POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO

Transcription

1 POINTWISE BOUNDS ON QUASIMODES OF SEMICLASSICAL SCHRÖDINGER OPERATORS IN DIMENSION TWO HART F. SMITH AND MACIEJ ZWORSKI Abstract. We prove optimal pointwise bounds on quasimodes of semiclassical Schrödinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical L p bounds conducted by Koch-Tataru- Zworski []. However, we show that the results of [] imply the two dimensional end-point estimate by scaling and localization. 1. Introduction Let g ij (x) be a positive definite Riemannian metric on R with the corresponding Laplace- Beltrami operator, g u := ḡ 1 ( xj g ij ḡ xj u ), (g ij ) := (g ij ) 1, ḡ := det(g ij ), i,j and let V C (R ) be real valued. We prove the following general bound which was already established (under an additional necessary condition) in higher dimensions in [], but which was open in dimension two: Theorem 1.1. Suppose that h 1, and u Hloc (R ). Suppose that u satisfies (1.1) h g u + V u L h, u L 1. Then for all K R, (1.) sup u(x) C K h 1, x K where the constant C K depends only on g, V, and K. A function u satisfying (1.1) is sometimes called a weak quasimode. It is a local object in the sense that if u is a weak quasimode then ψu, ψ Cc (R ) is also one, so the theorem is easily reformulated with g, V, and u defined on an open subset of R. The localization is also valid in phase space: for instance if χ Cc (R R ) then χ w (x, hd)u is also a weak HS is supported in part by NSF grant DMS and MZ is supported in part by NSF grant DMS

2 HART F. SMITH AND MACIEJ ZWORSKI quasimode see [1, Chapter 7] or [4, Chapter 4] for the review of the Weyl quantization χ χ w. If lim inf x V > 0, then h + V (defined on Cc (R )) is essentially self-adjoint and the spectrum of h +V is discrete in a neighbourhood of 0 see for instance [1, Chapter 4]. In this case weak quasimodes arise as spectral clusters: (1.3) w = c j w j, ( h + V )w j = E j w j, w j, w k L = δ jk, c j 1. E j Ch Then u = χw, for any χ Cc (R ), is a weak quasimode in the sense of (1.1). Since V (x) c > 0 for x R, Agmon estimates (see for instance [1, Chapter 6]) and Sobolev embedding show that u(x) e c/h, c > 0, for x R. Hence we get global bounds w(x) Ch 1, x R. It should be stressed however that a weak quasimode is a more general notion than a spectral cluster. The result also holds when R is replaced by a two dimensional manifold and, as in the example above, gives global bounds on spectral clusters (1.3) when the manifold is compact. If V < 0 this is also a by-product of of the Avakumovic-Levitan-Hörmander bound on the spectral function see [3], and for a simple proof of a semiclassical generalization see [, 3] or [4, 7.4]. In higher dimensions the theorem requires an additional phase space localization assumption and is a special case of [, Theorem 6]: Suppose p(x, ξ) is a function on R n R n satisfying α x β ξ p(x, ξ) = O( ξ m ) for some m. Suppose that K R n and χ C c (R n R n ), and that for (x, ξ) supp χ p(x, ξ) = 0, Then for u(h) such that d ξ p(x, ξ) = 0 = d ξp(x, ξ) is nondegenerate. (1.4) supp u(h) K, u(h) = χ w (x, hd)u + O S (h ), we have (1.5) u(h) C h n 1 ( u(h) L + 1 ) h pw (x, hd)u L, n 3. When n = the bound holds with (log(1/h)/h) 1, which is optimal in general if d ξp is not positive definite see [, 3, 6] and 3 below for examples. A small bonus for Schrödinger operators in dimension two is the fact that the frequency localization condition in (1.4) required for (1.5) is not necessary see (.6) below. And as noted already, in all dimensions the compact support condition on u is easily dropped when working with local estimates on u. j

3 BOUNDS ON QUASIMODES FOR SEMICLASSICAL SCHRÖDINGER OPERATORS 3 The proof of Theorem 1.1 is reduced to a local result presented in Proposition.1. That result follows in turn from a rescaling argument involving several cases, some of which use the following result that forms part of [, Corollary 1]. Theorem 1.. Suppose that u = u(h) satisfies (1.1), and that (1.4) holds. If V (x) 0 for x supp u, or if g ij is positive definite and dv (x) 0 for x supp u, then u L = O ( ) h n 1, n. This result is the basis for Propositions. and.3 used in our proof. The case of Theorem 1. with dv 0 is the most technically involved result of []. We do not know of any simpler way to obtain (1.).. Proof of Theorem 1.1 By compactness of K, it suffices to prove uniform L bounds on u over a small ball about each point in K, where in our case the diameter of the ball can be taken to depend only on C N estimates for g and V over a unit sized neighborhood of K, for some large N. Without loss of generality we consider a ball centered at the origin in R. Let B = {x R : x < 1}, B = {x R : x < }. After a linear change of coordinates, we may assume that (.1) g ij (0) = δ ij. Next, by replacing V (x) by cv (cx) and g ij (x) by g ij (cx), for some constant c 1 depending on the C norm of g and V over a unit neighborhood of K, we may assume that (.) sup V (x) + dv (x), sup d V (x) + dg ij (x).01. x B x B This has the effect of multiplying h by a constant in the equation (1.1), which can be absorbed into the constant C K in (1.). In general, we let (.3) C N = sup sup x B α N ( α V (x) + i,j=1 and will deduce Theorem 1.1 as a corollary of the following i,j=1 ) α g ij (x), Proposition.1. Suppose h 1, that g, V satisfy (.1) and (.), and that u satisfies (.4) h g u + V u L (B ) h, u L (B ) 1. Then (.5) u L (B) C h 1, where the constant C depends only on C N in (.3) for some fixed N.

4 4 HART F. SMITH AND MACIEJ ZWORSKI We start the proof of Proposition.1 by recording the following two propositions, which are consequences of Theorem 1.. Proposition.. Suppose that (.1)-(.) hold, and that 1 the following holds, and h 1, V (x) for x. If h g u + V u L (B ) h, u L (B ) 1, then u L (B) C h 1, where C depends only on C N in (.3) for some fixed N. Proposition.3. Suppose that (.1)-(.) hold, and that V (0) = 0 and dv (0) = 1. If the following holds, and h 1, h g u + V u L (B ) h, u L (B ) 1, then u L (B) C h 1, where C depends only on C N in (.3) for some fixed N. To see that these follow from Theorem 1., we first may assume that u is compactly supported in x < 3. Indeed, the assumptions imply du L ( x <3/) h 1, so that one may cut off u by a smooth function which is supported in x < 3 and equals 1 for x < 1 without affecting the hypotheses. We may then modify g and V outside B so that (.)- (.3) are global bounds. In Proposition.3 above, since d V.01, we have.98 dv (x) 1.0 for x, so since g is positive definite the conditions on g and V in Theorem 1. are met. We remark that the conditions of Proposition.3 guarantee that the zero set of V is a nearly-flat curve through the origin, although this is not strictly needed to apply the results of []. That the resulting constant C depends only on C N for some fixed finite N follows from the proofs in []. Finally, the condition (1.4) that u χ w (x, hd)u = O S (h ) for some χ Cc is not needed for Theorem 1. to hold for positive definite g ij in dimension two. To see this, we note that if V < and g ij (x) δ ij.0 on the ball x <, then if u is supported in x < 3 and ϕ(ξ) = 1 for ξ < 4, condition (1.1) implies that (hd) (u ϕ(hd)u) L = O(h). This follows by the semiclassical pseudodifferential calculus (see [4, Theorem 4.9]), since for ϕ 0 C c (R ) with supp ϕ 0 B, ϕ 0 (x)(1 ϕ(ξ)) ξ /( ξ g + V (x)) S(R R ).

5 BOUNDS ON QUASIMODES FOR SEMICLASSICAL SCHRÖDINGER OPERATORS 5 Hence, writing û(ξ) for the standard Fourier transform of u, u ϕ(hd)u L 1 1 ϕ(hξ) û(ξ) dξ (π) R C hξ 1 ϕ(hξ) û(ξ) (1 + hξ ) 1 dξ (.6) ( C (hd) (u ϕ(hd)u) L (1 + hξ ) dξ R Ch h 1 = C, an even better estimate than required. Hence we are reduced to proving estimates on ϕ(hd)u, which by compact support of u satisfies (1.4). We supplement Propositions. and.3 with the following two lemmas. Lemma.4. Suppose that (.1)-(.) hold, and that V (x) 99 h for x h 1. If the following holds, and h 1, h g u + V u L ( x <h 1/ ) h, u L ( x <h 1/ ) 1, then u L ( x <h 1/ ) C h 1, where C depends only on C N in (.3) for some fixed N. Proof. Consider the function ũ(x) = h 1 u(h 1 x), and g ij (x) = g ij (h 1 x). V u L ( x <h 1/ ) 99h, we have g ũ L ( x <) 100, ũ L ( x <) 1. ) 1 Then, since Since the spatial dimension equals, interior Sobolev estimates yield ũ L ( x <1) C, where we note that the conditions (.1) and (.) hold for g since h 1 1. Lemma.5. Suppose that (.1)-(.) hold, and that 1 c V (x) c for x c 1. If the following holds, and h c 1, h g u + V u L ( x <c 1/ ) h, u L ( x <c 1/ ) 1, then u L ( x <c 1/ ) C h 1, where C depends only C N in (.3) for some fixed N. Proof. Let ũ(x) = c 1 u(c 1 x), g ij (x) = g ij (c 1 x), and Ṽ (x) = c 1 V (c 1 x). Note that the assumptions on V (x) in the statement and in (.) imply that dv (x) c 1 for x < c 1/, so that Ṽ satisfies (.), and the constants C N in (.3) can only decrease for c 1. Then with h = c 1 h 1, h g ũ + Ṽ ũ L ( x <) h, ũ L ( x <) 1. By Proposition., we have ũ L ( x <1) C h 1, giving the desired result.

6 6 HART F. SMITH AND MACIEJ ZWORSKI Proof of Proposition.1. It suffices to prove that for each x 0 < 1 there is some 1 r > 0 so that u L ( x x 0 <r) C h 1, with a global constant C. Without loss of generality we take x 0 = 0. We will split consideration up into four cases, depending on the relative size of V (0) and dv (0). Since for h bounded away from 0 the result follows by elliptic estimates, we will assume h 1 4 so that h 1 below is at most 1. Case 1: V (0) h, dv (0) 8h 1. Since d V (x).01, then Lemma.4 applies to give the result with r = h 1. Case : V (0) h, dv (0) 8h 1. Since we may add a constant of size h to V without affecting (.4), we may assume V (0) = 0. By rotating we may then assume V (x) = βx 1 + f ij (x)x i x j, where β = dv (0) 8h 1. Dividing V by 4 if necessary we may assume β 1. Let ũ = βu(βx), g ij (x) = g ij (βx), and With h = β h < 1 we have Ṽ (x) = β V (βx) = x 1 + f ij (βx)x i x j. h g ũ + Ṽ ũ L ( x <) h, ũ L ( x <) 1. Proposition.3 applies, since g and Ṽ satisfy (.1)-(.), and the constants C N in (.3) for g and Ṽ are bounded by those for g and V. Thus ũ L ( x <1) C h 1, giving the desired result on u with r = dv (0). Case 3: V (0) h, dv (0) 9 V (0) 1. In this case, with c = V (0), it follows that 1 1 c V (x) c for x c 1. We may apply Lemma.5 with V replaced by 1 V to get the desired result with r = 1 V (0) Case 4: V (0) h, dv (0) 9 V (0) 1. Since d V (x).01, it follows that there is a point x 0 with x V (0) 1 where V (x 0 ) = 0. Since dv (x 0 ) 8 V (0) 1 8h 1, we may translate and apply Case to get L bounds on u over a neighborhood of radius dv (x 0 ) about x 0. This neighborhood contains the neighborhood about 0 of radius r =.9998 dv (0). 3. A counter-example for indefinite g. In [, Section 5], it was shown that there exist u h for which (3.1) h ( x 1 x )u h + (x 1 x )u h L h, u h L 1, for which u h L log h 1 h 1, showing that the assumption of definiteness of g cannot be relaxed to non-degeneracy in the main theorem. In [, Theorem 6] the positive result

7 BOUNDS ON QUASIMODES FOR SEMICLASSICAL SCHRÖDINGER OPERATORS 7 was established showing that this growth of u h L for indefinite, non-degenerate g in two dimensions is in fact worst case. The example of [] was produced using harmonic oscillator eigenstates. Here we present a different construction of such a u h with similar L growth to help illustrate the role played by the degeneracy of g. The idea is to produce a collection u h,j of functions satisfying (3.1) (or equivalent), for which u h,j (0) = h 1, and where j runs over log h different values. The examples will have disjoint frequency support, hence are orthogonal in L. Upon summation over j the L norm then grows as log h 1, whereas the L norm grows as log h h 1, yielding an example with worst case growth after normalization. We start by considering the form ξ 1 ξ with V = 0. To assure that h x1 x u h L h, we will take the Fourier transform of u h to be contained in the set ξ 1 ξ h 1, as well as ξ h 1 to satisfy the frequency localization condition [, (1.4)]. Our example is then based on the fact that one can find log h disjoint rectangles, each of volume h 1, within this region, as illustrated in the diagram. Each u h,j will be an appropriately scaled Schwartz function with Fourier transform localized to one of the rectangles. We now fix ψ, χ C c (R), with 0 ψ(x) and 0 χ(x) 1, with ψ = χ = 1, and where We additionally assume χ(0) = 1. Let u h,j (x) = h 1 supp ψ [1, ], supp χ [ 1, 1]. e ix 1ξ 1 +ix ξ χ( j h ξ 1 )ψ( j ξ ) dξ 1 dξ = h 1 ˇχ( j h 1 x 1 ) ˇψ( j x ). By the Plancherel theorem, u h,j L 1 and h D 1 D u h,j L h. Furthermore, u h,j (0) = h 1. By disjointness of the Fourier transforms, for i j we have u h,i, u h,j = 0, and similarly x1 x u h,i, x1 x u h,j = 0. We then form u h (x) = log h 1 1 j h 1 u h,j (x).

8 8 HART F. SMITH AND MACIEJ ZWORSKI Since there are log h terms in the sum, and the terms are orthogonal in L, it follows that u h L 1, h x1 x u h L (R ) h, u h (0) log h 1 h 1. Although the example is not compactly supported, it is rapidly decreasing (uniformly so for h < 1), and one may smoothly cutoff to a bounded set without changing the estimates. We observe that for this example it also holds that x 1 x u h L h. Hence, u h is also a counterexample for the form ξ 1 ξ ± x 1 x. Rotating by π/4 gives the form ξ 1 ξ ± (x 1 x ), including in particular the form considered in [, Section 6]. We also observe that x 1u h will be O L (h) if one restricts the sum in u h to 1 j h 1, which still has log h values of j, and thus exhibits the same L growth as u h. This idea does not however work to yield a counterexample for the form ξ 1 ξ + x 1 + x. References [1] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semiclassical Limit, Cambridge University Press, 1999 [] H. Koch, D. Tataru and M. Zworski, Semiclassical L p estimates, Ann. Henri Poincaré 8 (007), no. 5, [3] C. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics 105, Cambridge University Press, [4] M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics 138, AMS, 01. Department of Mathematics, University of Washington, Seattle, WA address: hart@math.washington.edu Department of Mathematics, University of California, Berkeley, CA address: zworski@math.berkeley.edu