Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

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1 Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type Angela Pasquale Université de Lorraine Joint work with Maddala Sundari Geometric Analysis on Euclidean and Homogeneous Spaces Tufts University January 8 9, 2012 Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 1 / 16

2 Qualitative uncertainty principle = a theorem which allows to conclude that f = 0 by giving quantitative conditions on f and its Fourier transform f In this talk: Initial value problem for the time-dependent Schrödinger equation on a Riemannian symmetric space of the noncompact type X: i tu(t, x) + u(t, x) = 0 u(0, x) = f (x) (S) where denotes the Laplace-Beltrami operator on X and f L 2 (X). We prove: The solution to (S) is identically zero at all times t whenever the initial condition f and the solution at a time t 0 > 0 are simultaneously very rapidly decreasing. This is an uncertainty principle because of the relation linking the Fourier-Helgason transforms of f and of the solution u t = u(t, ) The condition of rapid decrease we consider is of Beurling type. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 2 / 16

3 Previous work: Schrödinger equation on R n : Escauriaza, Kenig, Ponce, Vega (& Cowling) (2006, 2008, 2010, 2011) Also nonlinear Schrödinger equation, with a time-dependent potential,... Uncertainty conditions on the initial condition f and the solution u(t 0, ) at a time t 0 of different types: Hardy, Morgan, Beurling... On nilpotent Lie groups: Ben Said & Thangavelu (2010) for the Heisenberg group Uncertainty conditions of Hardy type On Riemannian symmetric spaces G/K of the noncompact type: Chanillo (2007) for G complex and K -invariant functions Uncertainty conditions of Hardy type Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 3 / 16

4 Beurling-type uncertainty principles on R n Theorem (Beurling s theorem on R, Hörmander (1991)) Let f L 1 (R). If f (x) f (y) e xy dx dy < R R then f = 0 almost everywhere. A higher dimensional version: Theorem (Beurling s theorem on R n, Bagchi & Ray (1998)) Let f L 1 (R n ). Suppose that R n R n f (x) f (y) e x y dx dy <. Then f = 0 almost everywhere. Sharpest version: Bonami, Demange & Jaming (2003) Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 4 / 16

5 Beurling s theorem Gelfand-Shilov type Let f L 2 (R n ) and 1 < p, q < with 1/p + 1/q = 1. Suppose a > 0, b > 0 so that f (x) e a p p x p dx < and f (x) e b q q x q dx < R n R n If ab 1, then f = 0 almost everywhere. Cowling-Price type Let f L 2 (R n ) and let 1 p, q <. Suppose a > 0, b > 0 so that ( 2 ) f (x) e a x pdx ( 2 ) < and f (x) e b x qdx < R R n n If ab > 1/4, then f = 0 almost everywhere. Hardy type Let f be measurable on R n. Suppose a > 0, b > 0 so that f (x) e a x 2 and f (x) e b x 2 If ab > 1/4, then f = 0 almost everywhere. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 5 / 16

6 The damped Schrödinger equation on R n Initial value problem for the time-dependent damped Schrödinger equation on R n : u(0, x) = f (x) where: = Laplace operator on R n c R = the damping parameter i tu(t, x) + ( c)u(t, x) = 0 Suppose f L 2 (R n ). Take Fourier transform in the x-variable of both sides of the equations. This gives rise to an ODE in the t-variable that can be solved directly. If f L 2 (R n ), then unique u C(R : L 2 (R n )) satisfying (S c) in the sense of distributions. The solution u t(x) = u(t, x) is characterized by the equation (S c) û t(λ) = e i( λ 2 +c)t f (λ). (*) More generally: (*) admits unique solution u t S (R n ) if f S (R n ). Explicitly: for t 0 and f L 2 (R n ) (or f L 1 (R n ) S (R n )): u(t, x) = (2π) n/2 (2 t ) n/2 e ict e πi(sign t)n/4 e i x 2 4t where h t(y) = e i y 2 4t f (y). Remark: If exists t 0 > 0 so that h t0 = 0 (i.e. u t0 = 0), then f = 0 ( x ) ĥ t 2t Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 6 / 16

7 Theorem (Cowling et al., P.-Sundari) Let u t(x) = u(t, x) be the solution to the damped Schrödinger equation (S c) with initial condition f L 2 (R n ) or f L 1 (R n ). If there is t 0 > 0 so that f (x) u(t 0, y) e x y 2t 0 dx dy < (BS) R n R n then f = 0. Hence u(t, ) = 0 for all t R. Proof. Suppose first f L 1 (R n ). Then: x y 1 2t + > f (x) u(t 0, y) e 0 dx dy = R n R n (2t 0 ) n/2 R n = (2t 0 ) n/2 h t0 (x) ĥt 0 (y) e x y dx dy R n R n i 2 4t Hence f = e 0 h t0 = 0 by Beurling s theorem. If f L 2 (R n ) satisfies (BS), then f L 1 (R n ). Indeed: u(t 0, y) f (x) e R n If u(t 0, y) = 0 a.e. y, then f = 0. If not, y 0 such that f 1 R n f (x) e R n h t0 (x)ĥt 0 ( y 2t 0 ) e x y 2t 0 dx < + for almost all y. x y 0 2t 0 dx < +. x y 2t 0 dx dy Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 7 / 16

8 The Schrödinger equation on X = G/K X = G/K Riemannian symmetric space of the noncompact type where G = noncompact connected semisimple Lie group with finite center K = maximal compact subgroup of G = Laplace-Beltrami operator on X Initial value problem for the time-dependent Schrödinger equation on X with f L 2 (X): i tu(t, x) + u(t, x) = 0 (S) u(0, x) = f (x) o = ek the base point of X σ(x) = d(x, o) = distance from x X to o wrt the Riemannian metric d Ξ(x) = the (elementary) spherical function of spectral parameter 0 Theorem Let u(t, x) C(R : L 2 (X)) denote the solution of (S) with initial condition f L 2 (X). If there is a time t 0 > 0 so that f (x) u(t 0, y) Ξ(x)Ξ(y)e σ(x)σ(y) 2t 0 dx dy <, (BS) then u(t, ) = 0 for all t R. X X Idea of proof: Helgason-Fourier transform + (careful) Euclidean reduction via Radon transform. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 8 / 16

9 The Helgason-Fourier and the Radon transforms g = k p Cartan decomposition of the Lie algebra g of G where k = Lie algebra of K a p max abelian subspace (Cartan subspace) A = exp a abelian subgroup of G G = KAN Iwasawa decomposition H(g) = Iwasawa projection of g G on a, i.e. g = k(g) exp H(g)n(g) M = centralizer of A in K, and B = K /M A : X B a by A(gK, km) := H(g 1 k) Σ a roots of (g, a) Σ + = choice of positive roots m α = multiplicity of the root α Σ ρ = 1/2 α Σ m αα + For λ a C and b B, define: e λ,b (x) = e (iλ+ρ)(a(x,b)), x X. The Helgason-Fourier transform of a (sufficiently regular) function f : X C is the function Ff : a C B C defined by Ff (λ, b) = f (x)e λ,b (x) dx = f (kan o)e ( iλ+ρ)(log a) da dn X (b = km; Haar measures da and dn suitably normalized) Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 9 / 16 AN

10 Theorem (Helgason, 1965; Eguchi, 1980) 1 (Plancherel theorem) F extends to an isometry of L 2 (X) onto L 2 (a + B, c(λ) 2 dλ db). Here c is Harish-Chandra s c-function. 2 (Inversion formula) If f L 1 (X) and Ff L 1 (a + B, c(λ) 2 dλ db), then f (x) = 1 dλ db Ff (λ, b)e λ,b (x) a.e. x X W c(λ) 2 a B The (modified) Radon transform of a (sufficiently regular) function f : X C is the function Rf : B A C defined by Rf (b, a) = e ρ(log a) f (kan o) dn, b = km. N The Euclidean Fourier transform of a (sufficiently regular) function f : A C is the function F A f : a C defined by (F A f )(λ) := f (a)e iλ(log a) da, λ a. A F and R (on suitable classes of functions on X) are linked by the Euclidean transform. For instance: if f L 1 (X), then Rf L 1 (B A) and ( ) Ff (λ, b) = F A Rf (b, ) (λ) = Rf (b, a)e iλ(log a) da. A for almost all b B and all λ a. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 10 / 16

11 Solution of the Schro dinger equation on X = G/K i tu(t, x) + u(t, x) = 0 u(0, x) = f (x) L 2 (X) (S) The Fourier-transform method used on R n carries out to (S) when the Fourier transform is replaced by the Helgason-Fourier transform F. The functions e λ,b appearing in the definition of F are eigenfunctions of Plancherel: F is a unitary equivalence between on L 2 (G/K ) and the multiplication operator M on L 2 (a dλ db B, ) defined by (Mf )(λ, b) = ( λ 2 + ρ 2 )f (λ, b). c(λ) 2 Hence: there is a unique u t(x) = u(t, x) C(R : L 2 (X)) satisfying (S) in the sense of distributions. It is characterized by the equation i.e. (Fu t)(λ, b) = e i( λ 2 + ρ 2 )t Ff (λ, b). u t = F 1( e i( λ 2 + ρ 2 )t Ff ). Remark: If f = 0, then u t = 0. Conversely, if t 0 R so that u t0 = 0, then e i( λ 2 + ρ 2 )t 0 Ff (λ, b) = 0 for all (λ, b). Hence f = 0 as F is injective on L 2 (X). Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 11 / 16

12 Fourier analysis on B = K /M K M = (equiv classes of) irreducible unitary reps of K with nonzero M-fixed vectors V δ = space of δ K M inner product, d(δ) = dim V δ {v δ 1,..., v δ d(δ)} = ON basis of V δ The Fourier coefficients of F L 1 (B) are Fi,j δ = δ(k 1 )vi δ, vj δ F(kM) dk. Then: K F = 0 if and only if F δ i,j = 0 for all δ K M and all i, j = 1,..., d(δ). Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 12 / 16

13 The space L 1 (X) C (with C 0) For measurable h : X C set: h 1,C := X h(x) Ξ(x)eCσ(x) dx L 1 (X) C = (equiv classes of a.e. equal) functions h : X C with h 1,C <. Motivation: If f, u t0 satisfy X X f (x) u(t 0, y) Ξ(x)Ξ(y)e σ(x)σ(y)/(2t0) dx dy <, then C > 0 so that f L 1 (X) C. Likewise for u t0. Remark: L 1 (X) C L 1 (X) if C >> 0 (e.g. C ρ ). Lemma Let h L 1 (X) C. Then: 1 The Radon transform Rh is a.e. defined and in L 1 (B A, db da) 2 For all λ a, the Helgason-Fourier transform Fh(λ, ) is a.e. defined and in L 1 (B) Proof for C = 0. > h 1,C=0 = h(g o) e ρ(h(g)) dg (by Ξ(g) = K e ρ(h(gk)) dk & K -inv of σ and h) G ) (e ρ(log a) h(kan o) dn da dk (by Iwasawa decomp & σ(an) σ(a)) K A N (Rh)(b, a) da db B A Moreover: Fh(λ, b) A N h(kan o) eρ(log a) da dn A (Rh)(b, a) da (b = km) Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 13 / 16

14 Consequence: If h L 1 (X) C then Rh(b, ( ) L 1 (A) ) for almost all b B we can consider F A Rh(b, ) Lemma F = F A R on L 1 (X) C. Proof (Sketch). For δ K M and i, j = 1,..., d(δ) define: (Rh) δ i,j (a) := ( Rh(, a) ) δ i,j (Fh) δ i,j (λ) := ( Fh(λ, ) ) δ i,j for almost all a A for all λ a ( Get: F A (Rh) δ i,j (a) ) : a C and (Fh) δ i,j : a C. { ( h FA (Rh) δ i,j (a) ) The functions L 1 (X) C L (a ) defined by h (Fh) δ i,j are continuous and equal on Cc (X). For all λ a : Fh(λ, ) δ i,j = (Fh) δ i,j (λ) = F ( A (Rh) δ i,j (a) ) (λ) = ( F A (Rh)(λ, ) ) δ i,j Fh(λ, ) = F A (Rh)(λ, ) Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 14 / 16

15 Proof of the main theorem: Theorem Let u(t, x) C(R : L 2 (X)) denote the solution of (S) with initial condition f L 2 (X). If there is a time t 0 > 0 so that σ(x)σ(y) 2t f (x) u(t 0, y) Ξ(x)Ξ(y)e 0 dx dy <, (BS) X X then u(t, ) = 0 for all t R. If (BS) holds, then f L 1 (X) C for some C > 0. For δ K M and i, j = 1,..., d(δ), condition (BS) yields (Rf ) δ i,j(a 1 ) (Ru t0 ) δ i,j(a 2 ) e A A log a 1 log a 2 2t 0 da 1 da 2 < Using F = F A R, we obtain from (Fu t)(λ, b) = e i( λ 2 + ρ 2 )t Ff (λ, b): F A ( (Rut) δ i,j) (λ) = e i( λ 2 + ρ 2 )t F A ( (Rf ) δ i,j ) (λ). Euclidean case: (Rf ) δ i,j = 0 a.e. on A. Conclusion: Rf = 0. Hence Ff = F A (Rf ) = 0. Thus f = 0 as F injective on L 2 (X). Corollary Let u(t, x) C(R : L 2 (X)) be the solution of (S) with initial condition f L 2 (X). Suppose that f has compact support. If t 0 > 0 so that u(t 0, ) has compact support, then u(t, ) = 0 for all t R. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 15 / 16

16 Applications: other uncertaninty principles Let u(t, x) C(R : L 2 (X)) be the solution of (S) with initial condition f L 2 (X). Corollary (Gelfand-Shilov type) Let 1 < p < and 1/p + 1/q = 1. Suppose α > 0, β > 0 and a time t 0 > 0 so that f (x) Ξ(x)e αp p σp (x) dx < and u(t 0, x) Ξ(x)e βq q σq (x) dx <. X X If 2t 0 αβ 1, then f = 0 and hence u(t, ) = 0 for all t R. Corollary (Cowling-Price type) Let 1 p, q <. Suppose α > 0, β > 0 and a time t 0 > 0 so that ( f ) (x) e aσ2 p ( ) (x) dx < and u(t 0, x) e bσ2 q (x) dx <. X X If 16t0 2 ab > 1, then f = 0 and hence u(t, ) = 0 for all t R. Corollary (Hardy type; S.Chanillo, 2007, for G complex & f K -inv) Suppose A > 0, α > 0 so that f (x) A e ασ2 (x) for all x X. Suppose t 0 > 0 and B > 0, β > 0 so that u(t 0, x) B e βσ2 (x) for all x X. If 16αβt0 2 > 1, then u(t, ) = 0 for all t R. Jan 8 9, 2012 (Tufts University) Uncertainty principle, Schrödinger equation 16 / 16