j,k =1 g jk ∂ x k u ∂ x j v + cuv ! dx = Z B x ∩O n X j,k =1 g jk ∂ x k u ∂ x j v + cu v ! dx + Z B x \O n X j,k =1 g jk ∂ x k u ∂ x j v + cu v ! dx = Z B x ∩O v Qu dx − Z ∂ O∩ B x v ∂ ν g u dS ( x ) + Z B x \O v Qu dx + Z ∂ O∩ B x v ∂ ν g u dS ( x )= Z B x ∩O z u v dx + Z B x \O z u v dx = Z B x z uv dx. Hence u is a weak solution of ( Q − z ) u = 0 in B x , the regularity results for secondorder elliptic differential equations show that u is H near x , thus u ∈ H (Γ θ ). It thenfollows from (4.9) that u solves the equation ( Q ε,θ − z ) u = 0, thus z ∈ Spec( Q ε,θ ),which gives a contradiction.To show surjectivity, we first choose a linear bounded operator L g : H / ( ∂ O ) → H ( O ) satisfying the following: L g ˜ ϕ := v, where v ∈ H ( O ) ∩ H ( O ) satisfiessupp v ⊂ O \ B (0 , R ) and ∂ ν g v = ˜ ϕ, ˜ ϕ ∈ H / ( ∂ O ) . (4.10) For any ˜ ϕ ∈ H / ( ∂ O ), let v := L g ˜ ϕ , f := ( Q O in − z ) v ∈ L ( O ) and we put u := ( Q ε,θ − z ) − ıf and ϕ := u | ∂ O ∈ H / ( O ) , where ı : L ( O ) ֒ → L (Γ θ ) denotes the extension by zero. Then u := 1 O u − v solvesthe boundary value problem ( Q − z ) u = 0 in O , u = ϕ on ∂ O ; u := 1 Γ θ \O u solves( Q ε,θ − z ) u = 0 in Γ θ \ O , u = ϕ on ∂ O . Hence we have N out ε,θ ( z ) ϕ − N in Q ( z ) ϕ = ∂ ν g Γ θ \O u − ∂ ν g (1 O u − v ) = ∂ ν g v = ˜ ϕ. In view of (4.8), it now suffices to show that N out ε,θ ( z ) −N out ε,θ ( z ) and N in P ( z ) −N in Q ( z )are compact: H / ( ∂ O ) → H / ( ∂ O ). Using (4.6) we have for any ϕ ∈ H / ( O ), N out ε,θ ( z ) ϕ − N out ε,θ ( z ) ϕ = ∂ ν g (( Q O ε,θ − z ) − ( Q ε,θ − z ) − ( Q O ε,θ − z ) − ( Q ε,θ − z )) E out ϕ = ( z − z ) ∂ ν g ( Q O ε,θ − z ) − ( I − ( Q O ε,θ − z ) − ( Q ε,θ − z )) E out ϕ ∈ H / ( ∂ O ) , thus N out ε,θ ( z ) − N out ε,θ ( z ) : H / ( ∂ O ) → H / ( ∂ O ) ⊂ H / ( ∂ O ) is compact since thelast inclusion map is compact. It remains to show that N in P ( z ) − N in Q ( z ) is compact: H / ( ∂ O ) → H / ( ∂ O ). Let ψ ∈ C ∞ c ( O ) be equal to 1 near B (0 , R ), ϕ ∈ H / ( O ),there exist u and v satisfying:( P − z ) u = 0 in O u = ϕ on ∂ O and ( Q − z ) v = 0 in O v = ϕ on ∂ O , recalling (2.13) that (1 − ψ ) u ∈ H ( O ), thus we have( N in P ( z ) − N in Q ( z )) ϕ = ∂ ν g ((1 − ψ ) u − (1 − ψ ) v ) . Using (1.5) we can show that (1 − ψ ) u − (1 − ψ ) v ∈ H ( O ) ∩ H ( O ) satisfies: Q ((1 − ψ ) u − (1 − ψ ) v ) = (1 − ψ ) P u − [ P, ψ ] u − (1 − ψ ) Qv + [ Q, ψ ] v = z (1 − ψ ) u − z (1 − ψ ) v − [ P, ψ ] u + [ Q, ψ ] v ∈ H ( O ) , then we conclude from the regularity results for second order elliptic differential equa-tions that (1 − ψ ) u − (1 − ψ ) v ∈ H ( O ), thus ( N in P ( z ) − N in Q ( z )) ϕ ∈ H / ( ∂ O ).Therefore, N in P ( z ) − N in Q ( z ) : H / ( ∂ O ) → H / ( ∂ O ) ⊂ H / ( ∂ O ) is compact, whichcompletes the proof. (cid:3) Remark:
The compactness of N out ε,θ ( z ) − N out ε,θ ( z ) and N in P ( z ) − N in Q ( z ) can also beproved using the facts that the principal symbols of N out ε,θ ( z ) and N out ε,θ ( z ) are identical,same for N in P ( z ) and N in Q ( z ) – see for instance Lee–Uhlmann [LeUh89] for a detailedaccount.In order to work on a single Hilbert space, we introduce b N ε,θ ( z ) := h D ∂ O i − N ε,θ ( z ) : H / ( ∂ O ) → H / ( ∂ O ) , (4.11) ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 19 where h D ∂ O i = (1 − ∆ ∂ O ) / is the standard isomorphism between Sobolev spaces H s ( ∂ O ) and H s − ( ∂ O ). Now we are ready to state the main results of this section: Lemma 4.2.
Suppose that ≤ θ < θ , ε ≥ and that Ω ⋐ { z : − θ < arg z < π/ θ } is disjoint from Spec( P O ) ∪ Spec( Q O ε,θ ) , z b N ε,θ ( z ) − , z ∈ Ω , is a meromorphic family of operators on H / ( ∂ O ) with poles of finite rank. Moreover, n ε,θ ( z ) := 12 πi tr I z b N ε,θ ( w ) − ∂ w b N ε,θ ( w ) dw = m ε,θ ( z ) , (4.12) where the integral is over a positively oriented circle enclosing z and containing nopoles other than possibly z and m ε,θ ( z ) is given by (3.8) (and by (2.11) when ε = 0 ).Proof.
1. Suppose that z ∈ Ω is an eigenvalue of P ε,θ , we choose u ∈ ker( P ε,θ − z )and let ϕ = u | ∂ O , then N out ε,θ ( z ) ϕ − N in P ( z ) ϕ = ∂ ν g u − ∂ ν g u = 0. Note that ϕ = 0since z / ∈ Spec( P O ), thus ker b N ε,θ ( z ) = { } . On the other hand, suppose that 0 = ϕ ∈ ker b N ε,θ ( z ), the same arguments as in the proof of Lemma 4.1 show that z ∈ Spec( P ε,θ ).Hence z ∈ Spec( P ε,θ ) ⇐⇒ ker b N ε,θ ( z ) = { } , (4.13)and we conclude from Lemma 4.1 that b N ε,θ ( z ) is invertible for z ∈ Ω \ Spec( P ε,θ ).Analytic Fredholm theory then shows that Ω ∋ z b N ε,θ ( z ) − is a meromorphicfamily of operators on H / ( ∂ O ) with poles of finite rank.2. Since (4.13) proves (4.12) in the case m ε,θ ( z ) = 0, we now assume that m ε,θ ( z ) = M ≥
1, and that P ε,θ has exactly one eigenvalue z in D ( z, r ) := { ζ ∈ C , | ζ − z | < r } .Since Ω ∩ Spec( P O ) = ∅ , z is not a compactly supported embedded eigenvalue of P ,that is, there does not exist 0 = u ∈ D with supp u ⊂ B (0 , R ) such that ( P − z ) u = 0.We claim that for any δ > V ∈ C ∞ ( O \ B (0 , R ); R ) with k V k ∞ < δ suchthat rank Z ∂D ( z,r ) ( P ε,θ + V − w ) − dw = M, and that the eigenvalues of P ε,θ + V in D ( z, r ) are all simple. This follows from theresults of Klopp–Zworski [KlZw95] (see also [DyZw19, Theorem 4.39]) and we omitthe proof here. Replacing P by P + V in (4.2), we can define b N Vε,θ for P ε,θ + V as in(4.7) and (4.11). Note that b N ε,θ has no kernel except at z in D ( z, r ) by (4.13), using(4.3) we can choose δ small enough such that for k V k ∞ < δ , k b N ε,θ ( w ) − ( b N ε,θ ( w ) − b N Vε,θ ( w )) k H / ( O ) → H / ( O ) < , ∀ w ∈ ∂D ( z, r ) . It then follows from the Gohberg–Sigal–Rouch´e theorem (see Gohberg–Sigal [GoSi71]and [DyZw19, Appendix C]) that12 πi tr Z ∂D ( z,r ) N Vε,θ ( w ) − ∂ w N Vε,θ ( w ) dw = n ε,θ ( z ) . Hence it is enough to prove (4.12) in the case m ε,θ ( z ) = 1 with P ε,θ replaced by P ε,θ + V .3. Now we assume that m ε,θ ( z ) = 1. In view of (4.13), b N ε,θ ( w ) − has a pole at z ,it remains to show that z is a simple pole. For any w near z and ˜ ϕ ∈ H / ( ∂ O ), werecall (4.10) that L g ˜ ϕ ∈ D O , then ( P O − w ) L g ˜ ϕ ∈ H O . Now we put u := ( P ε,θ − w ) − ı ( P O − w ) L g ˜ ϕ, ϕ := u | ∂ O , where ı : H O ֒ → H θ is the extension by zero. Following the arguments in the proof ofLemma 4.1 while P replacing Q there, we can show that N ε,θ ( w ) ϕ = ˜ ϕ , thus b N ε,θ ( w ) − ˜ ϕ = (( P ε,θ − w ) − ı ( P O − w ) L g ( h D ∂ O i ˜ ϕ )) | ∂ O , ∀ ˜ ϕ ∈ H / ( ∂ O ) . Since z is a simple pole of w ( P ε,θ − w ) − by our assumptions, it follows from theexpression above that z must be a simple pole of w b N ε,θ ( w ) − . (cid:3) Deformation of obstacles
We have shown that the eigenvalues of P ε,θ , ε ≥
0, can be identified with the poles of z
7→ N ε,θ ( z ) − . One problem of this characterization is that N ε,θ ( z ) can only be definedaway from Spec( P O ) and Spec( Q O ε,θ ). In this section we will show that the spectrumof P O and Q O θ can be moved by deforming the obstacle O . Hence for any resonance z of P , we can always assume that N θ ( z ) is well-defined in some neighborhood of z by selecting a proper obstacle.To describe the deformations of obstacles, we follow Pereira [Pe04] and introduce aset of smooth mappings which deforms the obstacle O :Diff( O ) := ( Φ ∈ C ∞ ( R n ; R n ) is a diffeomorphism : Φ( ∂ O ) = ∂ Φ( O ) , Φ( x ) = x, for all | x | ≤ R or | x | ≥ R . ) (5.1)We note that Φ ∈ Diff( O ) only deforms the region { x ∈ R n : R < | x | < R } , then italso defines a diffeomorphism of Γ θ , 0 ≤ θ < θ . The pullback Φ ∗ gives an isomorphismbetween L (Γ θ \ Φ( O )) and L (Γ θ \ O ), which also restricts to an isomorphism between D ( Q Φ( O ) θ ) and D ( Q O θ ) given in (4.5) since it preserves the Dirichlet boundary condition.Hence we can define the deformed operator of Q O θ associated with the deformation Φas follows: Q O θ, Φ := Φ ∗ Q Φ( O ) θ (Φ ∗ ) − , with D ( Q O θ, Φ ) = D ( Q O θ ) . (5.2)The Fredholm properties of Q Φ( O ) θ − z immediately show that Q O θ, Φ − z is a Fredholemoperator of index 0 for − θ < arg z < π/ θ , and (5.2) implies that the spectrum of ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 21 Q O θ, Φ in this region is identical to the spectrum of Q Φ( O ) θ . Moreover, Q O θ, Φ can be viewedas a restriction of Q θ, Φ := Φ ∗ Q θ (Φ ∗ ) − to Γ θ \ O with Dirichlet boundary condition. Adirect calculation shows that A Φ := Φ ∗ Q θ (Φ ∗ ) − − Q θ = Φ ∗ Q (Φ ∗ ) − − Q = X | α |≤ a α ( x ) ∂ αx , (5.3)where the coefficients a α are supported in B (0 , R ) \ B (0 , R ) ⊂ Γ θ . We note that k a α k ∞ ≤ C k Φ − id k C , thus A Φ = O ( k Φ − id k C ) : H (Γ θ ) → L (Γ θ ).Now we show that Spec( Q O θ ) can be moved by deforming the obstacle: Lemma 5.1.
Suppose that the obstacle
O ⊂ B (0 , R ) contains B (0 , R ) and that − θ < arg z < π/ θ , then for any δ > there exists Φ ∈ Diff( O ) with k Φ − id k C < δ such that z / ∈ Spec( Q Φ( O ) θ ) .Proof. We may assume that z ∈ Spec( Q O θ ), otherwise we can take Φ = id. Supposethat Q O θ has exactly one eigenvalue in D ( z , r ). For D := D ( z , r ) we defineΠ O ( D ) := − πi Z ∂D ( Q O θ − ζ ) − dζ , m O ( D ) := rank Π O ( D ) , (5.4)then m O ( D ) = m O ( z ), where m O ( z ) denotes the multiplicity of z ∈ Spec( Q O θ ).For δ > U δ ( O ) := { Φ ∈ Diff( O ) : k Φ − id k C ( R n \O ) < δ } . It follows from (5.3) that Q O θ, Φ − Q O θ = O ( k Φ − id k C ) : H (Γ θ \ O ) → L (Γ θ \ O ), thusfor Φ ∈ U δ ( O ) with δ sufficiently small,( Q O θ, Φ − ζ ) − = ( Q O θ − ζ ) − ( I + ( Q O θ, Φ − Q O θ )( Q O θ − ζ ) − ) − , ζ ∈ ∂D, exists and sup ζ ∈ ∂D k ( Q O θ, Φ − ζ ) − − ( Q O θ − ζ ) − k L (Γ θ \O ) → L (Γ θ \O ) < C (Ω) δ . We defineΠ Φ ( D ) := − πi Z ∂D ( Q O θ, Φ − ζ ) − dζ , m Φ ( D ) := rank Π Φ ( D ) , (5.5)then Π Φ ( D ) and Π O ( D ) have the same rank for any Φ ∈ U δ ( O ) if δ is sufficiently small.Since m Φ ( D ) = m Φ( O ) ( D ) by (5.2), we conclude that m Φ( O ) ( D ) is constant for Φ ∈ U δ ( O ) if δ is sufficiently small . (5.6)We note that for every z and O , one of the following cases has to occur: ∀ δ > , ∃ Φ ∈ U δ ( O ) such that m Φ( O ) ( z ) < m Φ( O ) ( D ) , (5.7)or ∃ δ > , such that ∀ Φ ∈ U δ ( O ) , m Φ( O ) ( z ) = m Φ( O ) ( D ) . (5.8)Assuming (5.7) we can prove the lemma by induction on m O ( z ). If m O ( z ) = 1, (5.6)shows that m Φ( O ) ( D ) = 1 for Φ ∈ U δ ( O ) with δ small. It then follows from (5.7) that we can find Φ ∈ U δ ( O ) such that m Φ( O ) ( z ) <
1, i.e. z / ∈ Spec( Q Φ( O ) θ ). Assuming thatwe proved the lemma in the case m O ( z ) < M , we now assume that m O ( z ) = M . Wenote that for any Φ ∈ Diff( O ) and Φ ∈ Diff(Φ ( O )), k Φ ◦ Φ − id k C ≤ C ( k Φ − id k C + k Φ − id k C ) , where C is a constant depending only on the dimension n . For any δ >
0, (5.7) impliesthat we can find Φ ∈ Diff( O ) with k Φ − id k C < δ/ C such that m Φ ( O ) ( z ) < M .It then follows from our induction hypothesis that there exists Φ ∈ Diff(Φ ( O )) with k Φ − id k C < δ/ C such that z / ∈ Spec( Q Φ (Φ ( O )) θ ). We now take Φ = Φ ◦ Φ , thenΦ ∈ U δ ( O ) and z / ∈ Spec( Q (Φ( O ) θ ).It remains to show that (5.8) is impossible. For that, we shall argue by contradiction,assume that m O ( D ) = M and that (5.8) holds. For Φ ∈ U δ ( O ), we define k (Φ) := min { k : ( Q O θ, Φ − z ) k Π Φ ( D ) = 0 } , then 1 ≤ k (Φ) ≤ M . It follows from (5.2) and (5.5) that if k Φ j − Φ k C M → Q O θ, Φ j − z ) k Π Φ j ( D ) = 0, then ( Q O θ, Φ − z ) k Π Φ ( D ) = 0. We now put k := max { k (Φ) : Φ ∈ U δ/ ( O ) } , and assume that the maximum is attained at Φ ∈ U δ/ ( O ) i.e. k (Φ ) = k , then thereexists δ ′ > k Φ − Φ k C M < δ ′ ⇒ k (Φ) = k . Henceforth, we can replaceour original obstacle O with Φ ( O ), decrease δ and then assume by (5.8) that( Q O θ, Φ − z ) k Π Φ ( D ) = 0 , ( Q O θ, Φ − z ) k − Π Φ ( D ) = 0 ,m Φ ( z ) = rank Π Φ ( D ) = M, ∀ Φ ∈ Diff( O ) , k Φ − id k C M < δ. (5.9)Before proving that (5.9) is impossible we introduce a family of deformations inDiff( O ) acting near a fixed point on ∂ O . For any fixed x ∈ ∂ O and some h > χ h ∈ C ∞ ( ∂ O ; [0 , ∞ )) depending continuouslyin h ∈ (0 , h ] with Z ∂ O χ h ( x ) dS ( x ) = 1 , supp χ h ⊂ B ∂ O ( x , h ) , ∀ h ∈ (0 , h ] , (5.10)where B ∂ O ( x , h ) denotes the geodesic ball on ∂ O with center x and radius h . Foreach h ∈ (0 , h ], we construct a smooth vector field V h ∈ C ∞ c ( R n ; R n ) with some smallconstant δ h = O ( h M + n − ) such that V h ( x ) = δ h χ h ( x ) ν g ( x ) , ∀ x ∈ ∂ O , k V h k C M < ε/ , supp V h ⊂ B R n ( x , Ch ) for some C > , (5.11)where ν ( x ) is the normal vector at x ∈ ∂ O pointing inward. Let ϕ th : R n → R n be theflow generated by the vector field V h . It follows from (5.11) that for every h ∈ (0 , h ] ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 23 there exists t > ϕ th ∈ Diff( O ) , k ϕ th − id k C M < δ, ∀ t ∈ ( − t , t ) . Assuming (5.9) we can find w ∈ L (Γ θ \ O ) so that u := ( Q O θ − z ) k − Π O ( D ) w = 0.For any fixed x ∈ ∂ O and h ∈ (0 , h ], we take Φ t := ϕ th , t ∈ ( − t , t ) and put u ( t ) := (Φ t − ) ∗ v ( t ) , v ( t ) := ( Q O θ, Φ t − z ) k − Π Φ t ( D ) w. In view of (5.2), ( Q O θ, Φ t − z ) v ( t ) = 0 implies that( Q θ − z ) u ( t ) = 0 in Γ θ \ Φ t ( O ) . (5.12)Since Φ t ( O ) ⊂ O for t ≥
0, we can restrict (5.12) to the region Γ θ \ O then differentiateit in t , by taking t = 0, we obtain that( Q θ − z ) u ′ (0) = 0 in Γ θ \ O . (5.13)Recalling that u ( t, x ) = v ( t, ϕ − th x ) and u (0) = v (0) = u , we conclude from the flowequation that u ′ (0) = v ′ (0) − ∂ x u · V h , thus by (5.11) we have u ′ (0) = − δ h χ h ( x ) ∂ ν g u, on ∂ O . (5.14)We now multiply (5.13) by u then integrate it on Γ θ \ O , then0 = Z Γ θ \O u ( Q θ − z ) u ′ (0)= Z Γ θ \O u ′ (0) ( Q θ − z ) u + Z Γ θ \O X j,k ∂ j ( u ′ (0) g jk ∂ k u − ug jk ∂ k u ′ (0))= Z ∂ O ( u ′ (0) ∂ ν g u − u ∂ ν g u ′ (0)) dS. (5.15)It then follows from u | ∂ O = 0 and (5.14) that0 = Z ∂ O χ h ( x )( ∂ ν g u ( x )) dS ( x ) , sending h → ∂ ν g u ( x ) = 0. We note that x ∈ ∂ O can be chosen arbitrarily, thus ∂ ν g u | ∂ O ≡
0. Putting ˜ u := 1 O · Γ θ \O · u , the samearguments as in the proof of Lemma 4.1 show that ˜ u ∈ H (Γ θ ) and ( Q θ − z )˜ u = 0on Γ θ . But unique continuation results for second order elliptic differential equationsshow that ˜ u ≡
0, thus a contradiction. (cid:3)
Now we consider the behavior of Spec( P O ) under the deformations of O . In thenotation of § ∈ Diff( O ), the pullback Φ ∗ gives an isomorphism between H Φ( O ) and H O , which also restricts to an isomorphism between D Φ( O ) and D O . Like(5.2) we define the deformed operator of P O associate with Φ: P O Φ := Φ ∗ P Φ( O ) (Φ ∗ ) − , with domain D O . (5.16) Since ( P Φ( O ) + i ) − is compact by Lemma 2.4, the same holds for P O Φ , it follows that P O Φ has a discrete spectrum. Moreover, Spec( P O Φ ) must be identical to Spec( P Φ( O ) ),which lies in R due to the self-adjointness of P Φ( O ) .Before stating the deformation results for Spec( P O ), we notice that unlike Lemma5.1, there is a subset of Spec( P O ) which is invariant under the deformations of theobstacle, that is the compactly supported embedded eigenvalues of P ,Spec comp ( P ) := { λ ∈ C : ∃ = u ∈ D comp such that ( P − λ ) u = 0 } , (5.17)where D comp := { u ∈ D : u | R n \ B (0 ,R ) ∈ H ( R n \ B (0 , R )) } . In view of the uniquecontinuation results for second order elliptic differential equations, u in (5.17) mustvanish on R n \ B (0 , R ), thus u ∈ D O for any O containing B (0 , R ), which impliesthat Spec comp ( P ) ⊂ Spec( P O ). The next lemma shows that any eigenvalue of P O other than those compactly supported embedded eigenvalues of P can still be movedby deforming the obstacle: Lemma 5.2.
Suppose that the obstacle
O ⊂ B (0 , R ) contains B (0 , R ) and z ∈ Spec( P O ) \ Spec comp ( P ) , then for any δ > there exists Φ ∈ Diff( O ) with k Φ − id k C <δ such that z / ∈ Spec( P Φ( O ) ) .Proof. The proof is similar to Lemma 5.1 except that we need a different approachfrom (5.15) since the integration by parts is not available in the black box. Supposethat z ∈ Spec( P O ) with multiplicity m P O ( z ) and that P O has exactly one eigenvaluein D ( z , r ). For D := D ( z , r ) we putΠ P O ( D ) := − πi Z ∂D ( P O − ζ ) − dζ , m P O ( D ) := rank Π P O ( D ) . Using (2.14) and (5.3) we can deduce that ∂D ∋ ζ ( P O Φ − ζ ) − exists for Φ ∈ U δ ( O )with δ small enough, then we defineΠ P Φ ( D ) := − πi Z ∂D ( P O Φ − ζ ) − dζ , m P Φ ( D ) := rank Π P Φ ( D ) = m P Φ( O ) ( D ) . We remark that m P O ( D ) is also invariant under small deformations of obstacles: m P Φ( O ) ( D ) is constant for Φ ∈ U δ ( O ) if δ is sufficiently small . (5.18)In view of the proof of Lemma 5.1, it is enough to exclude the following case: ∃ δ > , such that ∀ Φ ∈ U δ ( O ) , m P Φ( O ) ( z ) = m P Φ( O ) ( D ) . (5.19)Again we argue by contradiction, assume that (5.19) holds and m P O ( D ) = M ≥
1. Weremark that unlike the proof of Lemma 5.1, the self-adjointness of P Φ( O ) implies that( P Φ( O ) − z )Π P Φ( O ) ( D ) = 0 thus ( P O Φ − z )Π P Φ ( D ) = 0 for any Φ ∈ U δ ( O ). We nowchoose w ∈ H O such that u := Π P O ( D ) w = 0. For any fixed x ∈ ∂ O and h ∈ (0 , h ], ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 25 we set Φ t := ϕ th where ϕ th is the flow generated by V h given in (5.11), there exists t > t ∈ U δ ( O ) for all − t < t < t . Let v ( t ) := Π P Φ t ( D ) w ∈ D O , u ( t ) := (Φ − t ) ∗ v ( t ) , we have ( P O Φ t − z ) v ( t ) = 0, thus ( P Φ t ( O ) − z ) u ( t ) = 0. Recalling (2.14) we obtain thatfor some ψ ∈ C ∞ c ( O ), ψ = 1 near B (0 , R ) and t > ∀ t ∈ ( − t , t ) , P ( ψu ( t )) + Q ((1 − ψ ) u ( t )) − z u ( t ) = 0 in Φ t ( O ) . (5.20)Since Φ t ( O ) ⊃ O for t ≤
0, we can restrict (5.20) to O and differentiate it in t , bytaking t = 0, we have P ( ψu ′ (0)) + Q ((1 − ψ ) u ′ (0)) − z u ′ (0) = 0 in O . (5.21)Next we compute the inner product of the left hand side and u on the Hilbert space H O defined by (2.12). For that, choose ψ j ∈ C ∞ c ( O ), ψ j = 1 near B (0 , R ), so that ψ = 1 near supp ψ, ψ = 1 near supp ψ . (5.22)Then we have, using the self-adjointness of P , h P ( ψu ′ (0)) , u i H O = h P ( ψu ′ (0)) , ψ u i H = h ψu ′ (0) , P ( ψ u ) i H , and h Q ((1 − ψ ) u ′ (0)) , u i H O = h Q ((1 − ψ ) u ′ (0)) , (1 − ψ ) u i L ( O ) . Recalling (5.14),integration by parts as in (5.15) shows that h Q ((1 − ψ ) u ′ (0)) , (1 − ψ ) u i L ( O ) − h (1 − ψ ) u ′ (0) , Q ((1 − ψ ) u ) i L ( O ) = Z O X j,k ∂ j ((1 − ψ ) u ′ (0) g jk ∂ k ((1 − ψ )¯ u ) − (1 − ψ )¯ ug jk ∂ k ((1 − ψ ) u ′ (0)))= Z ∂ O − u ′ (0) ∂ ν g ¯ u + ¯ u∂ ν g u ′ (0) = Z ∂ O δ h χ h | ∂ ν g u | . It follows from (2.14) and (5.22) that h ψu ′ (0) , P ( ψ u ) i H = h u ′ (0) , ψ ( P O u − Q ((1 − ψ ) u )) i H O = h u ′ (0) , ψP O u i H O ;and that h (1 − ψ ) u ′ (0) , Q ((1 − ψ ) u ) i L ( O ) = h u ′ (0) , (1 − ψ )( P O u − P ( ψ u )) i H O = h u ′ (0) , (1 − ψ ) P O u i H O . We now conclude from (5.21) and all the calculation above that0 = h u ′ (0) , ( P O − z ) u i H O + Z ∂ O δ h χ h | ∂ ν g u | = Z ∂ O δ h χ h | ∂ ν g u | . It follows that ∂ ν g u ( x ) = 0. Since x ∈ ∂ O can be chosen arbitrarily, we obtain that ∂ ν g u | ∂ O ≡
0. Putting ˜ u := 1 O u +1 R n \O ·
0, the same arguments as in the proof of Lemma4.1 show that ˜ u ∈ D and ( P − z )˜ u = 0, which would imply that z ∈ Spec comp ( P ), acontradiction. (cid:3) Proof of convergence
Before proving the convergence of eigenvalues of P ε to resonances as ε → Q O θ off the diagonal { ( x, x ) : x ∈ Γ θ \ O} . For a detailed account see Shubin [Sh92] and references given there. Lemma 6.1.
Suppose that the obstacle
O ⊂ B (0 , R ) contains B (0 , R ) and that z / ∈ Spec( Q O θ ) with − θ < arg z < π/ θ . The Schwartz kernel of the resolvent ( Q O θ − z ) − : L (Γ θ \ O ) → L (Γ θ \ O ) is denoted by G ( z ; x θ , y θ ) , where x θ = f θ ( x ) is the parametrization on Γ θ . Then there exists β > such that for every δ > thereexists C δ > such that | G ( z ; f θ ( x ) , f θ ( y )) | ≤ C δ e − β | x − y | if | x − y | > δ. Proof.
Identifying Γ θ and R n by means of f θ , the pullback f ∗ θ gives an isomorphismbetween L (Γ θ \ O ) and L ( R n \ O ) since there exists C > C − < | det df θ ( x ) | = | x | − n | g θ ( | x | ) | n − | g ′ θ ( | x | ) | < C, for all x. Let ˜ Q O θ := f ∗ θ Q O θ ( f ∗ θ ) − : L ( R n \ O ) → L ( R n \ O ) then ˜ Q O θ is elliptic and equippedwith the domain H ( R n \ O ) ∩ H ( R n \ O ). Moreover, ( ˜ Q O θ − z ) − exists and we denoteits Schwartz kernel by ˜ G ( z ; x, y ), x, y ∈ R n \ O , i.e. ˜ G ( z ; x, y ) = [( ˜ Q O θ − z ) − δ y ( · )]( x )where δ y is the Dirac function supported at y .The same arguments as in [Sh92, Appendix 1] show that there exists β > δ > C δ > | ˜ G ( z ; x, y ) | ≤ C δ e − β | x − y | if | x − y | > δ. We remark that the assumption in [Sh92, Appendix 1.1] that the manifold M is com-plete can be dropped if we introduce ˜ d ( x, y ), the substitute with smoothness propertiesfor the distance | x − y | , on the whole R n then restrict it to R n \ O . The remainingarguments in [Sh92, Appendix 1.2] is still valid if we replace M by R n \ O .Using ( ˜ Q O θ − z ) − = f ∗ θ ( Q O θ − z ) − ( f ∗ θ ) − we obtain that G ( z ; f θ ( x ) , f θ ( y )) = (det df θ ( y )) − ˜ G ( z ; x, y ) , x, y ∈ R n \ O , the desired estimate of G ( z ; x θ , y θ ) then follows from the estimate of ˜ G ( z ; x, y ). (cid:3) We now state a more precise version of Theorem 1:
Theorem 2.
Suppose that Ω ⋐ { z : − θ < arg z < π/ θ } . Then exists δ = δ (Ω) > such that ∀ < δ < δ , ∃ ε δ > such that < ε < ε δ = ⇒ Spec( P ε ) ∩ Ω δ ⊂ J [ j =1 D ( z j , δ ) , (6.1) ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 27 where z , · · · , z J are the resonances of P in Ω and Ω δ := { z ∈ Ω : dist( z, ∂ Ω) > δ } .Furthermore, for each resonance z j with the multiplicity m ( z j ) given by (2.11) , P ε ) ∩ D ( z j , δ ) = m ( z j ) , ∀ < ε < ε δ , (6.2) where the eigenvalue in Spec( P ε ) is counted with multiplicity defined in (3.8) .Proof. First we put δ = min ≤ j ≤ J dist( z j , ∂ Ω) and fix θ ∈ [0 , θ ) such that Ω ⋐ { z : − θ < arg z < π/ θ } . To prove (6.1) we argue by contradiction. Suppose thatthere exist some δ < δ and a sequence ε k →
0+ such that ∃ z k ∈ Spec( P ε k ) ∩ Ω δ \ J [ j =1 D ( z j , δ ) , k = 1 , , · · · Then there exists a subsequence z n k → z , as k → ∞ , for some z ∈ Ω δ \ S Jj =1 D ( z j , δ ).Since z ∈ Ω, we see that z is not a resonance, thus P θ − z is invertible by definition.We may assume that D ( z , r ) is disjoint with Spec( P θ ) for some r >
0, it then followsfrom Lemma 3.5 that Spec( P ε,θ ) ∩ D ( z , r ) = ∅ for ε small enough. However, Lemma3.4 shows that Spec( P ε nk ,θ ) = Spec( P ε nk ) ∋ z n k → z while ε n k → z j , let V j := { u ∈ D comp : ( P − z j ) u = 0 } , then V j is finite dimensional and V j = { } if and only if z j ∈ Spec comp ( P ). We remarkthat V j is a subspace of H R given in (1.1), as a consequence of the unique continuationresults for second order elliptic equations. The self-adjointness of P implies that V ⊥· · · ⊥ V J in the Hilbert space H . Putting V := V ⊕ · · · ⊕ V J , H admits the followingorthogonal decomposition: H = V ⊕ ˜ H R ⊕ L ( R n \ B (0 , R )) . (6.3)Let Π : H → V be the orthogonal projection. Since V is an invariant subspace under P , we can introduce the restriction of P as follows:˜ P : ˜ H R ⊕ L ( R n \ B (0 , R )) → ˜ H R ⊕ L ( R n \ B (0 , R )) , ˜ P u := ( I − Π ) P u.
If we replace H R with ˜ H R and replace P by ˜ P , which is also self-adjoint with domain˜ D := ( I − Π ) D , it is easy to see that the assumptions (1.2) – (1.5) are still satisfied.Recalling the definition of resonances introduced in § P must alsobe a resonance of P and we have m ( z j ) = rank I z j ( z − ˜ P θ ) − dz + dim V j . Note that V j = { } implies that z j ∈ Spec( P ε ) for every ε >
0. Putting ˜ P ε :=˜ P − iε (1 − χ ( x )) x , it follows that P ε ) ∩ D ( z j , δ ) = P ε ) ∩ D ( z j , δ ) + dim V j , ∀ ε > , while both sides are counted with multiplicities. Hence it is enough to establish (6.2)for ˜ P . In other words, it suffices to prove (6.2) in the case that P has no compactlysupported embedded eigenvalues in Ω.Now we assume that Spec comp ( P ) ∩ Ω = ∅ . Lemma 5.1 and 5.2 show that there existsan obstacle O ⊂ B (0 , R ) containing B (0 , R ) such that χ in (1.8) is equal to 1 near O and that z j / ∈ Spec( P O ) ∪ Spec( Q O θ ), j = 1 , · · · , J . Then we can decrease δ suchthat Spec( P O ) and Spec( Q O θ ) are disjoint with S Jj =1 D ( z j , δ ). For each δ ∈ (0 , δ ),we can also decrease ε δ in (6.1) such that ∀ ≤ ε < ε δ , J [ j =1 D ( z j , δ ) ∩ Spec( Q O ε,θ ) = ∅ . This follows from Lemma 3.5 applied with P θ = Q O θ and Ω = S Jj =1 D ( z j , δ ). Hencethe Dirichlet-to-Neumann operators b N ε,θ ( z ) , ≤ ε < ε δ introduced in §
4, are well-defined for z ∈ S Jj =1 D ( z j , δ ). In view of (6.1), Lemma 3.4 and 4.2 we obtain that ∂D ( z j , δ ) ∋ w b N ε,θ ( w ) − exists and that for all 0 < ε < ε δ , j = 1 , · · · , J , P ε ) ∩ D ( z j , δ ) = 12 πi tr Z ∂D ( z j ,δ ) b N ε,θ ( w ) − ∂ w b N ε,θ ( w ) dw. (6.4)In order to apply the Gohberg–Sigal–Rouch´e theorem, we need the estimate: ∀ < ε < ε δ , k b N ε,θ ( w ) − b N θ ( w ) k H / ( ∂ O ) → H / ( ∂ O ) < , w ∈ ∂D ( z j , δ ) , (6.5)here we write b N θ ( · ) = b N ,θ ( · ) for simplicity. To obtain this estimate, we first choose E out in (4.6) such that χ = 1 near supp E out ϕ for any ϕ ∈ H / ( ∂ O ), then (4.6) reducesto N out ε,θ ( z ) ϕ = ∂ ν g ( E out ϕ − ( Q O ε,θ − z ) − ( Q − z ) E out ϕ ). Therefore,( b N ε,θ ( w ) − b N θ ( w )) ϕ = h D ∂ O i − ∂ ν g (( Q O θ − w ) − − ( Q O ε,θ − w ) − )( Q − w ) E out ϕ. Choosing ψ ∈ C ∞ c (Γ θ \ O ) such that ψ = 1 near supp E out ϕ , ∀ ϕ ∈ H / ( ∂ O ) and that χ = 1 near supp ψ , (6.5) then follows from the following estimate: for w ∈ ∂D ( z j , δ ),(( Q O θ − w ) − − ( Q O ε,θ − w ) − ) ψ = O δ ( ε ) : L (Γ θ \ O ) → H (Γ θ \ O ) . (6.6)To obtain (6.6), we denote the Schwartz kernel of the operator (1 − χ ) x θ ( Q O ε,θ − w ) − ψ by K ( w ; x θ , y θ ). In the notation of Lemma 6.1, we have K ( w ; f θ ( x ) , f θ ( y )) = (1 − χ ( x )) f θ ( x ) G ( w ; f θ ( x ) , f θ ( y )) ψ ( y ) . ESONANCES AS VISCOSITY LIMITS FOR BLACK BOX PERTURBATIONS 29
It follows from Lemma 6.1 that there exists β δ > w ∈ ∂D ( z j , δ ), j = 1 , · · · , J , | K ( w ; f θ ( x ) , f θ ( y )) | ≤ C | x | e − β δ | x − y | ψ ( y ), thussup x θ Z Γ θ \O | K ( w ; x θ , y θ ) || dy θ | ≤ C δ , sup y θ Z Γ θ \O | K ( w ; x θ , y θ ) || dx θ | ≤ C δ . The Schur test shows that (1 − χ ) x θ ( Q O ε,θ − w ) − ψ = O δ (1) : L (Γ θ \ O ) → L (Γ θ \ O ).Hence we can write(( Q O θ − w ) − − ( Q O ε,θ − w ) − ) ψ = − iε ( Q O ε,θ − w ) − (1 − χ ) x θ ( Q O θ − w ) − ψ. It remains to show that for ε δ > Q O ε,θ − w ) − = O δ (1) : L (Γ θ \ O ) → H (Γ θ \ O ) , w ∈ J [ j =1 ∂D ( z j , δ ) , < ε < ε δ . This follows from Lemma 3.5 with P θ = Q O θ and Ω = S Jj =1 ∂D ( z j , δ ). Using (6.6) wecan decrease ε δ such that (6.5) holds for j = 1 , · · · , J . Now we apply the Gohberg–Sigal–Rouch´e theorem to conclude that for all 0 < ε < ε δ and j = 1 , · · · , J ,12 πi tr Z ∂D ( z j ,δ ) b N ε,θ ( w ) − ∂ w b N ε,θ ( w ) dw = 12 πi tr Z ∂D ( z j ,δ ) b N θ ( w ) − ∂ w b N θ ( w ) dw. Finally, using Lemma 4.2, (6.4) and the equation above, we obtain (6.2). (cid:3)
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Email address : [email protected]@math.berkeley.edu