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Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.

1. Introduction. The perfectly matched layer (PML) method, originally introduced in [1], is in common use for the numerical analysis of a wide class of problems. For some of them stability and convergence of the method have been proved mathematically; see, e.g., [2,3,4,5,6,7,20]. In the present paper we develop the PML method for Dirichlet Laplacians in quasi-cylindrical domains G ⊂ R n+1 , see Fig. 1.1. These are unbounded domains that outside a bounded set can be transformed onto a semi-cylinder by suitable diffeomorphisms. Intuitively, one can understand G as a domain whose boundary asymptotically approaches at infinity the boundary of a semi-cylinder (0, ∞) × Ω, where the cross-section Ω of G at infinity is a bounded domain in R n . Dirichlet Laplacians ∆ model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. In order to characterize outgoing and incoming solutions of the Helmholtz equation (∆ − µ)u = f we establish a limiting absorption principle. Then we construct a uniquely solvable problem with PMLs of finite length. This is a Dirichlet problem in the domain G truncated at a finite distance. We prove that solutions of the latter problem locally approximate outgoing or incoming solutions of the Helmholtz equation with an error that exponentially tends to zero as the length of PMLs tends to infinity. In other words, we prove stability and exponential convergence of the PML method. We find that the rate of exponential convergence depends only on the spectral parameter µ and on the infinitely distant cross-section Ω. Thus the rate is the same as in the particular case of a domain G that coincides with the semi-cylinder (0, ∞) × Ω outside a bounded set.
As is known, construction of PMLs is closely related to the complex scaling. The complex scaling involves complex dilation of variables and has a long tradition in mathematical physics and numerical analysis; for a historical account see e.g. [16,10,34]. Although there are several papers utilizing different approaches to the complex scaling in waveguide-type of geometry, e.g. [8,11,12,20], the complex scaling has not been used in quasi-cylindrical domains before. Our approach to the complex scaling originates from the one developed in [18] for a Schrödinger operator in R 3 (see also [16]). Deformations of the Dirichlet Laplacian by means of the complex scaling give rise to an analytic family of non-selfadjoint operators in G. These operators correspond to a Dirichlet problem with infinite PMLs. Localization of the essential spectra of these operators together with certain relations between their resolvents justifies a limiting absorption principle. For locating the essential spectra we employ methods of the theory of elliptic boundary value problems [24,25,28]. Relations between the resolvents are obtained with the help of Hardy spaces of analytic functions. Note that Hardy spaces in context of the complex scaling were originally used in [37,38]. As we mention in Remark 6.2, our methods also make it possible to develop an analog of the celebrated Aguilar-Balslev-Combes-Simon theory of resonances.
As is typically the case, solutions to the Helmholtz equation satisfying the limiting absorption principle locally coincide with solutions to the problem with infinite PMLs. Moreover, under certain assumptions on the right hand side solutions to the latter problem are of some exponential decay at infinity. This allows us to prove unique solvability of the problem with finite PMLs and establish exponential convergence of the PML method by using the compound expansion technique [27,25]. In [20] we used a similar approach to study the PML method for inhomogeneous media. Here we study the PML method for a wide class of quasi-cylindrical domains.
This paper is organized as follows. Section 2 consists of preliminaries, where we introduce notations, formulate our assumptions on the quasi-cylindrical domains, and give a formal definition of operators corresponding to the problem with infinite PMLs. In Section 3 we demonstrate that the operators are well-defined and derive some estimates on their coefficients. As shown in Section 4, these operators give rise to an analytic family of m-sectorial operators. In Section 5 we introduce and study Hardy spaces of analytic functions. In Section 6 we formulate and prove a limiting absorption principle. In Section 7 we show that under certain assumptions on the right hand side solutions to the problem with infinite PMLs are of some exponential decay at infinity. Finally, in Section 9 we formulate and study the problem with finite PMLs and prove exponential convergence of the PML method.

Preliminaries.
In this section we introduce basic notations that are in use throughout the paper. We formulate our assumptions on the quasi-cylindrical domains and introduce differential operators corresponding to the problem with infinite PMLs. Recall that PMLs are artificial strongly absorbing layers designed so that waves coming from a non-PML medium to PMLs do not reflect at the interface.
Let G be a domain in R n+1 with smooth boundary ∂G. We suppose that the set {(ζ, η) ∈ G : ζ ≤ 0} is bounded, and the set {(ζ, η) ∈ G : ζ > 0} coincides with a quasi-cylinder C. (Extension of our results to the case of a domain G that coincides outside a bounded set with several quasi-cylinders is straightforward.) Following [25], we say that G is a quasi-cylindrical domain.
Consider the complex scaling x → x+ λs(x− r) with parameters r > 0 and λ ∈ C. Here s(x) is a smooth scaling function possessing the properties: is a selfdiffeomorphism of the semi-cylinder. Therefore is a selfdiffeomorphism of G. In other words, ϑ λ,r with λ ∈ (−1, 1) and r > 0 is a scaling of the quasi-cylinder C along the curvilinear coordinate x. Let (ϑ ′ λ,r ) ⊤ be the transpose of the Jacobian matrix ϑ ′ λ,r . Then e λ,r = (ϑ ′ λ,r ) ⊤ ϑ ′ λ,r is the matrix coordinate representation of a metric e λ,r on G, and is the Laplace-Beltrami operator on the Riemannian manifold (G, e λ,r ). As the parameter r increases, the equalities ϑ λ,r (ζ, η) = (ζ, η), e λ,r (ζ, η) = Id, where Id is the (n + 1) × (n + 1) identity matrix, and the equality ∆ λ,r = ∆ become valid on a larger and larger subset of G. In the case λ = 0 the scaling is not applied. Therefore e 0,r ≡ e is the Euclidean metric and ∆ 0,r ≡ ∆. In order to consider complex values of the scaling parameter λ, we impose additional assumptions on the diffeomorphism (2.1): i. the function R + ∋ x → κ(x, ·) ∈ C ∞ (Ω) has an analytic continuation from R + to some sector S α = {z ∈ C : | arg z| < α < π/4}; (2.7) ii. the elements κ ′ ℓm (z, ·) of the Jacobian matrix κ ′ tend to the Kronecker delta δ ℓm in the space C ∞ (Ω) uniformly in z ∈ S α as z → ∞. For instance, the assumptions i,ii are satisfied for the following quasi-cylinders: where as ϕ(x) and ψ(x) we can take the functions 1, 1 + e −x , 1 + (x + 1) −s with s > 0, 1 + 1/ log(x + 2), 1 + 1/ log(1 + log(x + 2)), and so on. These examples show that quasi-cylinders can have very different shapes comparing with the semi-cylinder.
In the next section we will show that for all sufficiently large r > 0 the assumptions i, ii on κ together with (2.2) and (2.3) lead to the analyticity of the coefficients of the differential operator (2.6) with respect to the scaling parameter λ in the disk Thus the equality (2.6) defines ∆ λ,r for all λ ∈ D α . Clearly, ∆ λ,r coincides with the Dirichlet Laplacian ∆ on the set We will show that on the set G \ G r the operator ∆ λ,r with λ ∈ D α \ R describes an infinite PML for ∆. In the case ℑλ > 0 (resp. ℑλ < 0) this PML is an artificial nonreflective strongly absorbing layer for the outgoing (resp. incoming) solutions. Remark 2.1. For simplicity we consider in this paper only Dirichlet Laplacians. However, similar methods can be used to develop and study PML method for the Schrödinger operator ∆ + V in G, where ∆ is the Dirichlet Laplacian and V ∈ C ∞ (G) is a real-valued potential with the following properties: for some r 0 > 0 and α > 0 the function x → V • κ(x, ·) ∈ L 2 (Ω) extends by analyticity to the sector {z ∈ C : | arg(z − r 0 )| < α}, where for all y ∈ Ω we have |V • κ(z, y)| ≤ C(|z|) → 0 as z → ∞. One can also include into consideration potentials with moderate local singularities and relatively bounded operator-valued potentials.
3. Construction of infinite PMLs. In this section we show that for all sufficiently large r > 0 the differential operator (2.6) is well-defined for complex values of the scaling parameter λ in the disk (2.8). We also obtain some estimates on the matrix e λ,r .
Let us consider complex values of the scaling parameter λ. We suppose that λ is in the complex disk D α , where α is the same as in our assumptions on the partial analytic regularity of the diffeomorphism κ; cf. (2.7), (2.8). The curve L λ,r = {z ∈ C : z = x + λs(x − r), x > 0} lies in the sector S α , see Fig. 3.1. We define the matrix g λ,r for all non-real λ in the disk by the equality (3.3), where g(x + λs(x − r), y) stands for the value of the analytic in z ∈ S α function g(z, y) at z = x + λs(x − r). By analyticity in λ we conclude that g λ,r is a complex symmetric matrix, the Schwarz reflection principle gives g λ,r = g λ,r , where the overline stands for the complex conjugation. If λ ∈ D α is non-real, then the matrix g λ,r does not correspond to a Riemannian metric. However, g λ,r is invertible for all λ ∈ D α provided r > 0 is sufficiently large. Indeed, s(x − r) = 0 and g −1 λ,r (x, y) = g −1 (x, y) for all x < r. On the other hand, the matrix g(x + λs(x − r), y) from (3.3) is invertible for x ≥ r with large r > 0 as it is only little different from the identity matrix; the latter fact is a consequence of (3.2) and the inequality |x + λs(x − r)| ≥ r.
We define the complex scaling ϑ λ,r for all λ in the disk D α by the equality (2.5), where κ • κ λ,r (x, y) is the value of the analytic in z ∈ S α function κ(z, y) at the point z = x + λs(x − r). Consider the matrix e λ,r = (ϑ ′ λ,r ) ⊤ ϑ ′ λ,r . It is clear that for all (ζ, η) ∈ G \ C the matrix e λ,r (ζ, η) coincides with the (n + 1) × (n + 1)-identity. For all real λ ∈ D α we have where g λ,r = κ * e λ,r is the pullback of the corresponding metric e λ,r on C by the diffeomorphism κ. Therefore e λ,r is analytic in λ ∈ D α and invertible for all sufficiently large r > 0. By analyticity in λ we conclude that e λ,r (ζ, η) is a complex symmetric matrix, the Schwarz reflection principle gives e λ,r = e λ,r . Differentiating the equality (3.6), we see that the matrices ∂ p ζ ∂ q η e λ,r and ∂ p ζ ∂ q η e −1 λ,r are analytic in λ ∈ D α . Moreover, from (3.4), (3.5), and our assumptions on κ we obtain where p + |q| ≤ 1, and c(r) → 0 as r → +∞. (3.7) Here (ζ, η) ∈ C and s ′ r (ζ, η) stands for the function s ′ (x − r) written in the coordinates (ζ, η). We extend s ′ r from C to G by zero. Note that the estimate (3.7) remains valid for all (ζ, η) ∈ G and the constant c(r) is independent of λ ∈ D α and (ζ, η) ∈ G. Now we see that for all sufficiently large r > 0 the differential operator (2.6) is well-defined for all λ in the disk D α , and its coefficients are subjected to the estimate (3.7).

Analytic families of operators.
In this section we study the unbounded operator ∆ λ,r in the Hilbert space L 2 (G) with the usual norm The operator ∆ λ,r is initially defined on the set C ∞ 0 (G). We show that the operator is closable, and the closure defines an analytic family D α ∋ λ → ∆ λ,r of type (B) [23]. In a standard way this implies that the resolvent (∆ λ,r − µ) −1 is an analytic function of λ and µ on some open subset of D α ×C. The latter fact will be used for justification of a limiting absorption principle in Section 6.
Taking into account (4.3) we complete the proof. Remark 4.2. Throughout the paper we say that r > 0 is sufficiently large if the matrix e λ,r (ζ, η) is invertible and its inverse meets the estimates (4.5) uniformly in (ζ, η) ∈ G and λ ∈ D α .
In the next lemma we show that in the right hand side of (4.2) the second term has an arbitrarily small relative bound with respect to the first term uniformly in λ ∈ D α . Lemma 4.3. For any ǫ > 0 and u ∈ C ∞ 0 (G) the estimate with arbitrarily smallǫ > 0 and C = sup (ζ,η) det e λ,r (ζ, η) 1/2 < ∞, cf. (3.7).
Thanks to (3.7) we have the uniform bound This together with (4.7) and (4.5) implies Now we make use of (4.6) and establish the assertion for C = C 2 cδ −1 (1 + tan ϕ) 2 and an arbitrarily small ǫ = Ccδ −1 (1 + tan ϕ) 2ǫ . As a consequence of the equality (4.2) and Lemmas 4.1, 4.3 for all sufficiently large r > 0 we obtain with some angle ϕ and some positive constants δ and γ, which are independent of u ∈ C ∞ 0 (G) and λ ∈ D α . The symmetric form q 0,r [u, u] = G ∇ ζη u, ∇ ζη u dζ dη is independent of r, it corresponds to the Dirichlet Laplacian ∆ ≡ ∆ 0,r . Clearly, q λ,r [u, u] = (∆ λ,r u, u). Estimate (4.8) implies that the numerical range is a subset of the sector {µ ∈ C : | arg(µ + γ)| ≤ ϕ < π/2}. By definition this means that the operator ∆ λ,r with the domain C ∞ 0 (G) is sectorial. We introduce the Sobolev space ii) the form q λ,r is closed, if every q λ,r -convergent sequence {u j } has a limit u in the domain of q λ,r , and q λ,r [u − u j , u − u j ] → 0. From (4.8), (4.9) it immediately follows that q λ,r with the domain • H 1 (G) is a closed densely defined sectorial form [23,34], and its sector {µ ∈ C : | arg(µ + γ)| ≤ ϕ} is independent of λ ∈ D α . As known [23, Chapter VI.2.1], to every closed densely defined sectorial form there corresponds a unique m-sectorial operator. Namely, to the form q λ,r there corresponds a unique m-sectorial operator ∆ λ,r in L 2 (M) such that its sector is the sector of q λ,r , the and v ∈ • H 1 (G). (Here and elsewhere m-sectorial means that the numerical range {µ = (Au, u) H ∈ C : u ∈ D(A), (u, u) H = 1} and the spectrum σ(A) of a closed unbounded operator A in a Hilbert space H with the inner product (·, ·) H both lie in some sector {µ ∈ C : arg(µ − γ) ≤ ϕ} with γ ∈ R and ϕ < π/2.) In particular, to the symmetric nonnegative form q 0,r there corresponds the selfadjoint Dirichlet Laplacian ∆ ≡ ∆ 0,r . The m-sectorial operator ∆ λ,r in L 2 (M) is the Friedrichs extension of the sectorial operator ∆ λ,r defined on C ∞ 0 (G), see [23, Chapter VI.2.3]. As we show in assertion 1 of the next proposition the set C ∞ 0 (G) is a core of the Friedrichs extension. Proposition 4.4. Assume that r > 0 is sufficiently large. Then the following assertions are valid.
is the closure of the operator ∆ λ,r defined on the set C ∞ 0 (G).

The family of m-sectorial operators
Consider the domain D(∆ λ,r ) as a Hilbert space endowed with the graph norm u; D(∆ λ,r ) = u; L 2 (G) + ∆ λ,r u; L 2 (G) . Let µ be a point outside of the sector of the m-sectorial operator ∆ λ,r . Then the set on the principal symbol of ∆ λ,r , where c > 0. Hence ∆ λ,r is a strongly elliptic operator. As is well-known, a strongly elliptic operator and the Dirichlet boundary condition set up an elliptic boundary value problem, e.g. [26]. The usual argument on the regularity of solutions to the elliptic boundary value problems [26,25] implies that the set C(∆ λ,r ) consists of smooth in G functions u with u ↾ ∂G = 0. Multiplying u ∈ C(∆ λ,r ) by appropriate cutoff functions χ j with expanding compact supports supp χ j ⊂ supp χ j+1 , it is easy to see that for any u ∈ C(∆ λ,r ) there is a sequence The family D α ∋ λ → q λ,r is analytic in the sense of Kato (i.e. q λ,r is a closed densely defined sectorial form, its domain ). By definition [23,34] this means that the family of m-sectorial operators D α ∋ λ → ∆ λ,r is an analytic family of type (B). This proves assertion 2.
As is well-known [23,34], any analytic family of type (B) is also an analytic family of operators in the sense of Kato. Now a standard argument justifies assertion 3; see, e.g., [34,Theorem XII.7].

Spaces of analytic functions.
We have shown that for all µ outside the sector of the family D α ∋ λ → ∆ λ,r of m-sectorial operators the resolvent (∆ λ,r −µ) −1 is an analytic function of λ, see Proposition 4.4.3. In order to get some relations between the resolvents (∆ − µ) −1 and (∆ λ,r − µ) −1 we will use a sufficiently large Hilbert space H α (G) of analytic functions here ϑ λ,r is the complex scaling (2.5). The goal of this section is two-fold: 1. To introduce a Hilbert space H α (G), which is sufficiently large in the sense that for any λ ∈ D α and r > 0 the set Below we cite some facts from the theory of Hardy classes.
2. The Hardy class H(S α ) endowed with the norm

For any
Proof. Assertions 1-3 are direct consequences of well-known facts from the theory of Hardy spaces of functions analytic in strips, e.g. [35]. In fact, the proof reduces to the conformal mapping of the sector S α to the strip {z ∈ C : −α < ℑz < α}, we omit the details.
Let us prove assertion 4. A standard argument, see e.g. [36], shows that any F ∈ H(S α ) can be represented by the Cauchy integral where we assume that the boundary limits F ± (x) are extended to x < 0 by zero. The integrals are absolutely convergent in the space L 2 (Ω). As is well-known [35], the first integral in (5.4) defines an element f + of the Hardy space H(C − α ) of L 2 (Ω)valued functions analytic in the half-plane C − α = {z ∈ C : ℑ(e −iα z) < 0}. As shown in [37,38], the norm in H(C − α ) can be defined by the equality As a consequence, for all z ∈ S α and ψ ∈ (−α, α) we have Consider the algebra E of all entire functions C ∋ z → F (z) ∈ C ∞ 0 (Ω) with the following property: in any sector |ℑz| ≤ (1 − ǫ)ℜz with ǫ > 0 the value F (z); L 2 (Ω) decays faster than any inverse power of ℜz as ℜz → +∞. Examples of functions F ∈ E are F (z) = e −γz 2 P (z), where γ > 0 and P (z) is an arbitrary polynomial in z with coefficients in C ∞ 0 (Ω). Clearly, E ⊂ H(S α ). The next lemma is an adaptation of [18, Theorem 3], we omit the proof.
for any λ ∈ D α and r > 0. Here F • κ λ,r (x, ·) stands for the value of the entire function z → F (z) at the point z = x + λs(x − r). Now we are in position to prove the following proposition.
holds uniformly in λ ∈ D α and F ∈ H(S α ). 2. The space H(S α ) is sufficiently large in the sense that for any λ ∈ D α and r > 0 the set is analytic. The proof is preceded by a discussion. By Proposition 5.3.1 for any r > 0 the complex scaling κ λ,r induces the uniformly bounded injection Any function F ∈ H(S α ) can be reconstructed from its trace x → F •κ λ,r (x, ·) ∈ L 2 (Ω) by analytic continuation from the curve {x + λs(x − r) ∈ C : x > 0} to the sector S α . Therefore we can always identify the space H(S α ) with the range of injection (5.6). By Proposition 5.3.2 the range is dense in x > 0} lies in the sector S α . Observe that this curve is differ from a ray only inside an independent of λ ∈ D α compact subset K r ⊂ S α . Now the uniform in λ ∈ D α and F ∈ H(S α ) estimate (5.5) follows from assertions 3 and 4 of Lemma 5.1.
2. The assertion is an immediate consequence of the embedding E ⊂ H(S α ), Lemma 5.2, and the estimate (5.5).
Introduce the Hilbert space H α (G) with the norm . As a consequence of Proposition 5.3 and definition (2.5) of the complex scaling ϑ λ,r we immediately get the following assertions.
. For any f ∈ H α (G) and r > 0 the function (5.1) is analytic. 0 (G)) as ǫ tends to zero from below (ǫ ↑ 0) or from above (ǫ ↓ 0). However the limits may exist in the space of bounded operators acting from a smaller source space to a larger target space. As a source space we take the Hilbert space H α (G) constructed in the previous section. As a target space we take the reflexive Fréchet space H 2 0,loc (G). The space H 2 0,loc (G) consists of all distributions u such that ̺u ∈ H 2 0 (G) with any ̺ ∈ C ∞ c (G), the topology in H 2 0,loc (G) is induced by the family of seminorms u → ̺u; H 2 0 (G) ; here C ∞ c (G) is the set of all smooth functions with compact supports in G.
1. For all sufficiently large r > 0 and λ ∈ D α \ R the resolvent is a bounded operator. 2. The resolvent (∆ − µ 0 − iǫ) −1 , ǫ ≷ 0, viewed as a bounded operator acting from H α (G) to H 2 0,loc (G), has limits as ǫ ↓ 0 and ǫ ↑ 0. 3. Suppose that r > 0 is sufficiently large, λ ∈ D α \ R, and f ∈ H α (G). Let u λ,r ∈ H 2 0 (G) be given by the equality u λ,r = (∆ λ,r −µ 0 ) −1 (f •ϑ λ,r ). Then the outgoing u − ∈ H 2 0,loc (G) and the incoming u + ∈ H 2 0,loc (G) solutions defined by the limiting absorption principle Here the bounded domain G r is the same as in (2.9). The proof is preceded by a discussion. From Theorem 6.1.3 we see that the equation (∆ λ,r − µ 0 )u λ,r = f • ϑ λ,r with non-real parameter λ ∈ D α describes infinite PMLs on G \ G r for all µ 0 satisfying the assumptions of the theorem. The layers are perfectly matched in the sense that for ℑλ > 0 (resp. for ℑλ < 0) u λ,r coincides in G r with the outgoing solution u − (resp. with the incoming solution u + ). The PMLs are absorbing because in contrast to u ± the function u λ,r decays at infinity in the mean as an element of H 2 0 (G). In the next section we will refine results of Theorem 6.1 by showing that under an additional assumption on f • ϑ λ,r the solution u λ,r is of some exponential decay at infinity. For instance, this assumption is a priori met for f ∈ L 2 (G) supported in G r . Then f • ϑ λ,r ≡ f and the operator ∆ λ,r completely describes infinite PMLs on G \ G r .
Proof. The proof consists of two steps.
Step 1. In Section 8 below we will show that the graph norm of ∆ λ,r is an equivalent norm in H 2 0 (G). This immediately implies that D(∆ λ,r ) = H 2 0 (G) as the set C ∞ 0 (G) is dense in both spaces, cf. Proposition 4.4.1. We will also localize the essential spectrum σ ess (∆ λ,r ) of the unbounded m-sectorial operator ∆ λ,r in L 2 (G). As is wellknown, the spectrum σ(∆ Ω ) consists of infinitely many positive isolated eigenvalues. It turns out that σ ess (∆ λ,r ) consists of an infinite number of rays emanating from every point ν ∈ σ(∆ Ω ), cf.  the Dirichlet Laplacian ∆. As λ varies, the ray {µ ∈ C : arg(µ−ν) = −2 arg(1+λ)} of the essential spectrum σ ess (∆ λ,r ) rotates about the threshold ν ∈ σ(∆ Ω ) and sweeps the sector {µ ∈ C : | arg(µ − ν)| < 2α}. In order to avoid several repetitions we organized the paper so that the proofs of these results in some greater generality are postponed to Section 8; here we take these results for granted.
Recall that µ is said to be a point of the essential spectrum σ ess (A) of a closed unbounded operator A in the space L 2 (G) with the domain H 2 0 (G), if the bounded operator A − µ : H 2 0 (G) → L 2 (G) is not Fredholm (a linear bounded operator between two Banach spaces is Fredholm, if its kernel and cokernel are finite dimensional, and its range is closed). Since the operator ∆ λ,r is m-sectorial, there exists a regular point of ∆ λ,r in the simply connected set C \ σ ess (∆ λ,r ). Therefore is a Fredholm holomorphic operator function, e.g. [24,Appendix]. Recall that the spectrum of a Fredholm holomorphic operator function consists of isolated eigenvalues of finite algebraic multiplicity, e.g. [24,Proposition A.8.4]. As a consequence, the resolvent is a meromorphic operator function. For f, g ∈ L 2 (G) and a sufficiently large r > 0 we define the quadratic form This form is bounded in L 2 (G). Indeed, thanks to the estimate (3.7) on e −1 λ,r we have 0 < c 1 ≤ | det e λ,r (ζ, η)| ≤ c 2 uniformly in λ ∈ D α and (ζ, η) ∈ G.
Assume that λ ∈ D α is real. Then the form (·, ·) λ,r is the inner product induced on G by the metric e λ,r , the norm (f, f ) λ,r is equivalent to the norm f ; L 2 (G) , and ∆ λ,r is the Laplace-Beltrami operator on (G, e λ,r ). We have Assume that µ is not in the sector of ∆ λ,r . Then (∆ λ,r − µ) −1 is a bounded operator and we can rewrite (6.4) in the form , and the operator ∆ − µ : H 2 0 (G) → L 2 (G) yields an isomorphism. It is clear that (f • ϑ λ,r , f • ϑ λ,r ) λ,r = (f, f ). As a consequence, the (real) scaling f → f • ϑ λ,r realizes an isomorphism in L 2 (G), and the equality (6.5) extends by continuity to all f ∈ L 2 (G). Taking the inner product of the equality (6.5) with g ∈ L 2 (G), and passing to the variables (ζ,η) = ϑ λ (ζ, η) in the right hand side, we obtain Now we assume that f, g ∈ H α (G). Then f •ϑ λ,r and g •ϑ λ,r are L 2 (G)-valued analytic functions of λ in the disk D α , see Corollary 5.4.3. This together with Proposition 4.4.4 implies that the right hand side of (6.6) extends by analyticity from λ ∈ D α ∩ R to all λ ∈ D α . The right hand side of (6.6) extends from all µ outside of the sector of ∆ λ,r to a meromorphic function of µ ∈ C \ σ ess (∆ λ,r ). In particular, for all λ ∈ D α \ R we have µ 0 ∈ C \ σ ess (∆ λ,r ), cf. Figure 6.1. Here µ 0 is the same as in the formulation of the theorem. Now we are in position to prove assertion 1. Consider the projection onto the eigenspace of the selfadjoint operator ∆. Suppose, by contradiction, that the resolvent (6.3) has a pole at the point µ 0 ∈ C \ σ ess (∆ λ,r ). By Corollary 5.4.2 there exist f and g in the space H α (G) such that µ 0 is a pole of the right hand side of (6.6). The equality (6.6) implies that (Pf, g) = 0, and thus ker(∆ − µ 0 ) = {0}. This is a contradiction. Assertion 1 is proven.
Step 2. We need to show that for any ̺ ∈ C ∞ c (G) the operator ̺(∆ − µ 0 − iǫ) −1 tends to some limits in the space of bounded operators B(H α (G), H 2 0 (G)) as ǫ ↓ 0 and ǫ ↑ 0. We take a sufficiently large r = r(̺) > 0 such that supp ̺ ⊂ G r . Then ̺ • ϑ λ,r = ̺ for all λ ∈ D α . Now we can pass from (6.5) to the equality For f ∈ H α (G) the equality (6.7) extends by analyticity to all λ ∈ D α . Consider, for instance, the case ℑλ > 0 (the case ℑλ < 0 is similar). Then the upper half-plane C + = {µ ∈ C : ℑµ > 0} and a complex neighborhood of the point µ 0 do not contain points of σ ess (∆ λ,r ), cf. Figure 6.1. Therefore the right hand side of (6.7) has a meromorphic continuation in µ to the union of C + and a complex neighborhood of µ 0 . Hence the left hand side of (6.7) has the same meromorphic continuation. Clearly, a pole at µ 0 may only appear due to a pole of the resolvent (6.3) at µ 0 , but it is a regular point by assertion 1. Since ̺ is an arbitrary smooth function supported in G r this proves assertion 3. In order to prove assertion 2 it remains to note that In the last inequality we used Corollary 5.4.1 and assertion 1.
In the following two remarks we collect some results that can be obtained by methods developed in the proof of Theorem 6.1. Although these results are not used in this paper, they provide additional insights of the problem. Remark 6.2. On the basis of the equality (6.6), Corollary 5.4, and the description of σ ess (∆ λ,r ) for λ ∈ D α , one can develop an analog of the celebrated Aguilar-Balslev-Combes-Simon theory of resonances [10,16,18,34]. We announce some results below, for the proof we refer to [21].
The results listed in Remarks 6.2 and 6.3 are new and might be of their own interest. Traditionally, when studying Laplacians in the waveguide-type of geometry, one imposes more or less restrictive assumptions on the rate of convergence of the metric g on (0, ∞) × Ω to its limit at infinity; see, e.g., [9,11,12,13,14,15,19,29,30,31]. Contrastingly, our assumptions on the diffeomorphism κ allow for arbitrarily slow convergence of the metric g = κ * e to the Euclidean metric e at infinity, see Section 3. As a substitution for the assumptions on the rate of convergence of the metric g at infinity we use assumptions on the analytic regularity of the diffeomorphism κ.
Let us also mention here the paper [33], where general elliptic problems whose coefficients slowly converge to their limits at infinity are considered. It is shown that any finite accumulation point of eigenvalues corresponding to exponentially decaying eigenfunctions is a threshold, these accumulations are characterized in terms of some non-classical "augmented scattering matrices." An additional investigation on decay of eigenfunctions at infinity is required in order to say whether these results describe accumulations of eigenvalues of the Dirichlet Laplacian ∆ or not. This goes beyond the scope of the present paper, we refer to [22].

Exponential decay of solutions in infinite PMLs.
In this section we prove the following theorem.
Theorem 7.1. Assume that µ 0 ∈ R\σ(∆ Ω ) is not an eigenvalue of the selfadjoint Dirichlet Laplacian ∆ in L 2 (G), the scaling parameter λ ∈ D α is not real, and Let s(ζ, η) stand for the scaling function R + × Ω ∋ (x, y) → s(x) written in the coordinates (ζ, η) ∈ C and extended to G by zero; see (2.2)-(2.4). Then for all sufficiently large r > 0 and all F ∈ L 2 (G) satisfying e βs F ∈ L 2 (G) the estimate is valid with a constant C independent of F . Theorem 7.1 together with Theorem 6.1 shows that under the additional assumption e βs (f • ϑ λ,r ) ∈ L 2 (G) on f ∈ H α (G) infinite PMLs absorb outgoing or incoming solutions (depending on the sign of ℑλ) so effectively that the function u λ,r in Theorem 7.1.3 exponentially decays at infinity in the mean.
Proof. Consider the conjugated operator e βs ∆ λ,r e −βs as an unbounded operator in L 2 (G) with the domain C ∞ 0 (G). With this operator we associate the quadratic form q β λ,r [u, u] = (e βs ∆ λ,r e −βs u, u) λ,r . Observe that where q λ,r is the same as in (4.1). Since the right hand side depends linearly on ∇ ζη u, similarly to the the proof of Lemma 4.3 one can deduce with an arbitrary small ǫ > 0 and a constant C independent of u ∈ C ∞ 0 (G). This estimate together with (4.8), (4.9), and Lemma 4.1 implies that for all sufficiently large r > 0 we have with some angle ϕ < π/2 and γ > 0, which are independent of u ∈ C ∞ 0 (G). Therefore e βs ∆ λ,r e −βs with the domain C ∞ 0 (G) is a densely defined sectorial operator in L 2 (G). Let D(e βs ∆ λ,r e −βs ) be the domain of its m-sectorial Friedrichs extension [23, Chapter VI.2]. The Friedrichs extension will also be denoted by e βs ∆ λ,r e −βs .
As in the proof of Proposition 4.4.1 we conclude that C ∞ 0 (G) is a core of the m-sectorial operator e βs ∆ λ,r e −βs . In Section 8 we will show that the graph norm of e βs ∆ λ,r e −βs is equivalent to the norm in H 2 0 (G); hence D(e βs ∆ λ,r e −βs ) = H 2 0 (G). Furthermore, we will localize the essential spectrum σ ess (e βs ∆ λ,r e −βs ) of the m-sectorial operator e βs ∆ λ,r e −βs , see Proposition 8.1. It turns out that the essential spectrum consists of an infinite number of parabolas, see Figure 7.1. In the case β = 0 the parabolas collapse to the dashed rays originating from the thresholds ν ∈ σ(∆ Ω ) and we obtain the essential spectrum σ ess (∆ λ,r ).
Let us note that for fixed λ and β the spectrum σ ess (e βs ∆ λ,r e −βs ) depends only on the behavior of the scaling function s and the matrix e λ,r outside any compact region of G. In order to control σ ess (e βs ∆ λ,r e −βs ) we imposed the condition (2.4).
Proposition 8.1. Assume that λ ∈ D α , β ∈ R, and r > 0 is sufficiently large. Then the following assertions hold.
1. The Hilbert spaces D(e βs ∆ λ,r e −βs ) and H 2 0 (G) are coincident and their norms are equivalent. 2. The bounded operator e βs ∆ λ,r e −βs − µ : is not Fredholm (or, equivalently, µ ∈ σ ess (e βs ∆ λ,r e −βs ) ) if and only if the parameters µ, λ, and β meet the condition Proof. The proof is essentially based on methods of the theory of elliptic nonhomogeneous boundary value problems [24,25,28,26]. We will rely on the following lemma due to Peetre, see e.g. [26,Lemma 5.1], [25,Lemma 3.4.1] or [32]: Let X , Y and Z be Banach spaces, where X is compactly embedded into Z. Furthemore, let L be a linear bounded operator from X to Y. Then the next two assertions are equivalent: (i) the range of L is closed in Y and dim ker L < ∞, (ii) there exists a constant C, such that u; X ≤ C( Lu; Y + u; Z ) ∀u ∈ X .
For an arbitrary positive weight w we have ̺ T u; L 2 (G) ≤ C wu; L 2 (G) with an independent of u ∈ H 2 0 (G) constant C. Thus the estimate (8.3) is a direct consequence of (8.11). By the Peetre's lemma we conclude that the range of the operator (8.1) is closed and the kernel is finite-dimensional.
We have proved that the operator (8.1) is Fredholm provided the condition (8.2) is not satisfied. Now we assume that the condition (8.2) is met, and show that the operator (8.1) is not Fredholm.
9. Problem with finite PMLs. Consider the truncated domain G R with piecewise smooth boundary, see (2.9). Introduce the Sobolev space H 2 0 (G R ) as the completion of the set C ∞ 0 (G R ) in the norm v; H 2 0 (G R ) = ( ℓ+|m|≤2 ∂ ℓ ζ ∂ m η v; L 2 (G R ) 2 ) 1/2 . In this section we study the problem with finite PMLs: Given g ∈ L 2 (G R ) find a solution v ∈ H 2 0 (G R ) of the equation where R > r. The next theorem presents a stability result for this problem.
Theorem 9.1. Assume that µ 0 ∈ R\σ(∆ Ω ) is not an eigenvalue of the selfadjoint Dirichlet Laplacian ∆ in L 2 (G). Take a sufficiently large r > 0 and λ ∈ D α \ R. Then there exists R 0 > r such that for all R > R 0 and g ∈ L 2 (G R ) the equation (9.1) has a unique solution v ∈ H 2 0 (G R ). Moreover, the estimate v; holds with an independent of R > R 0 and g constant C.
The proof of Theorem 9.1 will be carried out by the compound expansion technique. This requires construction of an approximate solution to the equation (9.1) compounded of solutions to limit problems. As the first limit problem we take the problem with infinite PMLs. As the second limit problem we take a Dirichlet problem in the semi-cylinder (−∞, R) × Ω studied in the next lemma.
Lemma 9.2. Introduce the weighted Sobolev space H 2 0,β (−∞, R)×Ω of functions satisfying the homogeneous Dirichlet boundary condition as the completion of the set Assume that λ ∈ D α \ R, µ 0 ∈ R \ σ(∆ Ω ), and β is in the interval (7.1). Then for any f ∈ L 2 β (−∞, R) × Ω there exists a unique solution u ∈ H 2 0,0 (−∞, R) × Ω to the equation Moreover, the estimate holds, where the constant C is independent of f and R.
Proof. It suffices to prove the assertion for R = 0. Then the general case can be obtained by the change of variables x → x − R.
Let us verify the condition ii. We have The support of the term [∆ λ,r , ̺ R ]u λ,r is a subset of the image of (R, R + 1/2) × Ω under the diffeomorphism κ. On this support the weight e βs is bounded from below by ce βR uniformly in R > 0. As a consequence we get the uniform in R estimates [∆ λ,r , ̺ R ]u λ,r ; L 2 (G R ) ≤ C 1 e −βR e βs u λ,r ; H 2 0 (G) ≤ C 2 e −βR e βs F ; L 2 (G) , (9.12) where we used Theorem 7.1. Now we estimate the second term in the right hand side of (9.11). On the support of [∂ 2 x , ρ R/3 ]u we have e −βx ≥ Ce −βR/3 . Relying on (9.4) we obtain Finally, consider the last term in the right hand side of (9.11). On the support of (1 − ρ R/3 )u the coefficients of the operator △ λ,r − ∆ Ω + (1 + λ) −2 ∂ 2 x tend to zero as R → +∞, cf. (3.4) and (8.6). This together with the estimate (9.4) for β = 0 gives where c R → 0 as R → +∞. From (9.10)-(9.14) it follows that w meets the condition ii. Thus w = w(R) is indeed an approximate solution to the problem with finite PML. Now we are in position to prove the assertion of the theorem. Observe that (∆ λ,r − µ 0 )w − g = O(R)g with some operator O(R) in L 2 (G R ), whose norm |||O(R)||| tends to zero as R → +∞ because of the condition ii on w. For all R > R 0 with a sufficiently large R 0 we have |||O(R)||| ≤ |||O(R 0 )||| < 1. Hence there exists the inverse (I + O(R)) −1 : L 2 (G R ) → L 2 (G R ) and its norm is bounded by the constant 1/(1 − |||O(R 0 )|||) uniformly in R > R 0 . We setg = (I + O(R)) −1 g. In the same way as before we construct the approximate solution w for the problem (9.1), where g is replaced byg. Then for v = w we have (∆ λ,r − µ 0 )v =g + O(R)g = g and v; H 2 0 (G R ) ≤ c g; L 2 (G R ) ≤ c/(1 − |||O(R 0 )|||) g; L 2 (G R ) , where C is independent of R > R 0 . Thus for R > R 0 and g ∈ L 2 (G R ) there exists a solution v ∈ H 2 0 (G R ) to the equation (9.1) satisfying the estimate (9.2), where the constant C is independent of R and g. In the remaining part of the proof we show that this solution is unique.
Let ∆ R λ,r be the unbounded operator in L 2 (G R ) such that for any v in its domain H 2 0 (G R ) we have ∆ R λ,r v = ∆ λ,r v. Note that ∆ R λ,r is the operator of the problem with finite PML. To the operator ∆ R λ,r there corresponds the quadratic form q R λ,r [v, v] = GR det e λ,r 1/2 e −1 λ,r ∇ ζη v, ∇ ζη det e λ,r −1/2 v dζ dη in L 2 (G R ) with the domain H 2 0 (G R ). In the same way as in Section 4 we conclude that the form q R λ,r admits a densely defined sectorial closure with the domain • H 1 (G R ) = ( ℓ+|m|≤1 ∂ ℓ ζ ∂ m η v; L 2 (G R ) 2 ) 1/2 . This together with the argument above implies that all µ < 0 with sufficiently large absolute value are regular points of the operator ∆ R λ,r . Hence the sectorial operator ∆ R λ,r coincides with its m-sectorial Friedrichs extension. Moreover, thanks to the argument above we know that under the assumptions of theorem for any g ∈ L 2 (G R ) there exists v ∈ H 2 0 (G R ) such that (∆ R λ,r − µ 0 )v = g. Similarly, one can study the adjoint m-sectorial operator (∆ R λ,r ) * . It turns out that under the assumptions of theorem for any g ∈ L 2 (G R ) there exists v in the domain H 2 0 (G R ) of (∆ R λ,r ) * such that (∆ R λ,r ) * − µ 0 v = g. Therefore the equation (9.1) is uniquely solvable in H 2 0 (G R ). In the next theorem we show that under some natural assumptions solutions v = v(R) of the problem with finite PMLs converge in the domain G r to outgoing or incoming solutions u ± with an exponential rate as R → +∞. In other words, we estimate the error produced by truncation of infinite PMLs. Solutions to the problem with finite PMLs can be found numerically with the help of finite element solvers; certainly, discretization produces yet another error that we do not estimate here. Theorem 9.3. Assume that µ 0 ∈ R\σ(∆ Ω ) is not an eigenvalue of the selfadjoint Dirichlet Laplacian ∆ in L 2 (G), the parameter r > 0 is sufficiently large in the sense of Remark 4.2, λ ∈ D α \ R, and β is in the interval (7.1). Let f ∈ H α (G) satisfy the inclusion e βs (f • ϑ λ,r ) ∈ L 2 (G), where s is the same function as in Theorem 7.1. In (9.1) we set g = f • ϑ λ,r . Then there exists R 0 > r such that for R > R 0 a unique solution v = v(R) ∈ H 2 0 (G R ) of the problem with finite PMLs converges in G r 1. to the outgoing solution u − ∈ H 2 0,loc (G) of the equation (∆ − µ 0 )u = f in the case ℑλ > 0 2. to the incoming solution u + ∈ H 2 0,loc (G) of the equation (∆ − µ 0 )u = f in the case ℑλ < 0 in the sense that as R → +∞ the estimate ℓ+|m|≤2 ∂ ℓ ζ ∂ m η (u ± − v R ); L 2 (G r ) 2 ≤ Ce −2βR e βs (f • ϑ λ,r ); L 2 (G) 2 (9.15) holds with a constant C independent of R > R 0 and f . Let us remark here that the assumptions of Theorem 9.3 on the right hand side f are a priori met for all f ∈ L 2 (G) such that f ↾ C = F • κ with some F ∈ E ; here E is the algebra defined in Section 5 and κ is the diffeomorphism (2.1). From Lemma 5.2 it follows that the set of functions f satisfying the assumptions of Theorem 9.3 is dense in L 2 (G). In particular, we can take any f ∈ L 2 (G) supported in G r .