Uniqueness Properties of Solutions to Schrödinger Equations
aa r X i v : . [ m a t h . A P ] D ec UNIQUENESS PROPERTIES OF SOLUTIONS TOSCHR ¨ODINGER EQUATIONS
L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA Introduction
To place the subject of this paper in perspective, we start out with a briefdiscussion of unique continuation. Consider solutions to(1.1) ∆ u ( x ) = n X j =1 ∂ u∂x j ( x ) = 0 , (harmonic functions) in the unit ball { x ∈ R n : | x | < } . When n = 2, these func-tions are real parts of holomorphic functions, and so, if they vanish of infinite orderat x = 0, they must vanish identically. We call this the strong unique continuationproperty (s.u.c.p.). The same result holds for n >
2, since harmonic functions arestill real analytic in { x ∈ R n : | x | < } . In fact, it is well-known that if P ( x, D ) is alinear elliptic differential operator with real analytic coefficients, and P ( x, D ) u = 0in a open set Ω ⊂ R n , then u is real analytic in Ω. Hence, the (s.u.c.p.) alsoholds for such solutions. Through the work of Hadamard [28] on the uniquenessof the Cauchy problem (which is closely related to the strong unique continuationproperty discussed earlier) it became clear (for applications in nonlinear problems)that it would be desirable to establish the strong unique continuation property foroperators whose coefficients are not necessarily real analytic, or even C ∞ . The firstresults in this direction were found in the pioneering work of Carleman [9] (when n = 2) and M¨uller [47] (when n > P ( x, D ) = ∆ + V ( x ) , with V ∈ L ∞ loc ( R n ) . In order to establish his result, Carleman introduced a method (the method of“Carleman estimates”) which has permeated the subject ever since. In this context,an example of a Carleman estimate is :
For f ∈ C ∞ ( { x ∈ R n : | x | < } − { } ) , α > and w ( r ) = r exp( Z r e − s − s ds ) , one has (1.2) α Z w − − α ( | x | ) f ( x ) dx ≤ c Z w − α ( | x | ) | ∆ f ( x ) | dx, with c independent of α Mathematics Subject Classification.
Primary: 35Q55.
Key words and phrases.
Schr¨odinger evolutions.The first and fourth authors are supported by MEC grant, MTM2004-03029, the second andthird authors by NSF grants DMS-0968472 and DMS-0800967 respectively.
For a proof of this estimate, see [26], [7]. The (s.u.c.p.) of Carleman-M¨ullerfollows easily from (1.2) (see [38] for instance).In the late 1950’s and 1960’s there was a great deal of activity on the subject of(s.u.c.p.) and the closely related uniqueness in the Cauchy problem, some highlightsbeing [1] and [8] respectively, both of which use the method of Carleman estimates.These results and methods have had a multitude of applications to many areas ofanalysis, including to non-linear problems. (For a recent example, see [39] for anapplication to energy critical non-linear wave equations).In connection with the Carleman-M¨uller (s.u.c.p.) a natural question is : Howfast is a solution u allowed to vanish, before it must vanish identically?By considering n = 2, u ( x , x ) = ℜ ( x + ix ) N , we see that to make sense ofthe question, a normalization is required, for instancesup | x | < / | u ( x ) | ≥ , k u k L ∞ ( | x | < < ∞ . We refer to questions of this type as “quantitative unique continuation”. It is also ofinterest to consider unique continuation type questions around the point at infinity.For instance, a conjecture of E. M. Landis [41] was : if∆ u + V u = 0 , x ∈ R n , with k V k ∞ ≤ , k u k ∞ < ∞ , and for some ǫ > | u ( x ) | ≤ c ǫ e − c ǫ | x | ǫ , then u ≡ V ( x ), this conjecture was disprovedby Meshkov [45] who constructed V, u, u | u ( x ) | ≤ c e − c | x | / , n ≥ . Meshkov also showed that if | u ( x ) | ≤ c ǫ e − c ǫ | x | / ǫ , for some ǫ > , then u ≡ R n , n ≥ ∂ t u − ∆ u = W · ∇ u + V u, with k W k ∞ + k V k ∞ < ∞ , (or equivalently | ∂ t u − ∆ u | ≤ M ( |∇ u | + | u | )). Using a parabolic analog of theCarleman estimate described earlier, one can show that if | ∂ t u − ∆ u | ≤ M ( |∇ u | + | u | ) , ( x, t ) ∈ { x ∈ R n : | x | < R } × [ t , t ] , R > , with | u ( x ) | ≤ A and u ≡ , ( x, t ) ∈ { x ∈ R n : R < | x | < R } × [ t , t ] , then u ≡ , ( x, t ) ∈ { x ∈ R n : | x | < R } × [ t , t ] . We call this type of result “unique continuation through spatial boundaries”,(see [26], [55] and references therein for this type of result and strengthenings of it).
NIQUE CONTINUATION 3
This result is closely related to the “elliptic” (s.u.c.p.) discussed before. On theother hand, for parabolic equations, there is also a “backward uniqueness” principle,which is very useful in applications to control theory (see [44] for an early result inthis direction) : Consider solutions to | ∂ t u − ∆ u | ≤ M ( |∇ u | + | u | ) , ( x, t ) ∈ R n × (0 , , with k u k ∞ ≤ A . Then, if u ( · , ≡
0, we must have u ≡
0. This result is alsoproved through Carleman estimates (see [44]).Recently, a strengthening of this result has been obtained in [25], where oneconsiders solutions only defined in R n + × (0 , R n + = { ( x , .., x n ) ∈ R n : x > } , without any assumptions on u at x = 0, and still obtains the “backwarduniqueness” result. This strengthening had an important application to non-linearequations, allowing the authors of [25] to establish a long-standing conjecture of J.Leray on regularity and uniqueness of solutions to the Navier-Stokes equations (seealso [52] for a recent extension).Finally, we turn to dispersive equations. Typical examples of these are the k -generalized KdV equation(1.3) ∂ t u + ∂ x u + u k ∂ x u = 0 , ( x, t ) ∈ R × R , k ∈ Z + , and the non-linear Schr¨odinger equation(1.4) ∂ t u = i (∆ u ± | u | p − u ) , ( x, t ) ∈ R n × R , p > . These equations model phenomena of wave propagation and have been extensivelystudied in the last 30 years or so.For these equations,“unique continuation through spatial boundaries ” also holds,as it was shown by Saut-Scheurer [51] for the KdV-type equations and by Izakov[36] for Shr¨odinger type equations. (All of these results were established troughCarleman estimates). These equations however are time reversible (no preferredtime direction) and so “backward uniqueness” is immediate, unlike in parabolicproblems. Once more in connection with control theory, this time for dispersiveequations, Zhang [56] showed, for solutions of(1.5) ∂ t u = i ( ∂ x u ± | u | u ) , ( x, t ) ∈ R × [0 , , that if u ( x, t ) = 0 for ( x, t ) ∈ ( −∞ , a ) × { , } (or ( x, t ) ∈ ( a, ∞ ) × { , } ) forsome a ∈ R , the u ≡
0. Zhang’s proof was based on the inverse scattering methodwhich uses that this is a completely integrable model, and did not apply to othernon-linearities or dimensions. This type of result was extended to the k -generalizedKdV (1.3) and the general non-linear Schr¨odinger equation in (1.4) in all dimensions(where inverse scattering is no longer available) using suitable Carleman estimates(see [40], [34], [35], and references therein).For recent surveys of the results presented so far, see [37], [38].Returning to “backward uniqueness” for parabolic equations, in analogy withLandis’ “elliptic” conjecture mentioned earlier, Landis-Oleinik [43] conjectured thatin the “backward uniqueness” result one can replace the hypothesis u ( · , ≡ | u ( x, | ≤ c ǫ e − c ǫ | x | ǫ , for some ǫ > . This is indeed true and was established in [18] and [49]. Similarly, one can conjecture(as it was done in [20]) that for Schr¨odinger equations, if | u ( x, | + | u ( x, | ≤ c ǫ e − c ǫ | x | ǫ , for some ǫ > , L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA then u ≡
0. This was established in [18].In analogy with the improvement of “backward uniqueness” in [25], one can showthat it suffices to deal with solutions in R n + × (0 ,
1] (for parabolic problems) andrequire | u ( x, | ≤ c ǫ e − c ǫ x ǫ , x > , for some ǫ > , to conclude that u ≡ u a solution in R n + × [0 , | u ( x, | + | u ( x, | ≤ c ǫ e − c ǫ x ǫ , x > , for some ǫ > , to conclude that u ≡
0, as we will prove in section 5 of this paper.In [16] it was pointed out for the first time (see also [10]) that both the resultsin [18] and in [16], in the case of the free heat equation ∂ t u = ∆ u, and the free Schr¨odinger equation ∂ t u = i ∆ u, respectively, are in fact a corollary of the more precise Hardy uncertainty principlefor the Fourier transform, which says : If f ( x ) = O ( e −| x | /β ) , b f ( ξ ) = O ( e − | ξ | /α ) and /αβ > / , then f ≡ , andif /αβ = 1 / , f ( x ) = ce −| x | /β as will be discussed below.Thus, in a series of papers ([16]-[23], [11]) we took up the task of finding the sharpversion of the Hardy uncertainty principle, in the context of evolution equations.The results obtained have already yielded new results on non-linear equations. Forinstance in [21] and [23] we have found applications to the decay of concentra-tion profiles of possible self-similar type blow-up solutions of non-linear Schr¨odngerequations and to the decay of possible solitary wave type solutions of non-linearSchr¨odinger equations.In the rest of this work we shall review some of our recent results concerningunique continuation properties of solutions of Schr¨odinger equations of the form(1.6) ∂ t u = i (∆ u + F ( x, t, u, ¯ u )) , ( x, t ) ∈ R n × R . We shall be mainly interested in the case where(1.7) F ( x, t, u, ¯ u ) = V ( x, t ) u ( x, t )is describing the evolution of the Schr¨odinger flow with a time dependent potential V ( x, t ), and in the semi-linear case(1.8) F ( x, t, u, ¯ u ) = F ( u, ¯ u ) , with F : C × C → C , F (0 ,
0) = ∂ u F (0 ,
0) = ∂ ¯ u F (0 ,
0) = 0.Let us consider a familiar dispersive model, the k -generalized Korteweg-de Vriesequation (1.3) and recall a theorem established in [17] : Theorem 1.
There exists c > such that for any pair u , u ∈ C ([0 ,
1] : H ( R ) ∩ L ( | x | dx )) of solutions of (1.3) such that if (1.9) u ( · , − u ( · , , u ( · , − u ( · , ∈ L ( e c x / dx ) , then u ≡ u . NIQUE CONTINUATION 5
Above we have used the notation: x + = max { x ; 0 } .Notice that taking u ≡ / ( ∂ t v + ∂ x v = 0 ,v ( x,
0) = v ( x ) , is given by the group { U ( t ) : t ∈ R } U ( t ) v ( x ) = 1 √ t Ai (cid:18) · √ t (cid:19) ∗ v ( x ) , where Ai ( x ) = c Z ∞−∞ e ixξ + iξ dξ, is the Airy function which satisfies the estimate | Ai ( x ) | ≤ c (1 + x − ) − / e − cx / . It was also shown in [17] that Theorem 1 is optimal :
Theorem 2.
There exists u ∈ S ( R ) , u and ∆ T > such that the IVPassociated to the k-gKdV equation (1.3) with data u has solution u ∈ C ([0 , ∆ T ] : S ( R )) , satisfying | u ( x, t ) | ≤ ˜ d e − x / / , x > , t ∈ [0 , ∆ T ] , for some constant ˜ d > . In the case of the free Schr¨odinger group { e it ∆ : t ∈ R } e it ∆ u ( x ) = ( e − i | ξ | t b u ) ∨ ( x ) = e i |·| / t (4 πit ) n/ ∗ u ( x ) , the fundamental solution does not decay. However, one has the identity(1.11) u ( x, t ) = e it ∆ u ( x ) = Z R n e i | x − y | / t (4 πit ) n/ u ( y ) dy = e i | x | / t (4 πit ) n/ Z R n e − ix · y/ t e i | y | / t u ( y ) dy = e i | x | / t (2 it ) n/ \ ( e i |·| / t u ) (cid:16) x t (cid:17) , where b f ( ξ ) = (2 π ) − n/ Z R n e − iξ · x f ( x ) dx. Hence, c t e − i | x | / t u ( x, t ) = \ ( e i |·| / t u ) (cid:16) x t (cid:17) , c t = (2 it ) n/ , L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA which tells us that e − i | x | / t u ( x, t ) is a multiple of the rescaled Fourier transform of e i | y | / t u ( y ). Thus, as we pointed out earlier, the behavior of the solution of thefree Schr¨odinger equation is closely related to uncertainty principles for the Fouriertransform. We shall study these uncertainty principles and their relation with theuniqueness properties of the solution of the Schr¨odinger equation (1.6). In the early1930’s N. Wiener’s remark (see [29], [33], and [46]):“a pair of transforms f and g ( b f ) cannot both be very small”,motivated the works of G. H. Hardy [29], G. W. Morgan [46], and A. E. Ingham[33] which will be considered in detail in this note. However, before that we shallreturn to a review of some previous results concerning uniqueness properties ofsolutions of the Schr¨odinger equation which we mentioned earlier and which werenot motivated by the formula (1.11).For solutions u ( x, t ) of the 1-D cubic Schr¨odinger equation (1.5) B. Y. Zhang[56] showed : If u ( x, t ) = 0 for ( x, t ) ∈ ( −∞ , a ) × { , } ( or ( x, t ) ∈ ( a, ∞ ) × { , } ) for some a ∈ R , then u ≡ . As it was mentioned before, his proof is based on the inverse scattering method,which uses the fact that the equation in (1.5) is a completely integrable model.In [40] it was proved under general assumptions on F in (1.8) that : If u , u ∈ C ([0 ,
1] : H s ( R n )) , with s > max { n/
2; 2 } are solutions of the equa-tion (1.6) with F as in (1.8) such that u ( x, t ) = u ( x, t ) , ( x, t ) ∈ Γ cx × { , } , where Γ cx denotes the complement of a cone Γ x with vertex x ∈ R n and opening < , then u ≡ u . (For further results in this direction see [40], [34], [35], and references therein).A key step in the proof in [40] was the following uniform exponential decayestimate: Lemma 1.
There exists ǫ > such that if (1.12) V : R n × [0 , → C , with k V k L t L ∞ x ≤ ǫ , and u ∈ C ([0 ,
1] : L ( R n )) is a strong solution of the IVP (1.13) (cid:26) ∂ t u = i (∆ + V ( x, t )) u + G ( x, t ) ,u ( x,
0) = u ( x ) , with (1.14) u , u ≡ u ( · , ∈ L ( e λ · x dx ) , G ∈ L ([0 ,
1] : L ( e λ · x dx )) , for some λ ∈ R n , then there exists c n independent of λ such that (1.15) sup ≤ t ≤ k e λ · x u ( · , t ) k L ( R n ) ≤ c n (cid:16) k e λ · x u k L ( R n ) + k e λ · x u k L ( R n ) + Z k e λ · x G ( · , t ) k L ( R n ) dt (cid:17) . NIQUE CONTINUATION 7
Notice that in the above result one assumes the existence of a reference L -solution u of the equation (1.13) and then under the hypotheses (1.12) and (1.14)shows that the exponential decay in the time interval [0 ,
1] is preserved.The estimate (1.15) can be combined with the subordination formula(1.16) e γ | x | p /p ≃ Z R n e γ /p λ · x −| λ | q /q | λ | n ( q − / dλ, ∀ x ∈ R n and p > , to get that for any α > a > ≤ t ≤ k e α | x | a u ( · , t ) k L ( R n ) ≤ c n (cid:16) k e α | x | a u k L ( R n ) + k e α | x | a u k L ( R n ) + Z k e α | x | a G ( · , t ) k L ( R n ) dt (cid:17) . Under appropriate assumptions on the potential V ( x, t ) in (1.7) one writes V ( x, t ) u = χ R V ( x, t ) u + (1 − χ R ) V ( x, t ) u = V ( x, t ) u + G ( x, t ) , with χ R ∈ C ∞ , χ R ( x ) = 1 , | x | < R , supported in | x | < R , and applies theestimate (1.17) by fixing R sufficiently large. Also under appropriate hypothesison F and u a similar argument can be used for the semi-linear equation in (1.8).The estimate (1.17) gives a control on the decay of the solution in the wholetime interval in terms of that at the end points and that of the “external force”.As we shall see below a key idea will be to get improvements of this estimate basedon logarithmically convex versions of it.We recall that if one considers the equation (1.6) with initial data u ∈ S ( R n )and a smooth potential V ( x, t ) in (1.7) or smooth nonlinearity F in (1.8), it followsthat the corresponding solution satisfies that u ∈ C ([ − T, T ] : S ( R n )). This can beproved using the commutative property of the operators L = ∂ t − i ∆ , and Γ j = x j + 2 t∂ x j , j = 1 , .., n, see [30]-[31]. From the proof of this fact one also has that the persistence propertyof the solution u = u ( x, t ) (i.e. if the data u ∈ X , a function space, then thecorresponding solution u ( · ) describes a continuous curve in X , u ∈ C ([ − T, T ] : X ), T >
0) with data u ∈ L ( | x | m ) can only hold if u ∈ H s ( R n ) with s ≥ m .Roughly speaking, for exponential weights one has a more involved argument wherethe time direction plays a role. Considering the IVP for the one dimensional freeSchr¨odinger equation(1.18) ( ∂ t u = i∂ x u, x, t ∈ R ,u ( x,
0) = u ( x ) ∈ L ( R ) , and assuming that e βx u ∈ L ( R ) , β >
0, then one formally has that v ( x, t ) = e βx u ( x, t )satisfies the equation ∂ t v = i ( ∂ x − β ) v. Thus, v ( x, ±
1) = e βx u ( x, ± ∈ L ( R ) if e ± βξ \ e βx u ∈ L ( R ) . However, if we knew that e βx u ( x, , e βx u ( x, − ∈ L ( R ) integrating forwardin time the positive frequencies of e βx u ( x, t ) and backward in time the negative L. ESCAURIAZA, C. E. KENIG, G. PONCE, AND L. VEGA frequencies of e βx u ( x, t ) one gets an estimate similar to that in (1.15) with λ = β and G = 0. This argument motivates the idea behind Lemma 1 and its proof.The rest of this paper is organized as follows: section 2 contains the resultsrelated to Hardy’s uncertainty principle including a short discussion on the versionof this principle in terms of the heat flow. Section 3 those concerned with Morgan’suncertainty principle. In section 4 we shall consider the limiting case in section3. Also, section 4 includes the statements of some related forthcoming results.Earlier in the introduction we have discussed uniqueness results obtained under theassumption that the solution vanishes at two different time in a semi-space (see [56],[34], [35], [20]). In section 2 similar uniqueness results will be established undera Gaussian decay hypothesis, in the whole space. In section 5 we shall obtain aunifying result, i.e. a uniqueness result under Gaussian decay in a semi-space of R n at two different times. The appendix contains an abstract lemma and a corollarywhich will be used in the previous sections.2. Hardy’s Uncertainty Principle
In [29] G. H. Hardy’s proved the following one dimensional ( n = 1) result: If f ( x ) = O ( e −| x | /β ) , b f ( ξ ) = O ( e − | ξ | /α ) and /αβ > / , then f ≡ .Also, if /αβ = 1 / , f ( x ) is a constant multiple of e −| x | /β . To our knowledge the available proofs of this result and its variants use complexanalysis, mainly appropriate versions of the Phragm´en-Lindel¨of principle. Therehas also been considerable interest in a better understanding of this result and onextensions of it to other settings: [5], [6], [12], [32], and [53]. In particular, theextension of Hardy’s result to higher dimension n ≥ (cid:26) ∂ t u = i △ u, ( x, t ) ∈ R n × (0 , + ∞ ) ,u ( x,
0) = u ( x ) , in the following manner : If u ( x,
0) = O ( e −| x | /β ) , u ( x, T ) = O ( e −| x | /α ) and T /αβ > / , then u ≡ .Also, if T /αβ = 1 / , u has as initial data u equal to a constant multiple of e − ( /β + i/ T ) | y | . The corresponding L -version of Hardy’s uncertainty principle was establishedin [13] : If e | x | /β f , e | ξ | /α b f are in L ( R n ) and /αβ ≥ / , then f ≡ . In terms of the solution of the Schr¨odinger equation it states : If e | x | /β u ( x, , e | ξ | /α u ( x, T ) are in L ( R n ) and T /αβ ≥ / , then u ≡ . More generally, it was shown in [13] that : If e | x | /β f ∈ L p ( R n ) , e | ξ | /α b f ∈ L q ( R n ) , p, q ∈ [1 , ∞ ] with at least one ofthem finite and /αβ ≥ / , then f ≡ . NIQUE CONTINUATION 9
In [20] we proved a uniqueness result for solutions of (1.6) with F as in (1.7) forbounded potentials V verifying that either, V ( x, t ) = V ( x ) + V ( x, t ) , with V real-valued andsup [0 ,T ] k e T | x | / ( αt + β ( T − t )) V ( t ) k L ∞ ( R n ) < + ∞ , or(2.1) lim R → + ∞ Z T k V ( t ) k L ∞ ( R n \ B R ) dt = 0 . More precisely, it was shown that the only solution u ∈ C ([0 , T ] , L ( R n )) to (1.6)with F = V ( x, t ) u , verifying(2.2) k e | x | /β u (0) k L ( R n ) + k e | x | /α u ( T ) k L ( R n ) < + ∞ with T /αβ > / V satisfying one of the above conditions is the zero solution.Notice that this result differs by a factor of 1 / L -version of the Hardy uncertainty principledescribed above ( T /αβ ≥ / L -norms, as established in [13], holds for solutions of(2.3) ∂ t u = i ( △ u + V ( x, t ) u ) , ( x, t ) ∈ R n × [0 , T ] , such that (2.2) holds with T /αβ > / V ( x, t ), while it fails for some complex-valued potentials in the end-point case, T /αβ = 1 / Theorem 3.
Let u ∈ C ([0 , T ]) : L ( R n )) be a solution of the equation (2.3) . Ifthere exist positive constants α and β such that T /αβ > / , and k e | x | /β u (0) k L ( R n ) , k e | x | /α u ( T ) k L ( R n ) < ∞ , and the potential V is bounded and either, V ( x, t ) = V ( x ) + V ( x, t ) , with V real-valued and sup [0 ,T ] k e T | x | / ( αt + β ( T − t )) V ( t ) k L ∞ ( R n ) < + ∞ or lim R → + ∞ k V k L ([0 ,T ] ,L ∞ ( R n \ B R ) = 0 . Then, u ≡ . We remark that there are no assumptions on the size of the potential in thegiven class or on the dimension and that we do not assume any decay of the gra-dient, neither of the solutions or of the time-independent potential or any a priori regularity on this potential or the solution.
Theorem 4.
Assume that
T /αβ = 1 / . Then, there is a smooth complex-valuedpotential V verifying | V ( x, t ) | .
11 + | x | , ( x, t ) ∈ R n × [0 , T ] , and a nonzero smooth function u ∈ C ∞ ([0 , T ] , S ( R n )) solution of (2.3) such that (2.4) k e | x | /β u (0) k L ( R n ) , k e | x | /α u ( T ) k L ( R n ) < ∞ . Our proof of Theorem 3 does not use any complex analysis, giving, in particular,a new proof (up to the end-point) of the L -version of Hardy’s uncertainty principlefor the Fourier transform. It is based on Carleman estimates for certain evolutions.More precisely, it is based on the convexity and log-convexity properties present forthe solutions of these evolutions. Thus, the convexity and log-convexity of appro-priate L -quantities play the role of the Phragm´en-Lindel¨of principle. We observethat the product of log-convex functions is log-convex which, roughly speaking,replaces the fact that the product of analytic functions is analytic.In [11] in collaboration with M. Cowling, we gave new proofs, based only on real variable techniques, of both the L -version of the Hardy uncertainty principleand the original Hardy’s uncertainty principle ( L ∞ ) n -dimensional version for theFourier transform as stated at the beginning of this section, including the end pointcase 1 /α β = 1 / Theorem 5.
Assume that u and V verify the hypothesis in Theorem 3 and T /αβ ≤ / . Then, (2.5) sup [0 ,T ] k e a ( t ) | x | u ( t ) k L ( R n ) + k p t ( T − t ) ∇ (cid:16) e ( a ( t )+ i ˙ a ( t )8 a ( t ) ) | x | u (cid:17) k L ( R n × [0 ,T ]) ≤ N h k e | x | /β u (0) k L ( R n ) + k e | x | /α u ( T ) k L ( R n ) i , where a ( t ) = αβRT αt + β ( T − t )) + 2 R ( αt − β ( T − t )) ,R is the smallest root of the equation Tαβ = R R ) and N depends on T , α , β and the conditions on the potential V in Theorem 3. One has that 1 /a ( t ) is convex and attains its minimum value in the interior of[0 , T ], when | α − β | < R ( α + β ) . To see the optimality of Theorem 5, we write(2.6) u R ( x, t ) = R − n (cid:18) t − iR (cid:19) − n e − | x | i ( t − iR ) = ( Rt − i ) − n e − ( R − iR t )4(1+ R t | x | , which is a free wave (i.e. V ≡
0, in (2.3)) satisfying in R n × [ − ,
1] the correspondingtime translated conditions in Theorem 5 with T = 2 and1 β = 1 α = µ = R R ) ≤ . Moreover R R t ) , is increasing in the R -variable, when 0 < R ≤ − ≤ t ≤ NIQUE CONTINUATION 11
We recall the conformal or Appell transformation: If u ( y, s ) verifies(2.7) ∂ s u = i ( △ u + V ( y, s ) u + F ( y, s )) , ( y, s ) ∈ R n × [0 , , and α and β are positive, then(2.8) e u ( x, t ) = (cid:16) √ αβα (1 − t )+ βt (cid:17) n u (cid:16) √ αβ xα (1 − t )+ βt , βtα (1 − t )+ βt (cid:17) e ( α − β ) | x | i ( α (1 − t )+ βt ) , verifies(2.9) ∂ t e u = i (cid:16) △ e u + e V ( x, t ) e u + e F ( x, t ) (cid:17) , in R n × [0 , , with(2.10) e V ( x, t ) = αβ ( α (1 − t )+ βt ) V (cid:16) √ αβ xα (1 − t )+ βt , βtα (1 − t )+ βt (cid:17) , and(2.11) e F ( x, t ) = (cid:16) √ αβα (1 − t )+ βt (cid:17) n +2 F (cid:16) √ αβ xα (1 − t )+ βt , βtα (1 − t )+ βt (cid:17) e ( α − β ) | x | i ( α (1 − t )+ βt ) . Thus, to prove Theorem 3 for free waves, it suffices to consider u ∈ C ([ − , , L ( R n ))being a solution of(2.12) ∂ t u − = i △ u, ( x, t ) ∈ R × [ − , , and(2.13) k e µ | x | u ( − k L ( R n ) + k e µ | x | u (1) k L ( R n ) < + ∞ , for some µ > u ≡ θ R : [ − , −→ [0 ,
1] such that(2.14) k e R | x | ( R t ) u ( t ) k L ( R n ) ≤ k e µ | x | u ( − k θ R ( t ) L ( R n ) k e µ | x | u (1) k − θ R ( t ) L ( R n ) , where R is the smallest root of the equation µ = R R ) . This gives the optimal improvement of the Gaussian decay of a free wave verifying(2.13) and we also see that if µ > /
8, then u is zero.The proof of these facts relies on new logarithmic convexity properties of freewaves verifying (2.13) and on those already established in [20]. In [20, Theorem 3],the positivity of the space-time commutator of the symmetric and skew-symmetricparts of the operator, e µ | x | ( ∂ t − i △ ) e − µ | x | , is used to prove that k e µ | x | u ( t ) k L ( R n ) is logarithmically convex in [ − , f ( x, t ) = e µ | x | u ( x, t ) = e it ∆ u ( x ) , it follows that e µ | x | ( ∂ t − i △ ) u = e µ | x | ( ∂ t − i △ ) ( e − µ | x | f ) = ∂ t f − S f − A f, where S is symmetric and A skew-symmetric with S = − iµ (4 x · ∇ + 2 n ) , A = i (∆ + 4 µ | x | ) , so that [ S ; A ] = − µ ( ∇ · I ∇ ) + 16 µ | x | . Formally, using the abstract Lemma 3 (see the appendix) and the Heisenberginequality k f k L ( R n ) ≤ n k | x | f k L ( R n ) k ∇ f k L ( R n ) , whose proof follows by integration by parts, one sees that H ( t ) = k f ( t ) k L ( R n ) = k e µ | x | u ( t ) k L ( R n ) is logarithmically convex so k e µ | x | u ( t ) k L ( R n ) ≤ k e µ | x | u ( − k − t L ( R n ) k e µ | x | u (1) k t L ( R n ) , when, − ≤ t ≤ a ≡ µ , we begin an iterative process, where at the k -th step, we have k smooth even functions, a j : [ − , −→ (0 , + ∞ ), 1 ≤ j ≤ k , such that µ ≡ a < a < · · · < a k ∈ ( − , ,F ( a i ) > , a j (1) = µ, j = 1 , . . . , k, where F ( a ) = 1 a (cid:18) ¨ a − a a + 32 a (cid:19) and functions θ j : [ − , −→ [0 , ≤ j ≤ k , such that for t ∈ [ . , k e a j ( t ) | x | u ( t ) k L ( R n ) ≤ k e µ | x | u ( − k θ j ( t ) L ( R n ) k e µ | x | u (1) k − θ j ( t ) L ( R n ) . These estimates follow from the construction of the functions a i , while themethod strongly relies on the following formal convexity properties of free waves:(2.16) ∂ t (cid:18) a ∂ t log H b (cid:19) ≥ − b | ξ | F ( a ) , (2.17) ∂ t (cid:18) a ∂ t H (cid:19) ≥ ǫ a Z R n e a | x | (cid:0) |∇ u | + | x | | u | (cid:1) dx, where H b ( t ) = k e a ( t ) | x + b ( t ) ξ | u ( t ) k L ( R n ) , H ( t ) = k e a ( t ) | x | u ( t ) k L ( R n ) ,ξ ∈ R n and a, b : [ − , −→ R are smooth functions with a > , F ( a ) > − , . Once the k -th step is completed, we take a = a k in (2.16) with a certain choiceof b = b k , verifying b ( −
1) = b (1) = 0 and then, a certain test is performed. Whenthe answer to the test is positive, it follows that u ≡
0. Otherwise, the logarithmicconvexity associated to (2.16) allows us to find a new smooth function a k +1 in[ − ,
1] with a < a < · · · < a k < a k +1 , ( − , , and verifying the same properties as a , . . . , a k .When the process is infinite, we have (2.15) for all k ≥ k → + ∞ a k (0) = + ∞ , or lim k → + ∞ a k (0) < + ∞ . NIQUE CONTINUATION 13
In the first case and (2.15) one has that u ≡
0, while in the second, the sequence a k is shown to converge to an even function a verifying(2.18) ( ¨ a − a a + 32 a = 0 , [ − , a (1) = µ. Because a ( t ) = R R t ) , R ∈ R + , are all the possible even solutions of this equation, a must be one of them and µ = R R ) , for some R >
0. In particular, u ≡
0, when µ > / V = V ( x, t )), is based on the extension of the above convexity propertiesto the non-free case.Theorem 4 establishes the sharpness of the result in Theorem 3 by giving anexample of a complex valued potential V ( x, t ) verifying (2.1) and a non-trivialsolution u ∈ C ([0 , T ] : L ( R n )) of (2.3) for which (2.2) holds with T /αβ = 1 / V ( x, t )verifying the same properties, i.e. satisfying (2.1) and having a non-trivial solution u ∈ C ([0 , T ] : L ( R n )) of (2.3) such that (2.2) holds with T /αβ = 1 / V = V ( x ). In this regard, we considerthe stationary problem(2.19) ∆ w + V ( x ) w = 0 , x ∈ R n , V ∈ L ∞ ( R n ) , and recall V. Z. Meshkov’s result in [45] : If w ∈ H loc ( R n ) is a solution of (2.19) such that (2.20) Z R n e a | x | / | w ( x ) | dx < ∞ , ∀ a > , then u ≡ . Moreover, it was also proved in [45] that for complex potentials V , the exponent4 / If w ∈ H loc ( R n ) is a solution of (2.19) with a complex valued potential V satisfying V ( x ) = V ( x ) + V ( x ) , such that (2.21) | V ( x ) | ≤ c (1 + | x | ) α/ , α ∈ [0 , / , and V real valued supported in { x : | x | ≥ } such that − ( ∂ r V ( x )) − < c | x | α , a − = min { a ; 0 } . Then there exists a = a ( k V k ∞ ; c ; c ; α ) > such that if (2.22) Z R n e a | x | r | w ( x ) | dx < ∞ , r = (4 − α ) / , then u ≡ . In addition, one can take the value r = 1 in (2.20) by assuming α > / α ∈ [0 , / φ ( x ) of the eigenvalue problem(2.23) ∆ φ + e V ( x ) φ = λφ, x ∈ R n , with λ ∈ R , then V ( x ) = e V ( x ) + λ satisfies the hypothesis of the previous resultand u ( x, t ) = e itλ φ ( x ) , solves the evolution equation(2.24) ∂ t u = i (∆ u + V ( x ) u ) , x ∈ R n , t ∈ R , one gets a lower bound for the value of the strongest possible decay rate of non-trivial solutions u ( x, t ) of (2.24) at two different times.As a direct consequence of Theorem 3 we have the following application con-cerning the uniqueness of solutions for semi-linear equations of the form (1.6) with F as in (1.8). Theorem 6.
Let u and u be strong solutions in C ([0 , T ] , H k ( R n )) , k > n/ ofthe equation (1.6) with F as in (1.8) such that F ∈ C k and F (0) = ∂ u F (0) = ∂ ¯ u F (0) = 0 . If there are α and β positive with T /αβ > / such that e | x | /β ( u (0) − u (0)) , e | x | /α ( u ( T ) − u ( T )) ∈ L ( R n ) , then u ≡ u . In Theorem 6 we did not attempt to optimize the regularity assumption on thesolutions u , u .By fixing u ≡ u of equation (1.6) with F as in (1.8). It isan open question to determine the optimality of this kind of result. More precisely,for the standard semi-linear Schr¨odinger equations(2.25) ∂ t u = i (∆ u + | u | γ − u ) , γ > , one has the standing wave solutions u ( x, t ) = e ω t ϕ ( x ) , ω > , where ϕ is the unique (up to translation) positive solution of the elliptic problem − ∆ ϕ + ωϕ = | ϕ | γ − ϕ, which has a linear exponential decay, i.e. ϕ ( x ) = O ( e − c | x | ) , as | x | → ∞ , NIQUE CONTINUATION 15 for an appropriate value of c > ∂ t u = ∆ u, t > , x ∈ R n , whose solution with data u ( x,
0) = u ( x ) can be written as u ( x, t ) = e t ∆ u ( x ) = Z R n e −| x − y | / t (4 πt ) n/ u ( y ) dy. More precisely, Hardy’s uncertainty principle can restated in the following equiv-alent forms : (i) If u ∈ L ( R n ) and there exists T > such that e | x | / ( δ T ) e T ∆ u ∈ L ( R n ) for some δ ≤ , then u ≡ .(ii) If u ∈ S ( R n ) (tempered distribution) and there exists T > such that e | x | / ( δ T ) e T ∆ u ∈ L ∞ ( R n ) for some δ < , then u ≡ . Moreover, if δ = 2 , then u is a constant multiple of the Dirac delta measure. In fact, applying Hardy’s uncertainty principle to e T △ u one has that e | x | δ T e T △ u and e T | ξ | \ e T △ u = b u in L ( R n ) with 2 δ ≤ e △ u ≡
0. Then, backwarduniqueness arguments (see for example [44, Chapter 3, Theorem 11]) shows that u ≡ Theorem 7.
Let u ∈ C ([0 ,
1] : L ( R n )) ∩ L ([0 , T ] : H ( R n )) be a solution of theIVP ( ∂ t u = △ u + V ( x, t ) u, in R n × (0 , ,u ( x,
0) = u ( x ) , where V ∈ L ∞ ( R n × [0 , . If u and e | x | δ u (1) ∈ L ( R n ) , for some δ < , then u ≡ . It is natural to expect that Hardy’s uncertainty principle holds in this contextwith bounded potentials V and with the parameter δ verifing the condition of thefree case, i.e. δ ≤ Uncertainty Principle of Morgan type
In [46] G. W. Morgan proved the following uncertainty principle: If f ( x ) = O ( e − ap | x | pp ) , < p ≤ and b f ( ξ ) = O ( e − ( b + ǫ ) q | ξ | qq ) , /p + 1 /q = 1 , ǫ > , with ab > (cid:12)(cid:12)(cid:12) cos (cid:16) p π (cid:17) (cid:12)(cid:12)(cid:12) , then f ≡ . In [32] Beurling-H¨ormander showed : If f ∈ L ( R ) and (3.1) Z R Z R | f ( x ) || b f ( ξ ) | e | x ξ | dx dξ < ∞ , then f ≡ . This result was extended to higher dimensions n ≥ If f ∈ L ( R n ) , n ≥ and (3.2) Z R n Z R n | f ( x ) || b f ( ξ ) | e | x · ξ | dx dξ < ∞ , then f ≡ . We observe that from (3.1) and (3.2) it follows that : If p ∈ (1 , , /p + 1 /q = 1 , a, b > , and (3.3) Z R n | f ( x ) | e ap | x | pp dx + Z R n | b f ( ξ ) | e bq | ξ | qq dξ < ∞ , ab ≥ ⇒ f ≡ . Notice that in the case p = q = 2 this gives us an L -version of Hardy’s un-certainty result discussed above, and for p < n -dimensional L -version ofMorgan’s uncertainty principle.In the one-dimensional case ( n = 1), the optimal L -version of Morgan’s resultin (3.3),(3.4) Z R | f ( x ) | e ap | x | pp dx + Z R | b f ( ξ ) | e bq | ξ | qq dξ < ∞ , ab > (cid:12)(cid:12)(cid:12) cos (cid:16) p π (cid:17) (cid:12)(cid:12)(cid:12) ⇒ f ≡ . was established in [6] and [2] (for further results see [5] and references therein).A sharp condition for a, b, p in (3.4) in higher dimension seems to be unknown.However, in [6] it was shown : If f ∈ L ( R n ) , < p ≤ and /p + 1 /q = 1 are such that for some j = 1 , .., n , (3.5) Z R n | f ( x ) | e ap | xj | pp dx < ∞ + Z R n | b f ( ξ ) | e bq | ξj | qq dξ < ∞ . If ab > (cid:12)(cid:12)(cid:12) cos (cid:0) p π (cid:1) (cid:12)(cid:12)(cid:12) , then f ≡ .If ab < (cid:12)(cid:12)(cid:12) cos (cid:0) p π (cid:1) (cid:12)(cid:12)(cid:12) , then there exist non-trivial functions satisfying (3.5).Using (1.11) the above result can be stated in terms of the solution of the freeSchr¨odinger equation. In particular, (3.3) can be re-written as : If u ∈ L ( R ) or u ∈ L ( R n ) , if n ≥ , and for some t = 0(3.6) Z R n | u ( x ) | e ap | x | pp dx + Z R n | e it ∆ u ( x ) | e bq | x | qq (2 t ) q dx < ∞ , with ab > (cid:12)(cid:12)(cid:12) cos (cid:16) p π (cid:17) (cid:12)(cid:12)(cid:12) if n = 1 , and ab > if n ≥ , then u ≡ . NIQUE CONTINUATION 17
Related with Morgan’s uncertainty principle one has the following result due toGel’fand and Shilov. In [27] they considered the class Z pp , p >
1, defined as thespace of all functions ϕ ( z , .., z n ) which are analytic for all values of z , .., z n ∈ C and such that | ϕ ( z , .., z n ) | ≤ C e P nj =1 ǫ j C j | z j | p , where the C j , j = 0 , , .., n are positive constants and ǫ j = 1 for z j non-real and ǫ j = − z j real, j = 1 , .., n , and showed that the Fourier transform of thefunction space Z pp is the space Z qq , with 1 /p + 1 /q = 1.Notice that the class Z pp with p ≥ e ic | x | . Thus, if u ∈ Z pp , p ≥
2, then by (1.11) one has that | e it ∆ u ( x ) | ≤ d ( t ) e − a ( t ) | x | q , for some functions d, a : R → (0 , ∞ ).In [21] the following results were established: Theorem 8.
Given p ∈ (1 , there exists M p > such that for any solution u ∈ C ([0 ,
1] : L ( R n )) of ∂ t u = i ( △ u + V ( x, t ) u ) , in R n × [0 , , with V = V ( x, t ) complex valued, bounded (i.e. k V k L ∞ ( R n × [0 , ≤ C ) and (3.7) lim R → + ∞ k V k L ([0 , L ∞ ( R n \ B R )) = 0 , satisfying that for some constants a , a , a > Z R n | u ( x, | e a | x | p dx < ∞ , and for any k ∈ Z + (3.9) Z R n | u ( x, | e k | x | p dx < a e a k q/ ( q − p ) , /p + 1 /q = 1 , if (3.10) a a ( p − > M p , then u ≡ . Corollary 1.
Given p ∈ (1 , there exists N p > such that if u ∈ C ([0 ,
1] : L ( R n )) is a solution of ∂ t u = i (∆ u + V ( x, t ) u ) , with V = V ( x, t ) complex valued, bounded (i.e. k V k L ∞ ( R n × [0 , ≤ C ) and lim R →∞ Z sup | x | >R | V ( x, t ) | dt = 0 , and there exist α, β > such that (3.11) Z R n | u ( x, | e α p | x | p /p dx + Z R n | u ( x, | e β q | x | q /q dx < ∞ , /p + 1 /q = 1 , with (3.12) α β > N p , then u ≡ . As a consequence of Corollary 1 one obtains the following result concerning theuniqueness of solutions for the semi-linear equations (1.6) with F as in (1.8)(3.13) i∂ t u + △ u = F ( u, u ) . Theorem 9.
Given p ∈ (1 , there exists N p > such that if u , u ∈ C ([0 ,
1] : H k ( R n )) , are strong solutions of (3.13) with k ∈ Z + , k > n/ , F : C → C , F ∈ C k and F (0) = ∂ u F (0) = ∂ ¯ u F (0) = 0 , and there exist α, β > such that (3.14) e α p | x | p /p ( u (0) − u (0)) , e β q | x | q /q ( u (1) − u (1)) ∈ L ( R n ) , /p + 1 /q = 1 , with (3.15) α β > N p , then u ≡ u . Notice that the conditions (3.10) and (3.12) are independent of the size of the po-tential and there is not any a priori regularity assumption on the potential V ( x, t ).The result in [6], see (3.5), can be extended to our setting with an non-optimalconstant. More precisely, Corollary 2.
The conclusions in Corollary 1 still hold with a different constant N p > if one replaces the hypothesis (3.11) by the following one dimensionalversion (3.16) Z R n | u ( x, | e α p | x j | p /p dx < ∞ + Z R n | u ( x, | e β q | x j | q /q dx < ∞ , for some j = 1 , .., n . Similarly, the non-linear version of Theorem 9 still holds, with different constant N p >
0, if one replaces the hypothesis (3.14) by e α p | x j | p /p ( u (0) − u (0)) , e β q | x j | q /q ( u (1) − u (1)) ∈ L ( R n ) , for j = 1 , .., n .In [21] we did not attempt to give an estimate of the universal constant N p .The limiting case p = 1 will be considered in the next section.The main idea in the proof of these results is to combine an upper estimate witha lower one to obtain the desired result. The upper estimate is based on the decayhypothesis on the solution at two different times (see Lemma 1). In previous workswe had been able to establish these estimates from assumptions that at time t = 0and t = 1 involving the same weight. However, in our case (Corollary 1) we havedifferent weights at time t = 0 and t = 1. To overcome this difficulty, we carryout the details with the weight e a j | x | p , < p < j = 0 at t = 0 and j = 1 at t = 1, with a fixed and a = k ∈ Z + as in (3.9). Although the powers | x | p inthe exponential are equal at time t = 0 and t = 1 to apply our estimate (Lemma1) we also need to have the same constant in front of them. To achieve this weapply the conformal or Appell transformation described above, to get solutions andpotentials, whose bounds depend on k ∈ Z + . Thus we have to consider a family ofsolutions and obtain estimates on their asymptotic value as k ↑ ∞ .The proof of the lower estimate is based on the positivity of the commutatoroperator obtained by conjugating the equation with the appropriate exponentialweight, (see Lemma 3 in the appendix) NIQUE CONTINUATION 19 Paley-Wiener Theorem and Uncertainty Principle of Ingham type
This section is concerned with the limiting case p = 1 in the previous section.It is easy to see that if f ∈ L ( R n ) is non-zero and has compact support, then b f cannot satisfy a condition of the type b f ( y ) = O ( e − ǫ | y | ) for any ǫ >
0. However,it may be possible to have f ∈ L ( R n ) a non-zero function with compact support,such that b f ( ξ ) = O ( e − ǫ ( y ) | y | ), ǫ ( y ) being a positive function tending to zero as | y | → ∞ .In the one-dimensional case ( n = 1) soon after Hardy’s result described above,A. E. Ingham [33] proved the following : There exists f ∈ L ( R ) non-zero, even, vanishing outside an interval such that b f ( y ) = O ( e − ǫ ( y ) | y | ) with ǫ ( y ) being a positive function tending to zero at infinity ifand only if Z ∞ ǫ ( y ) y dy < ∞ . In a similar direction the Paley-Wiener Theorem [50] gives a characterization ofa function or distribution with compact support in term of analyticity propertiesof its Fourier transform.Regarding our results discussed above it would be interesting to identify a classof potentials V ( x, t ) for which a result of the following kind holds:If u ∈ C ([0 ,
1] : L ( R n )) is a non-trivial solution of the IVP(4.1) (cid:26) ∂ t u = i ( △ u + V ( x, t ) u ) , ( x, t ) ∈ R n × [0 , ,u ( x,
0) = u ( x ) , with u ∈ L ( R n ) having compact support, then e ǫ | x | u ( · , t ) / ∈ L ( R n ) for any ǫ > t ∈ (0 , Theorem 10.
Assume that u ∈ C ([0 ,
1] : L ( R n )) is a strong solution of the IVP (2.4) with (4.2) supp u ⊂ B R (0) = { x ∈ R n : | x | ≤ R } , (4.3) Z R n e a | x | | u ( x, | dx < ∞ , a > , and (4.4) k V k L ∞ ( R n × [0 , = M , with (4.5) lim R → + ∞ k V k L ([0 , L ∞ ( R n \ B R )) = 0 . Then, there exists b = b ( n ) > (depending only on the dimension n ) such that if a R (1 + M ) ≥ b, then u ≡ . A similar question can be raised for results of the type described above due toA. E. Ingham in [33] and possible extensions to higher dimensions n ≥ u ( x, t ) to the equation (1.6) with F as in (1.8) associatedto data u ∈ L ( R n ) with compact support or with u ∈ C ∞ ( R n ). In this direction,some results can be deduced as a consequence of Theorem 10, see [24].5. Hardy’s Uncertainty Principle in a half-space
In the introduction we have briefly reviewed some uniqueness results establishedfor solutions of the Schr¨odinger equation vanishing at two different times in a semi-space of R n , (see [56], [15], [34], [35], [20]). In section 2, we have studied uniquenessresults gotten under the hypothesis that the solution of the Schr¨odinger equationat two different times has an appropriate Gaussian decay, in the whole space R n .In this section, we shall deduce a unified result, i.e. a uniqueness result under thehypothesis that at two different times the solution of the Schr¨odinger equation hasGaussian decay in just a semi-space of R n . Theorem 11.
Assume that u ∈ C ([0 ,
1] : L ((0 , ∞ ) × R n − )) is a strong solutionof the IVP (5.1) (cid:26) ∂ t u = i (∆ + V ( x, t )) u,u ( x,
0) = u ( x ) , with (5.2) Z Z / / | ∂ x u ( x, t ) | dx dt < ∞ , (5.3) V : R n × [0 , → C , V ∈ L ∞ ( R n × [0 , , and (5.4) lim R → + ∞ Z k V ( t ) k L ∞ ( { x >R } ) dt = 0 . Assume that (5.5) Z x > e c | x | | u ( x, | dx < ∞ , Z x > e c | x | | u ( x, | dx < ∞ , with c , c > sufficiently large. Then u ≡ . Remarks : (a) Note that in Theorem 11, the solution does not need to be definedfor x ≤
0. In this sense, this is a stronger result that the uniqueness results in [56],[40], [34], [35], and [15], which required that the solution be defined in R n × [0 , C ([0 ,
1] : L ( R n )).On the other hand, we need to assume the condition (5.2). Note that [40] alsoneeds an extra assumption on ∇ u , stronger that (5.2), but that in [34], whichamong other things removed any extra assumption on ∇ u , but still required thesolution to be defined in R n × [0 ,
1] and be in C ([0 ,
1] : L ( R n )). If in the setting of NIQUE CONTINUATION 21
Theorem 11 we know that u is a solution in R n × [0 ,
1] and is in C ([0 ,
1] : L ( R n )),then we can dispose the hypothesis (5.2) as follows:First as in the first step of the proof of Theorem 11, we can use the Appell trans-formation to reduce to the case c = c = 2 γ . Then, using ϕ ( x ) a “regularized”convex function which agrees with x +1 for x > x < −
1, an application ofLemma 3 and Corollary 3 in the appendix yields the estimatesup ≤ t ≤ Z e γ ( x +1 ) | u ( x, t ) | dx + Z Z x > t (1 − t ) |∇ u ( x, t ) | e γ ( x +1 ) dxdt < ∞ . Once this is obtained, by restricting our attention to(2 , ∞ ) × R n − × [ δ, − δ ] , for each δ >
0, we are in the situation of Theorem 11, and hence u ≡ { x > } × [0 , u ≡ ∇ u to exist for − < x < . (b) We have seen that Theorem 11 includes many of the uniqueness results forsolutions vanishing at two different times in a semi-space. In comparison with theresults in section 2, since the extra assumption (5.2) can be recovered as in remark(a) when the solution is defined in R n × [0 ,
1] and is in C ([0 ,
1] : L ( R n )), the onlyweakness is that the provide an optimal estimate for the constants c , c , but onthe other hand deals with solutions only defined in (0 , ∞ ) × R n − × [0 , ~e can be replaced by any other ω ∈ S n − .Proof of Theorem 11: The strategy of the proof follows closely the one in [16].We divide the proof into three steps.First Step : Reduction to the case c = c = 2 γ .This follows by using the conformal or Appell transformation introduced in sec-tion 2 (see (2.7)-(2.11)), combined with the observation that the set { x > } remains invariant.Second Step : Upper Bounds.We define v ( x, t ) = θ ( x ) u ( x, t ) , with θ ∈ C ∞ ( R ), non-decreasing with θ ( x ) ≡ x > /
2, and θ ( x ) ≡ x < /
2. Therefore,(5.6) ∂ t v = i ∆ v + i V ( x, t ) v + i F ( x, t ) , F ( x, t ) = 2 ∂ x u θ ′ ( x ) + u θ ′′ ( x ) . Using (5.2) we can apply Lemma 1 to get that(5.7) sup ≤ t ≤ k e λ · x v ( · , t ) k L ( R n ) ≤ c n (cid:16) k e λ · x v (0) k L ( R n ) + k e λ · x v (1) k L ( R n ) + Z k e λ · x F ( · , t ) k L ( R n ) dt + Z k e λ · x V χ { x NIQUE CONTINUATION 23 We recall the following result which is a slight variation of that proven in detailin [16] (Lemma 3.1, page 1818) : Lemma 2. Assume that R > and ϕ : [0 , → R is a smooth function. Then,there exists c = c ( n ; k ϕ ′ k ∞ + k ϕ ′′ k ∞ ) > such that the inequality (5.13) α / R (cid:13)(cid:13)(cid:13) e α | x − x R + ϕ ( t ) | g (cid:13)(cid:13)(cid:13) L ( dxdt ) ≤ c (cid:13)(cid:13)(cid:13) e α | x − x R + ϕ ( t ) | ( i∂ t + ∆) g (cid:13)(cid:13)(cid:13) L ( dxdt ) holds when α > cR and g ∈ C ∞ ( R n +1 ) is supported in the set { ( x, t ) = ( x , .., x n , t ) ∈ R n +1 : | x − x R + ϕ ( t ) | ≥ } . Now, we will chose x = R/ 2, 0 ≤ ϕ ( t ) ≤ a, with a = 3 / − /R , ϕ ( t ) = a, on 3 / ≤ t ≤ / ϕ ( t ) = 0 , for t ∈ [0 , / ∪ [3 / , θ R ∈ C ∞ ( R ) with θ R ( x ) = 1 on 1 < x < R − 1, and θ R ( x ) = 0 for x < / x > R .Also we chose η ∈ C ∞ ( R ) with η ( x ) = 0 , x ≤ η ( x ) = 1 , x ≥ / R .We notice that up to translation we can assume that(5.14) Z / / Z 8) one has ϕ = a , η (cid:16) x − R/ R + a (cid:17) = 1 and θ R = 1, hence in this domain g ( x, t ) = u ( x, t ) . Thus, from (5.16) it follows that | x − R/ R + ϕ ( t ) | ≥ /R, so we have the lower bound of (5.13) α / R b e α (1+1 /R ) , with b as in (5.14). Now we shall estimate the right hand side of (5.13). Thus,(5.17) ( i∂ t − ∆) g = − θ R ( x ) η (cid:16) x − R/ R + ϕ ( t ) (cid:17) V ( x, t ) u ( x, t )+ η (cid:16) x − R/ R + ϕ ( t ) (cid:17) (2 θ ′ ( x ) ∂ x u + u θ ′′ R ( x ))+ ( iη ′ ( · ) ϕ ′ ( t ) + η ′′ ( · ) 1 R ) θ R ( x ) u ( x, t ) ≡ E + E + E . Choosing R >> k V k ∞ , and recalling the fact that α > cR we see that thecontribution of the term E involving the potential V can be absorbed by the termin the left hand side of (5.13).Next, we notice that the terms in E involve derivatives of θ R ( θ ′ R or θ ′′ R ) so theyare supported in the ( x, t ) ∈ R n × [0 , 1] such that1 / < x < , or R − < x < R. But, if 1 / < x < 1, it follows that x − R/ R + ϕ ( t ) ≤ /R − / / − /R = 1 , so η (cid:16) x − R/ R + ϕ ( t ) (cid:17) = 0 . Thus, we only get contribution from the ( x, t ) ∈ R n × [0 , 1] such that R − < x < R ,which can be bounded by c Z / / Z R − Above we have used the following abstract results established in [20]: Lemma 3. Let S be a symmetric operator, A be a skew-symmetric one, both allowedto depend on the time variable. Let G be a positive function, f ( x, t ) a reasonablefunction, H ( t ) = ( f, f ) = k f k L ( R n ) = k f k , D ( t ) = ( S f, f ) ,∂ t S = S t and N ( t ) = D ( t ) H ( t ) . 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