The Klein-Gordon operator

Transcription

1 The Klein-Gordon operator with Atiyah-Patodi-Singer-type boundary conditions on asymptotically Minkowski spacetimes joint work w. Christian Gérard arxiv: Michał Wrochna Université Grenoble-Alpes

2 INVERSES O THE KLEIN-GORDON OPERATOR On Minkowski space R 1,d, η), P = t 2 x, has four natural inverses P 1 I, I = {+,,, }: 1 advanced/retarded P ± τ, k) = τ i0) 2 k 2 ) 1, 1 eynman/anti-eynman P τ, k) = τ 2 k 2 ± i0) 1 / On globally hyperbolic spacetimes M, g) P = g + m 2, m 0 has four classes of parametrices P 1) I, described by singularities of P 1) I x, x ) [Duistermaat, Hörmander 72] Let N = p 1 {0}) = N + N. or instance eynman: W P 1) ) = diag T M ) t 0 Φ tdiag T M ) π 1 N ) here Φ t bicharacteristic flow, π projects to left leg) In applications in QT & non-linear PDEs one needs approximate inverses in a stronger sense!

3 REDHOLM SETUP OR EYNMAN PROPAGATOR Two recent results give redholm property of P : X Y with X, Y Hilbert spaces with built-in generalized Atiyah - Patodi - Singer-type boundary conditions: P Dirac operator on [t 1, t 2 ] Σ, Σ closed manifold [Bär, Strohmaier 15] index theorem available index related to particle creation and anomalies P wave operator on asymptotically Minkowski spacetime [Gell-Redman, Haber, Vasy 15] positivity i 1 P 1 P 1 ) ) 0 [Vasy 15] KerP X C, [Vasy, W. 15] P 1 closest possible analogue of + m 2 ) 1!! In this talk: redholm problem for Klein-Gordon operator P = g + m 2, m > 0 on asymptotically Minkowski spacetimes

4 GEOMETRICAL SETUP We assume M, g) asymptotically Minkowski spacetime, i.e. M = R 1+d and g Lorentzian metric s.t. am) g µν η µν S δ R 1+d ) for δ > 1, where f S δ means α x f O 1 + x 2 ) δ α 2 ), time function t s.t. t t S 1 ɛ R 1+d ), ɛ > 0. In practice: reduction to model case P = 2 t + rt) t + at, x, D x ), and a out/in x, D x ) elliptic, a out/in x, D x ) C > 0 s.t. at, x, D x ) = a out/in D x ) + Ψ 2, δ std, δ > 1, rt) Ψ 0, 1 δ std. This means decay of P P out/in, where P out/in = 2 t + a out/in.

5 IN/OUT STATES Let Ut, s), U free t, s) Cauchy evolution for P, P free = 2 t x + m 2 i.e. maps Cauchy data ϱ s u = u, i 1 t u) t=s to ϱ t u In/out covariances: t-dependent projections c ± out t) = lim t + Ut, t +)c ± free Ut +, t), c ± in t) = lim t Ut, t )c ± free Ut, t), where c ± free = 1 ± ) x + m 2 ± x + m 2 1 Theorem [Gérard, W.]) or all Cauchy data h, WUt, t 0 )c ± t 0 )h) N ± : In QT terms, the in/out states are Hadamard. = R ± spectral projection of generator of U free t, s)

6 MAIN RESULT ix t 0 R and define eynman/anti-eynman scattering data: ϱ u = lim c + t ± ± free U freet 0, t + )ϱ t+ u + c free U freet 0, t )ϱ t u ), ϱ u = c + free U freet 0, t )ϱ t u + c free U freet 0, t + )ϱ t+ u ). lim t ± ± Theorem [Gérard, W.]) Let Y m = t γ L 2 R; H m R d )), 1 2 γ < δ and X m = {u C 0 R; H m R d )) C 1 R; H m+1 R d )) : Pu Y m, ϱ u = 0}. Then P : X m Y m is redholm of index ind P = ind c free W 1 out + ) c+ free W 1 in, where W 1 out/in = lim U free0, t ± )Ut ±, 0). t ± ± urthermore, there exists P 1 : Y X with eynman wave front set and s.t. 1 PP 1 and 1 P 1 P are compact with smooth kernel.

7 1 st STEP APPROXIMATE DIAGONALIZATION Suppose we have bt) C R, Ψ 1 R d )) s.t. t + ibt) + rt)) t ibt)) = 2 t + at) + rt) t Hence also t ib + r) t + ib ) = t 2 + a + r t Set ψt) t + ib = ) t) φt). Now t ibt) t 2 + a + r t)φt) = 0 i 1 t ψt) = Bt) ψt), Bt) = b t) + irt) 0 0 bt) + irt) ) U b This way, U B t, s) = +irt, s) 0 and 0 U b+ir t, s) Ut, s) = Tt)U B t, s)t 1 s) where ψt) = Tt) ψt). In practice we get the factorization modulo smooth errors, but we can correct U B afterwards. ).

8 1 st STEP APPROXIMATE DIAGONALIZATION Suppose we have bt) C R, Ψ 1 R d )) s.t. t + ibt) + rt)) t ibt)) = 2 t + at) + rt) t Hence also t ib + r) t + ib ) = t 2 + a + r t Set ψt) t + ib = ) t) φt). Now t ibt) t 2 + a + r t)φt) = 0 i 1 b t ψt) = Bt) ψt), t) + irt) 0 Bt) = 0 bt) + irt) ) U b This way, U B t, s) = +irt, s) 0 and 0 U b+ir t, s) Ut, s) = Tt)U B t, s)t 1 s) where ψt) = Tt) ψt). We construct bt) precisely enough to control decay of bt) a 1/2 t), bt) a 1/2 using Ψ m,δ std -calculus) ).

9 2 nd STEP MØLLER OPERATORS ϱ out u = ϱ u = ϱ u = lim t ± ± lim t ± ± c + free U freet 0, t + )ϱ t+ u + c free U freet 0, t )ϱ t u ), c + free U freet 0, t )ϱ t+ u + c free U freet 0, t + )ϱ t u ), lim U freet 0, t + )ϱ t+ u, ϱ in u = lim U freet 0, t )ϱ t u. t + + t or solutions, ϱ t+ u = Ut +, t 0 )ϱ t0 u, so ϱ I u = W I ut 0), where W out/in = s lim Ut +, t 0 )U free t 0, t ± ), W t ± ± out/in W out/in = 1, W = W out c + free + W inc free, W = W inc + free + W outc free, So Cauchy data of KerD Kerϱ is simply KerW. Lemma [Gérard, W.]) or α < δ/2, = in, out, c ± t 0) = W c + free W 1 = c ± ref t 0)+Tt 0 ) x α Ψ R d ) x α T 1 t 0 ).... so W W = 1+ compact, smoothing term, same for W W!

10 3 rd STEP INHOMOGENEOUS CAUCHY PROBLEM Need Hilbert space Y s.t. { Pu = f, f Y ϱ t0 u = v, v H 1 R d ) L 2 R d ) has unique solution, eg. Y = t γ L 2 R; H 1 R d )), γ > 1 2. This gives u X = {u C 0 R; H 1 R d )) C 1 R; L 2 R d )) : Pu Y}. redholm analysis: ϱ t0 P : X H 1 R d ) L 2 R d )) Y isomorphism ϱ P : X H 1 R d ) L 2 R d )) Y redholm reduction to ϱ : KerP X H 1 R d ) L 2 R d )) redholm P : Kerϱ X }{{} X Y redholm! The index is ind W.

11 REDHOLM INVERSE irst guess: eynman propagator associated to in or out) state P 1,in f )t) = iπ 0 Ut, t 0 )c + in t 0) 1 t<s )Ut 0, s)ϱ s f ds with π 0 proj. to first component of Cauchy data). P 1,in does not map Y X recall X = X Kerϱ ) Theorem [Gérard,W.]) One can find P 1 : Y X that has eynman wave front set, satisfies i 1 P 1 P 1 ) ) 0, and 1 PP 1 and 1 P 1 P are compact with smooth kernel. In practice: P 1 = analogue of P 1,in constructed using diagonalizable part of the dynamics.

12 OUTLOOK The new: redholm property of Klein-Gordon operator on asymptotically Minkowski spacetimes Hadamard property of in/out states works also for asymptotically static spacetimes with Cauchy surface of bounded geometry Questions: Index formula as in [Bär, Strohmaier 15]? Dirac case? Applications of the latter to anomalies? Relation to [Gell-Redman,Haber,Vasy 15]? Complex powers?? More general spacetimes: non-static free dynamics Dereziński s conjecture: P 1 = boundary value of resolvent of a self-adjoint extension of P? Thank you for your attention!

13 APPENDIX Ψ m,δ std CLASSES or m, δ R, S m,δ std R; T R d ) = functions at, x, k) s.t. γ t α x β k at, x, k) O t + x )δ γ α k m β ), γ N, α, β N d. Let W std R; Rd ) space of operator-valued functions at) s.t. D 2 x + x 2 ) m γ t at)d2 x + x 2 ) m BL 2 R d )) O t n ), m, n N. inally Op w Weyl quantization, ph k-polyhomogeneity): Ψ m,δ std R; Rd ) = Op w S m,δ std,ph R; T R d )) + W std R; Rd ). Theorem [Gérard,W.]) Let a Ψ m,0 std R; Rd ) be elliptic, selfadjoint with at) c 0 1 with c 0 > 0. Then a α C b) R; Ψmα Σ)) for any α R and σ pr a α )t) = σ pr at)) α. proof by reduction to [Ammann, Lauter, Nistor, Vasy 05]