Propagation Through Trapped Sets and Semiclassical Resolvent Estimates Kiril Datchev and András Vasy Let P D h 2 C V.x/, V 2 C0 1 estimates of the form.rn /. We are interested in semiclassical resolvent k.p E i0/ 1 k L 2.R n /!L 2.R n / a.h/ h ; h 2.0; h 0 ; (1) for E>0, 2 C 1.R n / with j.x/j hxi s, s>1=2. We ask: how is the function a.h/ for which (1) holds affected by the relationship between the support of and the trapped set at energy E,definedby K E Df 2 T R n W9C>0;8t >0;j exp.th p / j C g Here p Djj 2 C V.x/ and H p D 2 r x rv r. We have (1) with.x/ Dhxi s and a.h/ D C for all E in a neighborhood of E 0 >0ifand only if K E0 D;([6, 7]). For general V and, the optimal bound is a.h/ D exp.c=h/,butburq[1] and Cardoso-Vodev [2] prove that for any given V, if vanishes on a sufficiently large compact set, for any E>0there exists C such that (1) holds with a.h/ D C. In our main theorem we improve the condition on and obtain a shorter proof at the expense of an a priori assumption. Theorem 1 ([3]). Fix E>0. Suppose that (1) holds for.x/ Dhxi s with s> 1=2 and with a.h/ D h N for some N 2 N. Then if we take instead such that K E \ T supp D;, we have (1) with a.h/ D C. K. Datchev ( ) Mathematics Department, MIT, Cambridge, MA, 02139-4307, USA e-mail: datchev@math.mit.edu A. Vasy Mathematics Department, Stanford University, Stanford, CA, 94305-2125, USA e-mail: andras@math.stanford.edu D. Grieser et al. (eds.), Microlocal Methods in Mathematical Physics and Global Analysis, Trends in Mathematics, DOI 10.1007/978-3-0348-0466-0 2, Springer Basel 2013 7
8 K. Datchev and A. Vasy In fact our result holds for more general operators, and the cutoff can be replaced by a cutoff in phase space whose microsupport is disjoint from K E.In certain situations it is even possible to take a cutoff whose support overlaps K E :see [3] for more details and references. The a priori assumption that (1) holds for.x/ Dhxi s with a.h/ D h N is not present in [1, 2] and is not always satisfied, but there are many examples of hyperbolic trapping where it holds: see e.g. [5, 8]. To indicate the comparative simplicity of our method, we prove a special case of the Theorem, under the additional assumption that supp V fjxj <R 0 g and supp fr 0 < jxj <R 0 C 1g. In other words, suppose.p /u D f, with Re D E, and supp f fr 0 < jxj <R 0 C 1g, kf k1. We must prove that kuk Ch 1, uniformly as Im! 0 C. Here and below all norms are L 2 norms. Let S denote functions in C 1.T R n / which are bounded together with all derivatives, and for a 2 S define Op.a/u.x/ D.2h/ n Z exp.i.x y/ =h/a.x;/u.y/dyd: Because P has a semiclassical elliptic inverse away from p 1.E/ (see for example [4, Chap. 4]), we have k Op.a/uk C whenever supp a \ p 1.E/ D;. Consequently it is enough to show that k Op.a/uk Ch 1 for some a 2 S with a nowhere vanishing on T supp \ p 1.E/. We will prove this inductively: we will show that if there is a 1 with this nowhere vanishing property such that k Op.a 1 /uk Ch k, then there is a 2 with the same nowhere vanishing property such that k Op.a 2 /uk Ch kc1=2, provided k 3=2. The base case follows from the a priori assumption that kuk h N 1, so it suffices to prove the inductive step. Take ' D '.jxj/ 0 a smooth function such that ' D 1 when jxj R 0, ' D 0 when jxj R 0 C1, ' 0 D 2 with smooth. We require further that T supp be contained in the set where a 1 is nonvanishing, and in the end we will take a 2 D. We will now use a positive commutator argument with ' as the commutant: ihœp; ' u; ui Dihu;'fi ih'f; ui 2 Im kuk 2 C k ukkf k; (2) where we used first.p /u D f and then Im 0 and supp f f 0g.The semiclassical principal symbol of iœp;' is hh p ' D 2h' 0 D 2h 2 ; where is the dual variable to jxj in T R n. We now define an open cover and partition of unity of T supp according to the regions where this commutator does and does not have a favorable sign (the favorable sign is H p '<0, because of the direction of the inequality in (2)). Take c > 0 small enough that for < 2c, jxj > R 0, t < 0 we have x C 2t 62 supp V.LetK be a neighborhood of p 1.E/ \ T supp with compact closure in T fr 0 < jxj <R 0 C 1g, andleto be a neighborhood of K with compact closure
Propagation Through Trapped Sets and Semiclassical Resolvent Estimates 9 in T fr 0 < jxj <R 0 C 1g,andlet U C Df 2 OW >cg; U Df 2 OW <2cg[.T R n n K/: Take 2 C0 1.O/ with 2 C C 2 D 1 on T supp and with supp U. Then H p ' D b 2 2 2 2 ; where b D p 2 C ; and if B D Op.b/ and ˆ D Op. / iœp;' D hb B C hˆ R 1ˆ C h 2 R 2 C O.h 1 /; where R 1;2 D Op.r 1;2 / for r 1;2 2 S with supp r 1;2 supp. Combining with (2), and using L 2 boundedness of R 1, we obtain hkbuk 2 Chkˆ uk 2 C h 2 hr 2 u; uicc k ukkf kco.h 1 /: Since hr 2 u; ui Ch 2k by inductive hypothesis, we have kbuk 2 C.kˆ uk 2 C h 2kC1 C h 1 k ukkf k/ C.kˆ uk 2 C h 2kC1 C ı 1 h 2 C ık uk 2 /; where we used kf k1,andwhereı>0will be specified presently. Since at least one of B and ˆ is elliptic at each point in the interior of T supp, wehave k uk 2 C.kˆ uk 2 CkBuk 2 /; (3) fromwhich we concludethat, if ı is sufficiently small, kbuk 2 C ı.kˆ uk 2 C h 2 C h 2kC1 /: (4) Because c was chosen small enough that all backward bicharacteristics through supp stay in T fjxj >R 0 g,wherep D h 2,wehave kˆ uk Ch 1 ; by standard nontrapping estimates (see, for example, [3, Sect. 6]). This, combined with (3)and(4), gives k uk 2 C ı.h 2 C h 2kC1 /; after which taking a 2 D completes the proof of the inductive step. Acknowledgements The first author is partially supported by a National Science Foundation postdoctoral fellowship, and the second author is partially supported by the NSF under grant DMS-0801226, and a Chambers Fellowship from Stanford University. The authors are grateful for the hospitality of the Mathematical Sciences Research Institute, where part of this research was carried out.
10 K. Datchev and A. Vasy References 1. Nicolas Burq. Lower bounds for shape resonances widths of long range Schrödinger operators. Amer.J.Math., 124:4, 677 735, 2002. 2. Fernando Cardoso and Georgi Vodev. Uniform estimates of the resolvent of the Laplace- Beltrami operator on infinite volume Riemannian manifolds II. Ann. Henri Poincaré, 3:4, 673 691, 2002. 3. Kiril Datchev and András Vasy. Propagation through trapped sets and semiclassical resolvent estimates. To appear in Ann. Inst. Fourier. Preprint available at arxiv:1010.2190. 4. Lawrence C. Evans and Maciej Zworski. Lecture notes on semiclassical analysis. Available online at http://math.berkeley.edu/ zworski/semiclassical.pdf. 5. Stéphane Nonnenmacher and Maciej Zworski. Quantum decay rates in chaotic scattering. Acta Math. 203:2, 149 233, 2009. 6. Didier Robert and Hideo Tamura. Semiclassical estimates for resolvents and asymptotics for total scattering cross-sections. Ann. Inst. H. Poincaré Phys.Théor. 46:4, 415 442, 1987. 7. Xue Ping Wang. Semiclassical resolvent estimates for N -body Schrödinger operators. J. Funct. Anal. 97:2, 466 483, 1991. 8. Jared Wunsch and Maciej Zworski. Resolvent estimates for normally hyperbolic trapped sets. To appear in Ann. Inst. Henri Poincaré (A).Preprint available at arxiv:1003.4640.
http://www.springer.com/978-3-0348-0465-3