Essential self-adjointness of real principal type operators

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.


Introduction
In this paper, we consider formally self-adjoint real principal type operator P = Op(p) on the Euclidean space R n with n ≥ 1, where Op(•) denotes the Weyl quantization.A typical example is the Klein-Gordon operator with variable coefficients (see Remark 1.2), and the propagation of singularities plays an essential role in the proof of the essential self-adjointness.
Our main theorem is the following: 1 Theorem 1.1.Suppose Assumption A and B. Then P = Op(p) is essentially self-adjoint on C ∞ c (R n ).Remark 1.2.(Klein-Gordon operators on asymptotically Minkowski spaces) Let g 0 be the Minkowski metric on R n : g 0 = dx 2  1 − dx 2 2 − ... − dx 2 n and g −1 = (g ij 0 ) n i,j=1 be its dual metric.A Lorentzian metric g on R n is called asymptotically Minkowski if g −1 (x) = (g ij (x)) n i,j=1 satisfies, for any α ∈ Z n + there is C α > 0 such that satisfies Assumption A. The essential self-adjointness for this model is studied by Vasy [17].
Remark 1.3.In this paper, we only deal with operators with order greater than 1.The essential self-adjointness of first order operators on C ∞ c (R n ) can be proved by Nelson's commutator theorem with its conjugate operator N = −∆ + |x| 2 + 1 ([14, Theorem X.36]).We also note that if P commutes with the complex conjugation: P u = P u, then, it is enough to assume the forward null non-trapping condition only instead of null non-trapping condition (cf.[14,Theorem X.3]).
The study of essential self-adjointness has a long history but mostly on operators of elliptic type (see [14] Chapter X and reference therein).For the construction of solutions to evolution equation with real principal type operators, we refer the classical paper [3] by Duistermaat and Hörmander, and the textbook by Hörmander [8].Chihara [2] studies the well-posedness and the local smoothing effects of the Schrödinger-type equations : ∂ t u(t, x) = −iP u(t, x) under the globally non-trapping condition.The well-posedness implies essential self-adjointness of P if the operator P is symmetric.We assume the non-trapping condition only for null trajectories, since the microlocally elliptic region should not be relevant.
Recently, the scattering theory for Klein-Gordon operators on Lorenzian manifolds has been studied by several authors (see, e.g., [1,4,17] and references therein).We also mention related work on Strichartz estimates for Lorenzian manifolds ( [7,12,16]), nonlinear Schrödinger-type equations with Minkowski metric ([6, 15, 19]), and quantum field theory on Minkowski spaces ( [18,5]).In order to study spectral properties of such equations or operators, self-adjointness is fundamental.We note a sufficient condition for the essential self-adjointness is discussed in Taira [16].The essential selfadjointness for Klein-Gordon operators on scattering Lorentzian manifolds is proved by Vasy [17] under the same null non-trapping condition.We had independently found a proof of the essential self-adjointness using different method for compactly supported perturbations (we discuss the basic idea in Appendix C).Inspired by discussions with Vasy during 2017, we generalized the model to include long-range perturbations, and also to higher order real principal type operators.Our proof is considerably different from [17], relatively self-contained, and hopefully simpler even though our result is more general than [17] for the R n case.
This paper is constructed as follows: In Section 2, we prepare several notations and basic lemmas.Our main result is proved in Section 3. In Subsection 3.1 we show that (P − i)u = 0 implies u is smooth.The basic idea of the proof is analogous to Nakamura [13] on microlocal smoothing estimates, and relies on the construction of time-global escaping functions (see also Ito, Nakamura [9] for related results for scattering manifolds).The technical detail is given in Appendix B. In Subsection 3.2, we show the local smoothness implies an weighted Sobolev estimate, which is sufficient for the proof of the essential self-adjointness.The idea is analogous to the radial point estimates of Melrose [11], and also related to the positive commutators method of Mourre.Here we construct weight functions explicitly to show necessary operator inequalities.The proof relies on the standard pseudodifferential operator calculus.In Appendix A, we prove non-trapping estimates for the classical trajectories generated by p m (x, ξ), which are necessary in Appendix B. The main lemma (Lemma A.2) is a generalization of a result by Kenig, Ponce, Rolvung and Vega [10], though the proof is significantly simplified.In Appendix C, we give a simplified proof of the essential selfadjintness for the compactly supported perturbation case.In this case the relatively involved argument of Subsection 3.2 is not necessarily.
Acknowledgment.SN is partially supported by JSPS grant Kiban-B 15H03622.KT is supported by JSPS Research Fellowship for Young Scientists, KAKENHI Grant Number 17J04478 and the program FMSP at the Graduate School of Mathematics Sciences, the University of Tokyo.We are grateful to Andraś Vasy for stimulating discussions during RIMS meeting at Kyoto in 2017.

Preliminary
We set x = (1 + |x| 2 ) 1/2 and D x = −i∂/∂x.We denote the weighted Sobolev spaces by for s, ℓ ∈ R, and their norms are given by We use the following notation of pseudo-differential operators.For any symbol a ∈ C ∞ (R 2n ), we define the Weyl quantization of a (at least formally) by Op(a)u(x) = (2π) −n e i(x−y)•ξ a((x + y)/2, ξ)u(y) dy dξ, u ∈ S(R n ).
We set the symbol classes S, S k,ℓ and S(m, g) by where g is a slowing varying metric and m is a g-continuous function (see [8, §18.4]).We denote the Poisson bracket of symbols a and b by {a, b} = The proofs of the following lemmas are standard, and we omit the proofs.
Lemma 2.2.Let k, ℓ ∈ R. Assume a j ∈ S k,ℓ is a bounded sequence in S k,ℓ and a j → 0 in S k+δ,ℓ+δ for some δ > 0.Then, for each s, t ∈ R and u ∈ H s,t Op(a j )u H s−k,t−ℓ → 0 as j → ∞.

Proof of Theorem 1.1
By the basic criterion for the essential self-adjointness ([14, Theorem VII.3]), it is sufficient to show Ker (P * ± i) = 0 to prove Theorem 1.1.Since )} where P acts on u in the distribution sense.We hence show: We only consider "−" case.The "+" case is similarly handled.Moreover, we note if u satisfies (P − i)u = 0 and u ∈ H m−1 2 ,− 1 2 (R n ), then u = 0 follows from a simple argument in [17].Namely, we take a real-valued function We note that [Op(ψ R ), P ] is uniformly bounded in OpS m−1,−1 and converges to 0 in OpS m−1+δ,−1+δ as R → ∞ for any δ > 0. We obtain u = 0 by using Lemma 2.2.Thus, in order to prove Theorem 1.1, it suffices to prove The proof of Proposition 3.1 is divided into two parts.In Subsection 3.1, we prove the local smoothness of u.In Subsection 3.2, using the local smoothness of u, we prove weighted Sobolev properties of u.

Local regularity.
The main result of this subsection is the following proposition.We note that we need the null non-trapping condition only for this proposition.
).Now we use the following proposition.Proposition 3.3.There exists a family of bounded operators (ii) There exists C > 0 such that for 0 < h ≤ 1, Proposition 3.3 can be proved similarly as [13,Lemma 9].For the completeness, we give a proof of Proposition 3.3 in the Appendix B. Now we set u(t, x) := e −t u(x).Then u(t, x) satisfies where the first equality is in the distributional sense.We set F ℓ (t) = F (h ℓ , t).Then, we have where all the inner products and norms here are in L 2 (R n ), and O(h ∞ ℓ ) is uniformly in t.Now, we take t = h −1 ℓ then we conclude a contradiction.Thus, we obtain u ∈ H k loc (R n ) for any k > 0. This completes the proof of Proposition 3.2 3.2.Uniform regularity outside a compact set.In this subsection, we prove a priori sub-elliptic estimates near infinity.The following estimates are based on the radial points estimates in [11], where the radial points estimates are used for scattering theory on scattering manifolds.By the classical propagation of singularities, the singularities of a solution to P u = 0 (provided P is real-valued real principal type) propagate along the Hamilton flow associated with p.At points where the Hamilton vector filed vanishes, we may use the so-called radial points, which implies u is rapidly decaying at a radial source if u has a threshold regularity at the radial source.
In our case, the radial points estimates are analogous to the Mourre estimate microlocally near outgoing or incoming regions, which is used commonly in scattering theory.We give a self-contained proof of the radial point estimate based on an explicit construction of escaping functions.We note the operator theoretical framework of the Mourre theory is not applicable here since we do not have the self-adjointness of P at this point.
We set where We use the following smooth cut-off functions: Let χ ∈ C ∞ (R) be such that and supp χ ′ ⋐ (1, 2).We write χ (t) = 1 − χ (t), and with M > 0. A main result of this subsection is the following theorem.
Now we show Proposition 3.1 follows from Theorem 3.4.
. This completes the proof of Proposition 3.1.
Weight functions.We choose ρ(t) ∈ C ∞ (R) such that For δ ∈ (1/2, 7/8), we set We use the following notation: Then we set with M, ν > 0. We also write The next lemma is a key of the proof of Theorem 3.4.
If M is sufficiently large, then: There are pseudodifferential operators Moreover, For any N > 0 and k ≥ 0 there is C > 0 such that Remark 3.6.The constant C in the lemma is independent of ϕ and z ∈ C\R.We note we assume Bϕ ∈ H k+(m−1)/2,−1/2 for technical reasons, though only the norm of Bϕ in H k,−1 appears in the RHS of (3.1).Theorem 3.4 follows from Lemma 3.5.
We write Lemma 3.8.Under the above assumptions, there are pseudodifferential operators S, T, U, V and a constant δ 4 > 0 such that where (i) S ∈ OpS(1, g) and the symbol is supported in Ω 1 (M, ν); (ii) T ∈ OpS(1, g) and the symbol is supported in Ω 2 (M, ν); In the proof of Lemma 3.8, we use the following estimate: Lemma 3.9.Suppose a be a symbol such that supp [a] ⊂ Ω ′ , where where s, ℓ ∈ R. Then for any N , there is C, C N > 0 such that Proof.We note, for any α, β ∈ Z n + , We write g = x −2+4γ dx 2 + x 4γ ξ −2 dξ 2 .Using the above estimate and the assumption on a, and following the construction of parametrices for elliptic operators, we can construct a symbol h(x, ξ) ∈ S(1, g) such that Proof of Lemma 3.8.By the standard pseudodifferential operator calculus, we can find f1 , f2 such that fj ∈ S(1, g), supp [ fj ] ⊂ Ω j (M, ν), j = 1, 2, and where R j are smoothing operators.We set S = Op( f1 ) and T = Op( f2 ).If we write We note, by the construction, Hence by the sharp Gårding inequality, we have with some C > 0. Then by the asymptotic expansion, we learn where R 3 ∈ S( x −3 ξ 2k+m−3 , g), and the symbol is supported in supp [b ′ ] modulo S(R 2d ).Using Lemma 3.7, we can estimate R 3 and other error terms from below by −C B D x k−1+m/2 x −2 D x k−1+m/2 B, modulo smoothing operators, and these will be included in U to complete the proof.Lemma 3.10.For ϕ ∈ S(R n ), the inequality (3.1) holds, where S = Op(f 1 ), Proof.We compute the commutator to obtain quadratic inequalities.For ϕ ∈ S(R n ), we have Combining this with Lemma 3.8, we have Thus we have Now we note, by Lemma 3.9, with any N .These imply (3.1) for ϕ ∈ S(R n ).
We now extend Lemma 3.10 to more general ϕ to prove Lemma 3.5.We choose M ′ and ν ′ so that M < M ′ < M , ν < ν ′ < ν, δ < δ ′ < δ, and set and we denote their symbols by a ε , ãε and a ′ ε , respectively.By the same computation as in the proof of Lemma 3.7, we have modulo S(R n )-terms, where constants are independent of ε, and f 1 and f 2 are independent of ε.Then, as well as Lemma 3.8, we have where the symbol of U ε has the property: and symbols of U ε and V ε are bounded in the respective symbol classes.It follows that , where the constant is independent of ε.Thus we have, as well as Lemma 3.10, for ϕ ∈ S(R n ), with any N , where C and C N are independent of ε ∈ (0, 1].
We observe that the symbol of [X L , A ε ] is bounded by C x −1 ξ −1 a ′ ε (x, ξ), modulo S(R 2d )-terms, uniformly in L, and also it converges to 0 locally uniformly as L → ∞.These imply By the same argument, using Bϕ ∈ H k−1+m/2,−1/2 , we learn We have similar estimates for Sϕ H k+(m−1)/2 and T ϕ L 2 .Concerning the estimate for Combining these with (3.5) for X L ϕ, we learn lim , and this implies the assertion.
Proof of Lemma 3.5.It remains to take the limit ε → 0 in (3.5).We note and hence

Thus we have
We note this holds without assuming Bϕ ∈ H s,ℓ , and if the right hand side is finite, we obtain Bϕ ∈ H s,ℓ .By the same argument, we also have and similarly, Substituting these to (3.5), we have H −N,−N , and this completes the proof of Lemma 3.5.and we write η 0 = |η(t 0 , x 0 , ξ 0 )| > 0. We set By Lemma A.1, we have and hence for t 0 ≤ t ≤ T .Thus we have .
We set , where # denotes the composition of the Weyl quantization ( [20, (4.3.6)] with h = 1) and |A| 2 = A * A for an operator A.
By the sharp Gårding inequality, there exists r 0,h,t ∈ S h x −1−µ , g h such that (iv) holds.