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Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces

We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 when acting on an appropriate subdomain.


Introduction
In this short note we discuss the self-adjointness of the wave operator on generalizations of Minkowski space, answering a question of Jan Dereziński. More precisely, the setting is that of non-trapping sc-metrics, an extension of Lorentzian scattering metrics introduced in [3] and studied in more detail in [9], [6] and [18], with the Feynman propagator, whose role is discussed below, being particularly closely examined in the latter two papers. These are Lorentzian analogues of the Riemannian scattering metrics introduced by Melrose in [11]. The non-trapping condition is a condition on null-geodesics on M , namely they should converge to a replacement of the light cone at infinity in both the forward and backward directions. In fact, the signature of the metric makes no difference; the same conclusion is true for non-trapping semi-Riemannian metrics of any signature, as the proof goes through without any changes. Later on, we also discuss the limiting absorption principle, for which there is also a 'non-trapping at energy λ' condition which is a condition on the limiting geodesics at infinity corresponding to the spectral parameter being considered, see Section 2 for detail.
For the statement of the first result recall that for a (densely defined) unbounded operator L self-adjointness is a symmetry plus an invertibility (for the operator L ± ι), or equivalently surjectivity statement; essential self-adjointness amounts to a symmetry plus a dense range statement for the operator L ± ι. The author is very grateful to Jan Dereziński for suggesting this line of investigation and for very interesting discussions and comments on an earlier version of the manuscript. He is also grateful to Shu Nakamura, Kouichi Taira and Micha l Wrochna for stimulating discussions on this matter, and also to Micha l Wrochna for a careful reading of an earlier version of the manuscript. The author thanks the NSF for partial support under grant number DMS-1361432, the Simons Foundation for partial support via a Simons fellowship grant, and the Kyoto University RIMS for support during the conference at which the final conceptual ingredients of this project were completed.
As far as the author is aware, the first general mathematically precise result in this direction is that of Dereziński and Siemssen [5], see also [4], who assumed time-translation invariance, though there is a long history in the physics literature of treating the wave operator as at least a potentially self-adjoint operator. Note that this time translation invariance means that the overlap of the present paper with [5] is minimal. Our result also relates to recent/ongoing work of Nakamura and Taira, see [14], with details of the directly relevant aspects being parts of works in progress. In the work of Nakamura and Taira [14], the non-trapping condition (for the purpose of essential self-adjointness) is replaced by a positive energy condition.
The key part of proving the theorem is to show that ( g ± ı)u = f is solvable when f ∈ C ∞ c (M • ). Concretely, we take the weighted scattering Sobolev spaces being recalled below, and then a straightforward regularization argument shows that C ∞ c (M • ) is dense in it and g is symmetric on it. The main point is thus to show that C ∞ c (M • ) ⊂ ( g ± ι)D ⊂ L 2 sc and hence ( g ± ι)D is dense in L 2 sc . We do so by using a Fredholm framework for inverting g − λ on appropriate variable order Sobolev spaces discussed below; this in fact works uniformly to the real axis in λ, thus giving the limiting absorption principle. We then show additional regularity of the solution, proving the Theorem 1.
The aforementioned Fredholm framework gives rise to the massive Feynman propagators via the limiting absorption principle. Gérard and Wrochna studied these in a different Fredholm setting in [8,7], based in part on earlier work of Bär and Strohmaier [1].
We finally comment on some generalizations. Considering electromagnetic potentials A amounts to working with (−ıd − A) * g (−ıd − A) + V . If A, V are real and symbols of negative order (thus decaying) with values in one-forms, resp. scalars, all our results and arguments are unaffected. If A, V are real and symbols of order 0 then essential self-adjointness (including its proof) is unaffected.
Working with the wave operator on differential forms of any (or all) form degree or on tensors again does not affect the Fredholm theory; again adding to these symmetric (on C ∞ c (M • ) with values in forms or tensors, with the L 2 -inner product) first order operators with symbolic of negative order coefficients (such as the electromagnetic terms above) does not affect the Fredholm theory either. In particular, the limiting absorption principle holds in the sense of Theorem 5, i.e. a limiting resolvent exists, assuming that at λ ∈ R the a priori finite dimensional nullspace of P = g − λ and P * is trivial. (Both of these have to be assumed to be trivial: in Theorem 5, index 0 considerations mean that only one of these need to be assumed.) However, the argument for showing the triviality of the a priori finite dimensional nullspace of P = g − λ, λ / ∈ R, and its adjoint in Lemma 3 would be affected since the inner product with respect to which the operator is symmetric is no longer positive definite. However, if the operator has additional structure, the nullspace may be shown to be trivial by other arguments, e.g. Wick rotations work in the case of translation invariant metrics on R n ; the perturbation stability of the Fredholm framework implies the same conclusion on perturbations of these in the scattering category. Hence, in these cases (when the conclusion of Lemma 3 holds), the essential self-adjointness also holds.

Background
We now recall some background. We refer to [11] for the introduction of scattering, or sc-, structures, and to [16,17] for another discussion which emphasizes an R n -based perspective localizing to asymptotic cones. Recall that on a manifold with boundary M , the space of b-vector fields V b (M ) is the Lie algebra of smooth vector fields tangent to ∂M (indeed, this is the definition even for manifolds with corners, which will be used below for the compactified cotangent bundle), while the space of scattering vector fields or sc-vector fields is V sc (M ) = xV b (M ), where x is any boundary defining function, i.e. a non-negative C ∞ function on M , with zero set exactly ∂M such that dx is non-zero at ∂M . Such vector fields are exactly all smooth sections of a vector bundle, sc T M , over M , called the scattering tangent bundle, which over the interior M • is naturally identified with T M • . Indeed, notice that in a local coordinate chart, in which x is one of the coordinates, and the other coordinates (coordinates on ∂M ) are y 1 , . . . , y n−1 , V ∈ V sc (M ) means exactly that V = a 0 (x 2 ∂ x ) + n−1 j=1 a j (x∂ yj ) with a j smooth in the chart, so x 2 ∂ x , x∂ y1 , . . . , x∂ yn−1 give a local basis of smooth sections, and thus a local basis for the fibers of the vector bundle sc T M . Hence, the a j are coordinates on the fibers of sc T M (locally), and thus x, y 1 , . . . , y n−1 , a 0 , a 1 , . . . , a n−1 are local coordinates on the bundle sc T M . There is a dual vector bundle, sc T * M , called the scattering cotangent bundle, with local basis dx x 2 , dy1 x , . . . , dyn−1 x . A sc-metric of signature (k, n − k) is then a smooth (in the base point p) non-degenerate symmetric bilinear map sc T p M × sc T p M → R of signature (k, n−k). Equivalently, it is a smooth section of sc T * M ⊗ s sc T * M (symmetric tensor product) of the appropriate signature. Then L 2 sc is the L 2 -space of the metric density of any sc-metric (either Lorentzian or Riemannian, or of another definite signature), with all choices being equivalent in that they define the same space and equivalent norms, and H s,r sc is the corresponding weighted Sobolev space, so if s ≥ 0 integer then H s,0 sc = {u ∈ L 2 sc (M ) : ∀k ≤ s ∀V 1 , . . . V k ∈ V sc (M ), V 1 . . . V k u ∈ L 2 sc }, and H s,r sc (M ) = x r H s,0 sc (M ). As an example, M could be the radial compactification R n of R n , in which a sphere S n−1 is attached as the ideal boundary of R n , so the compactification is diffeomorphic to a closed ball. Concretely, a neighborhood of the boundary is diffeomorphic to [0, ǫ) x × S n−1 , ǫ > 0, whose interior, (0, ǫ) x × S n−1 is identified with the subset {z ∈ R n : |z| > ǫ −1 } via the reciprocal spherical coordinate map, where the sphere is regarded as a submanifold of R n to make sense of the map. Then any translation invariant metric of any signature is (i.e. can be naturally identified with) a sc-metric of the same signature. In fact, C ∞ (M ) is then the space of classical (one-step polyhomogeneous) symbols of order 0 on R n ,Ċ ∞ (M ) (the space of C ∞ functions on M vanishing to infinite order at ∂M ) is the space of Schwartz functions S(R n ), V sc (M ) is spanned by the lift of translation invariant vector fields ∂ zj , j = 1, . . . , n, over C ∞ (M ), i.e. an element of V sc (M ) is of the form a j ∂ zj with a j ∈ C ∞ (M ), i.e. a classical symbol of order 0, and similarly sc T * M ⊗ s sc T * M is spanned by the lifts of dz i ⊗ s dz j with C ∞ (M ) coefficients. Correspondingly, L 2 sc is the standard Lebesgue space, H s,r sc the standard weighted Sobolev space.
More generally, a coordinate neighborhood of a point on the boundary of a manifold with boundary M can be identified with a similar coordinate neighborhood of a point on the boundary of R n . Note that from the perspective of R n , such a neighborhood is asymptotically conic. Rather than following the above intrinsic definitions, one could transplant the notions discussed above directly from R n via such an identification, exactly how standard notions on manifolds without boundary are defined by identifying coordinate charts with open subsets of R n ; this is the approach taken by [17,16].
In particular, this gives a convenient way of introducing scattering pseudodifferential operators Ψ m,l scc (M ) by reducing to the case of R n , or equivalently to an appropriate uniform structure in the interior, R n . (The notation Ψ scc stands for 'scattering conormal'; [11] uses Ψ sc for 'classical' scattering; classical symbols are the one-step polyhomogeneous ones.) In the present case these are simply quantizations of (product-type) symbols of order (m, l), a ∈ S m,l , i.e. C ∞ functions on R n z × R n ζ such that for all α, β Note that as in [16,17], the second, decay order, uses the opposite sign convention than Melrose's original definition [11]; thus, the space Ψ m,l scc (M ) gets bigger with increasing m, l. One also has variable order pseudodifferential symbols and operators. In this paper the relevant order is the second, decay order, which we must allow to vary, thus l is in S 0,0 ; for this purpose we also need to relax the type of the symbol estimate and allow small power (more optimally logarithmic) losses: thus the estimates for a ∈ S m,l δ are, for fixed δ ∈ (0, 1/2), which could be taken small as one wishes for our purposes, The standard concepts, such as the principal symbol, still work; in the present case it lies in S m,l δ /S m−1+2δ,l−1+2δ δ , and it is multiplicative, i.e. the principal symbol of a product is the product of the principal symbols, of course with the appropriate orders as usual. Since the choice of δ is usually irrelevant, we typically suppress it in the subscripts. One can thus define the ellipticity, etc., as usual. Then the elements of Ψ 0,0 scc are bounded operators on all weighted Sobolev spaces H s,r sc , and one can define variable order Sobolev spaces (with just r variable for notational simplicity) by taking r 0 < inf r, and A ∈ Ψ s,r scc elliptic, and saying with the norm whose square is sc ; see [16,17] for details.
It is also important that in addition to principal symbols of products, we can compute principal symbols of commutators.
and with a, resp. b, denoting the principal symbols of A, resp. B, its principal symbol is the Poisson bracket 1 i {a, b} (which, recall, arises from the symplectic structure on T * M • ). In the case of M being the radial compactification of R n , this is simply Writing coordinates on the fibers of the scattering cotangent bundle as τ, µ 1 , . . . , µ n−1 , sc-dual to the coordinates x, y 1 , . . . , y n−1 discussed above, i.e. sc-covectors are written as see [11,Equation (5.24)], as follows by a change of variables computation. Here H a is called the Hamilton vector field of a. Finally, we need to discuss microlocalization. Since there are two different important behaviors, in the case of T * R n these being |z| → ∞ and |ζ| → ∞, it is even more useful to compactify phase space than in the usual microlocal analysis setting, where using homogeneity in dilations of the fibers of the cotangent bundle is an effective substitute. In the case of T * R n , this compactified phase space is sc T * R n = R n z × R n ζ . i.e. we compactify the position and the momentum space separately using the above radial compactification. Thus sc T * R n is the product of two closed balls, and hence it is a manifold with corners. The two boundary hypersurfaces are R n × ∂R n , which is 'fiber infinity', where standard microlocal analysis takes place, and ∂R n × R n , i.e. 'base infinity'; these intersect in the corner ∂R n × ∂R n . The locus of microlocalization is then thus the elliptic set, the characteristic set and the wave front set are subsets of this.
These notions immediately extend to general manifolds via the local coordinate identifications. The general compactified phase space is the fiber-radial compactification sc T * M of sc T * M ; the locus of microlocalization is its boundary where sc S * M is fiber infinity, i.e. the boundary of the fiber compactification. One can take x as a boundary defining function of base infinity sc T * ∂M M and (with the local coordinate notation from above) ρ ∞ = (τ, µ) −1 = (|(τ, µ)| 2 + 1) −1/2 as a defining function of fiber infinity, where |.| is the length with respect to any Riemannian sc-metric. Relating to the above discussion of variable order spaces, one may have e.g. the decay order as smooth function on sc T * ∂M M ; in order to match with the previous definition one extends it to a smooth function on all of sc T * M , with all potential extensions resulting in exactly the same Sobolev space as can be seen immediately from the definition.
Note that in view of the vanishing factor x on the right hand side of (1), as well as the one order lower than that of a homogeneity of one may want to rescale H a , factoring out this vanishing. In order to obtain a well-behaved, namely smooth and at least potentially non-degenerate, vector field on the compactification, one may consider If a is classical this becomes a smooth vector field tangent to the boundary of the compactified space sc T * M , and thus defines a flow on it (the Hamilton flow); for general a the vector field is conormal of order (0, 0) as a vector field tangent to the boundary, i.e. is in S 0,0 V b ( sc T * M ). Note that the defining functions we factored out are defined only up to a smooth positive multiple (smooth on the compact space, thus this means bounded from above and below by positive constants in particular), and hence the rescaled vector field is only so defined, but such a change merely reparameterizes the flow, which is irrelevant for considerations below. Radial points of H a are then points on the boundary of sc T * M at which the H a,s,r vanishes (as a vector in T sc T * M ), thus are critical points of the flow. Note that at such points H a,s,r need not vanish as a b-vector field, so e.g. it may have a non-trivial x∂ x component (or the analogous statement at fiber infinity); indeed this non-vanishing is what makes the radial point non-degenerate: in analytic estimates x∂ x hitting the weight x m of a commutant is what can give a contribution of a definite sign.
After these differentiable manifold structure type discussion in the sc-category, we briefly discuss the geometry, namely metrics. For the purposes of the present paper what we need is that the metric g is a Lorentzian (or more general pseudo-Riemannian) sc-metric for which the Hamilton flow has a source/sink structure. Thus, we need that there are submanifolds of sc T * ∂M M which are transversal to sc S * ∂M M and are normal sources (meaning normally to the submanifold they are sources) L − , resp. sinks L + , of the Hamilton flow of the dual metric function, G (the principal symbol of g ). The definition of a non-trapping sc-metric g is then that g is a sc-metric (of some signature) such that all integral curves of H G inside the characteristic set {G = 0} at sc S * M (i.e. fiber infinity), except those contained in L ± , tend to L + (indeed, necessarily to L + ∩ sc S * M ) in the forward and L − in the backward direction. We call a sc-metric non-trapping at energy λ if it is nontrapping in the sense above, and in addition, all integral curves of H G inside the λ characteristic set {G = λ} at sc T * ∂M M (i.e. base infinity), except those contained in L ± , tend to L + (indeed, necessarily to L + ∩ sc T * ∂M M ) in the forward and L − in the backward direction.
The class of non-trapping Lorentzian sc-metrics is a much larger class of metrics than that of Lorentzian scattering metrics introduced in [3], for the latter class also demands that L ± be (scattering) conormal bundles of submanifolds S ± of ∂M at which the metric has a certain model form, generalizing that of the Minkowski metric on the radial compactification, and also that S ± are (non-degenerate) zero sets of a certain function v ∈ C ∞ (∂M ), and {v > 0} has two connected components C ± ('spherical caps' in Minkowski space, and though they may not be spherical in any sense, we continue calling them so in general) with S ± as their respective boundaries, while {v < 0} = C 0 has two boundary components S ± . The 'nontrapping at energy λ' condition played no role in [3], since that paper considered the wave equation (λ = 0) and rescaled the wave operator to a b-operator, for which the non-trapping condition is exactly the above sc-non-trapping condition; this rescaling is possible and the non-trapping claim holds since the dual metric G is necessarily a homogeneous quadratic polynomial on the fibers of sc T * M , unlike G − λ for non-zero λ.

Fredholm theory and essential self-adjointness
In our proof of the main theorem we focus on the Lorentzian case of signature (1, n − 1) for the sake of being definite in terminology; the general pseudo-Riemannian case barely differs, except that the only reasonable problems are the Feynman and anti-Feynman problems, but they are also the only ones that matter below. In all of our discussions below we assume that the metrics are non-trapping sc-metrics.
In order to get started, we first recall that (assuming M is connected -otherwise the statement is for each connected component) for λ > 0 the Klein-Gordon operator P = g − λ has four Fredholm problems, see [21,17,16], corresponding to the characteristic set having two connected components. (This follows from the characteristic set {ζ : G(ζ, ζ) = λ}, G = g −1 , having two connected components fiberwise: this fiberwise characteristic set is the two-sheeted hyperboloid.) Indeed, in each connected component of the characteristic set one can choose the direction in which one propagates estimates for P , and then for P * using dual spaces one propagates the estimates in the opposite direction, resulting in 2 2 possibilities. Concretely, these are the retarded, advanced, Feynman and anti-Feynman, Fredholm problems; the direction of propagation is encoded by the use of appropriate weighted Sobolev type spaces. Concretely, these are based on variable order Sobolev spaces H s,r sc , where s is constant, r variable, a function on sc T * ∂M M , monotone along the rescaled Hamilton flow, and satisfies the inequalities r > −1/2 at the radial points from which estimates are propagated, r < −1/2 at the radial points to which estimates are propagated, and Fredholm. Corresponding to the above discussion, this fact relies on the nontrapping nature of the bicharacteristic flow within the characteristic set, which has two parts: the part at fiber infinity, in sc S * M , which is independent of λ, and the part at spatial infinity, sc T * ∂M M , which does depend on λ. In particular, these non-trapping conditions are perturbation stable, and hold for the Minkowski (as well as translation invariant pseudo-Riemannian on R n ) metrics. Note that the choice of s, r with r satisfying the above constraints is irrelevant; and the nullspace automatically lies in the intersection of all these spaces; thus in particular for elements u of the nullspace of P , WF sc (u) is a subset of the radial points towards which the estimates are propagated. If λ < 0 (and n = dim M ≥ 3), then due to the behavior of the characteristic set at base infinity, the characteristic set has only one connected component (the characteristic set {ζ : G(ζ, ζ) = λ} has one connected component fiberwise: this fiberwise characteristic set is the one-sheeted hyperboloid.), and correspondingly only two of these problems remain: Feynman and anti-Feynman problems. Note that the Cauchy problem (or retarded/advances problems) is still solvable, but the solution will typically grow exponentially, thus it does not exist as far as polynomially weighted Sobolev spaces (our world in this paper) are concerned. (For λ = 0 we still have the four problems, but in weighted b-Sobolev spaces, see [6], which we do not discuss here.) Now, if λ is made complex, of course the usual principal (and even the subprincipal!) symbol are not affected, and correspondingly estimates at fiber infinity, sc S * M , are unchanged over M • . However, the estimates at sc T * ∂M M become more delicate.
Namely, in this region one has a non-real principal symbol (since λ is part of it). Thus, by the usual propagation estimates (these are the ones used for 'complex absorption') one can propagate estimates in the forward direction along the H Re p flow when Im λ ≥ 0, and in the backward direction when Im λ ≤ 0. (See [17,Section 5.4.5] and [16], as well as the usual microlocal analysis version in [15,Section 2.5].) Of course, the operator is elliptic at finite points (not at sc S * M ) of sc T * ∂M when Im λ > 0; the point is that the estimates work at the corner (fiber infinity at ∂M ), and they work uniformly in λ even as Im λ → 0. This propagation works for any s, r (s a priori relevant only when one is at fiber infinity at ∂M ), including variable r, when r is monotone decreasing in the direction in which the estimates are propagated.
Notice that corresponding to the ellipticity at finite points, for these estimates, as well as the ones below, the only relevant non-trapping condition is the basic one concerning bicharacteristics at fiber infinity, sc S * M , i.e. the 'non-trapping at energy λ' condition is only relevant when one wants to let λ to the real axis, as we do below for the limiting absorption principle (but not for the essential self-adjointness discussion), and then the relevant condition is non-trapping at the limiting energy λ.
Most importantly though one needs to get radial point estimates, however. For real λ = 0 these are two estimates, see [17,Section 5.4.7] and [16], as well as the usual microlocal analysis version in [15,Section 2.4]. The first, high 'regularity' (where here this means decay), which gives a 'free' estimate at the radial point, is of the form when r > r ′ > −1/2, s, s ′ , M, N arbitrary (one usually considers s ′ , M, N very large and negative; these are background error terms with relatively compact properties), Q elliptic at the radial set, with wave front set in a small neighborhood, Q 1 elliptic on WF ′ (Q) and bicharacteristics from all points in the intersection of WF ′ (Q) and the characteristic set tend to L in the appropriate (forward/backward) direction depending on the sink/source nature, remaining in Ell(Q 1 ). The second, low 'regularity' (where again here this means decay), which allows one to propagate estimates into the radial point from a punctured neighborhood, is of the form when r < −1/2, s, s ′ , r ′ , M, N arbitrary (one considers s ′ , r ′ , M, N very large and negative), Q elliptic at the radial set, with wave front set in a small neighborhood, Q 1 elliptic on WF ′ (Q) and bicharacteristics from all points in WF ′ (Q) intersected with the characteristic set which are not in L, tend to L in the appropriate (forward/backward) direction depending on the sink/source nature, remaining in Ell(Q 1 ), and intersect Ell(Q 2 ) at some point in the opposite direction along the flow, still remaining in Ell(Q 1 ). Now allowing λ complex, say Im λ ≥ 0, one can only propagate estimates in the forward direction along the H Re p -flow, and correspondingly one obtains the high regularity estimates only at the sources, the low regularity ones at the sinks (with sources and sinks reversed for Im λ ≤ 0). These estimates in fact become stronger than the ones above, cf. the complex absorption arguments in [17,Section 5.4.5] and [16], namely one can in addition control a term Qu H s−1/2,r+1/2 sc , i.e. one that is stronger in the sense of decay (though not differentiability) than that on Qu above. This results from an extra term Ǎ * Im λǍu, u = Im λ Ǎ u 2 in the estimate, in addition to the commutator terms, [Ǎ * Ǎ , ]u, u , withǍ ∈ Ψ m ′ /2,l ′ /2 scc , s = (m ′ + 1)/2, r = (l ′ − 1)/2. Thus the estimates are when r < −1/2, s, s ′ , r ′ , M, N arbitrary. Now, taking r with −1/2 < r at the sources, −1/2 > r > −1 at the sinks, monotone along the flow, s > 1/2, this in particular gives: Then for s, r as above corresponding to either the Feynman spaces (Im λ ≥ 0) or anti-Feynman spaces (Im λ ≤ 0), the operator sc , X s,r = {u ∈ H s,r sc : P u ∈ H s−1,r+1 sc }.
One can interpret the estimates (2)-(3), as well as the analogous real principal type estimates in the characteristic set between the radial points as additional regularity estimates giving that in fact In particular, this lets one solve P u = f , f ∈Ċ ∞ (M ), up to finite dimensional obstacles, namely one gets that the solution u (which exists in the complement of a finite dimensional subspace) is almost in L 2 , namely u ∈ H 0,−ǫ sc for all ǫ > 0. Indeed, we have that if P u = f , f ∈Ċ ∞ (M ), with u ∈ X s,r as above, then u ∈ Hs −1/2,r+1/2 sc for alls and for allr < −1/2, thus in H ∞,−ǫ sc for all ǫ > 0. This is not quite sufficient, however, since we want to conclude u ∈ L 2 sc , and also that there are no finite codimension issues (i.e. we have invertibility and not just Fredholmness) so, for Im λ > 0, one needs to do a borderline estimate, with r = −1/2 at the sink (everywhere else one is in L 2 already), which corresponds to l ′ = 0. Note that such an estimate cannot work when Im λ = 0, and thus cannot be uniform in Im λ when Im λ > 0. The key point is that in this case the commutator [Ǎ * Ǎ , ] will have principal symbol at sc S * M for which the normally main term (arising from the weight) vanishes at L.
It suffices for us to consider m ′ = l ′ = 0, in which case the situation is very simple: we will takeǍ to be microlocally the identity near the sinks, i.e. to have WF ′ sc (Id−Ǎ) disjoint from the sink. (Such microlocalizers play an important role in the proof of asymptotic completeness in the N -body setting; a partially microlocal version is the work of Sigal and Soffer [13] and Yafaev [22], see [20] for a discussion.) Since WF sc (u) is in the sink when f ∈Ċ ∞ (M ), for u ∈ H s ′ ,r ′ sc (with e.g. r ′ = r + 1/2 from above, so < 0 but close to 0 allowed) the pairing u, [Ǎ * Ǎ , ]u makes sense (and a regularized version remains bounded: the regularizer gives the correct sign as it behaves exactly the same way as if one had a more decaying weight, i.e. as if l ′ < 0) if 2r ′ − l ′ + 2 ≥ 0 and 2s ′ − m ′ − 1 ≥ 0, which holds with l ′ = m ′ = 0 if r ′ < 0 is close to 0 and s ′ = 1, say. Then the Im λ term gives an estimate for Ǎ u 2 , which is an estimate for u in H m ′ /2,l ′ /2 sc = L 2 sc as desired. In particular, if u ∈ X s,r and P u ∈Ċ ∞ (M ) then u ∈ L 2 sc .
Lemma 3. Suppose that Im λ > 0, and consider the Feynman Fredholm problem X s,r → Y s−1,r . Then KerP and KerP * (on the dual space) are trivial. Analogous statements hold for Im λ < 0 for the anti-Feynman Fredholm problem.
Proof. We have already seen that elements of KerP and KerP * lie in L 2 sc . Thus, formally the lemma follows from 0 = P u, u − u, P u = (P − P * )u, u = −2ı Im λu, u = −2ı Im λ u 2 , but the issue is that P * u, u does not actually make sense a priori due to the too weak a priori differentiability of u when the unweighted spaces are used (all we know is that u ∈ L 2 sc , so P * u ∈ H −2,0 sc only, unless we use P − P * ∈ Ψ 0,0 scc , but even then we need to justify the integration by parts (because the adjoint a priori puts us in dual spaces)!), so we need to have a more careful, if standard, regularization argument.
Namely, we take Λ t ∈ Ψ −∞,0 scc , t ∈ [0, 1] such that the family is uniformly bounded in Ψ 0,0 scc and converges to Id in Ψ ǫ,0 scc , ǫ > 0, as t → 0, and thus strongly on L 2 sc . Then we have for t > 0, if u ∈ L 2 sc and P u ∈ L 2 sc , 0 = P u, u − u, P u = lim , then the last term vanishes, and we conclude that u = 0. But we have seen that in the Feynman spaces this holds, namely u ∈ H ∞,−ǫ sc for all ǫ > 0, so we conclude that u 2 = 0 and thus u = 0 as well.
The analogous argument also holds for P * on the anti-Feynman space, which proves that P * is also injective.
Corollary 4. Suppose Im λ = 0. The operator P : X s,r → Y s−1,r+1 is indeed invertible (not just Fredholm) and moreover we have for f ∈Ċ ∞ (M ) that u = P −1 f ∈ L 2 sc as well. We take D = {u ∈ H 1,−1/2 sc ∩ L 2 sc : g u ∈ L 2 sc }. Actually H 1,−1/2 sc in this definition could be replaced by H s ′ ,r ′ sc for any s ′ ∈ [1, 2], r ′ ∈ [−1/2, 0), as is immediate from the following argument; slightly increasing the requirements on the above choice of r is then needed for the statement of the next sentence. Then for either sign of Im λ, and corresponding choices of s, r (only r depends on Im λ) as above with the slightly stronger requirements s ≥ 2, while r > −1, with r < −1/2 at the sinks, we have X s,r ∩ L 2 sc ⊂ D since r takes values in (−1, ∞), so r + 1/2 in (−1/2, ∞) and since ( g − λ)u ∈ H s−1,r+1 Furthermore, g is symmetric on this domain since Indeed, as noted above [ g ,Λ t ] is uniformly bounded in Ψ Thus, g : D → L 2 sc is a continuous map, C ∞ c (M • ) is dense in D (by virtue oḟ C ∞ (M ) being so), and g is a symmetric operator. In order to prove that g is essentially self-adjoint, it suffices to prove that for λ / ∈ R, g − λ has a dense range in L 2 sc . ButĊ ∞ (M ) is dense in L 2 sc , so it suffices to show that for Im λ = 0 and f ∈Ċ ∞ (M ) there exists u ∈ D such that ( g − λ)u = f . But we have seen above that under these conditions there exists u ∈ X s,r such that ( g − λ)u = f , and moreover u ∈ L 2 sc , so as X s,r ∩L 2 sc ⊂ D in view of (4), the desired conclusion follows. (Here the limitations on s ′ , r ′ are strongly relevant and required for the generalized version of D discussed above, together with the corresponding strengthening of the requirements on s, r.) This proves that g is essentially selfadjoint on C ∞ c (M • ), namely proves Theorem 1.

The limiting absorption principle
The limiting absorption principle is an immediate consequence of our discussion. Namely, under the assumption of g being non-trapping at energy λ for the limiting λ (or interval of λ's, if one wishes), the estimates for g − λ on the Feynman spaces are uniform in Im λ ≥ 0, and similarly on the anti-Feynman spaces in Im λ ≤ 0; and indeed, for λ ∈ R, g − λ is Fredholm on either one of these spaces. Furthermore, when Im λ = 0, the operator is invertible, thus index 0, and this is stable under perturbations (even of the kind we discussed), cf. [15,Section 2.7], which also discusses continuity in the weak operator topology. In particular, the limit is the (anti-)Feynman propagator, up to finite dimensional nullspace issues on the limiting space. Thus, Theorem 5. Suppose λ ∈ R \ {0}, g is a non-trapping sc-metric which is nontrapping at energy λ and g − λ has trivial nullspace on either the Feynman or the anti-Feynman function spaces. Then lim ǫ→0 ( g − (λ ± ıǫ)) −1 exist in the weak operator topology on the Feynman (+), resp. anti-Feynman (−) function spaces, and is the Feynman, resp. anti-Feynman propagator, i.e. the inverse of g − λ on the appropriate function spaces. Remark 6. The argument of [18, Proposition 3.1] applies with minor notational changes (corresponding to the b-setting employed there and the sc-setting employed here) to prove that the primed wave front set of the Schwartz kernel of ( g − (λ ± ı0)) −1 is in the backward/forward flowout of the diagonal of the cotangent bundle T * M • over the interior of M .
Of course, it is still a question whether the nullspace of g −λ is trivial; the set of λ for which it is, is necessarily open by stability. Again, this stability is true even for the relatively drastic kind of perturbations we have which change the domain space (since the space depends strongly on λ via the condition ( g −λ)u ∈ H s−1,r+1 sc ); the point is that the relevant estimates on fixed spaces, with fixed relatively compact error terms are perturbation stable. Interestingly, cf. the discussion of [21,Section 4], adopting arguments of Isozaki [10] from N -body scattering as done in [19, Proof of Proposition 17.8], which are valid after minor modification in this setting as we discuss below, any element of the nullspace of g − λ in either the Feynman or the anti-Feynman spaces in fact lies inĊ ∞ (M ): Proposition 7. On both the Feynman and anti-Feynman function spaces the nullspace of P = g − λ is a subspace ofĊ ∞ (M ).
Proof. The arguments following [10] rely on using the commutant χ ǫ (x) = ǫ −2r−1 x/ǫ 0 φ(s) 2 s −2r−2 ds, are either all sources or all sinks, thus there is a single definite sign in (5), arising for the common source, or sink, nature of them. For other Fredholm problems, the elements of the nullspace are not necessarily in the 'trivial space',Ċ ∞ (M ).
Thus, the absence of embedded eigenvalues depends on a unique continuation argument at infinity, namely that the rapid decay (infinite order vanishing) at ∂M of an element of KerP implies its vanishing nearby.
In the case of non-trapping Lorentzian scattering metrics (possibly long range), as in [3,2], if one assumes that the there is a boundary defining function ρ of M such that, say, near the past 'spherical cap' C − , dρ ρ 2 , is timelike (which for instance is true on perturbations of Minkowski space), for λ > 0 energy estimates imply that, being an element ofĊ ∞ (M ), an element of this nullspace vanishes identically at first near C − , and then the non-trapping condition implying global hyperbolicity, see [9,Section 5] in this setting for this implication, vanishes globally, so the nullspace is indeed trivial. An analogous conclusion holds by a Wick rotation argument, see [6], for the Minkowski metric, as well as pseudo-Riemannian translation invariant metrics, and again the perturbation stability implies that the conclusion also holds for their perturbations in the sc-category.
We finally remark that the λ = 0 Fredholm problem was studied in [6]; one can also discuss the limiting absorption principle there, under somewhat stronger conditions than we needed here, but we defer it to future work.