Evolution of semiclassical Wigner function (the higher dimensio

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1 Evolution of semiclassical Wigner function (the higher dimensional case) Workshop on Fast Computations in Phase Space, WPI-Vienna, November 2008 Dept. Appl. Math., Univ. Crete & IACM-FORTH

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3 -1 The limit Wigner measure of a WKB function satisfies a simple transport equation in phase space and is well suited for capturing oscillations at scale of order O(ɛ), but it fails, for instance, to provide the correct amplitude on caustics where different scales appear. We define the semi-classical Wigner function of an N dimensional WKB function, as a suitable formal approximation of its scaled Wigner function.

4 -2 The semi-classical Wigner function is an oscillatory integral that provides an ɛ dependent regularization of the limit Wigner measure, it obeys a transport dispersive evolution law in phase space, and it is well defined at least on simple caustics.

5 : WKB method-1 Schrödinger equation and oscillatory initial data iɛψ ɛ t(x, t) = ɛ2 2 ψɛ (x, t) + V (x)ψ ɛ (x, t), x R N, t > 0 with WKB initial data ψ ɛ (x, 0) = ψ0(x) ɛ = A 0 (x) exp ( is 0 (x)/ɛ ) WKB method looks for solutions ψ ɛ (x, t) = A(x, t) exp ( is(x, t)/ɛ )

6 : WKB method-2 where S and A solve the eikonal S t xs 2 + V (x) = 0, S(x, 0) = S 0 (x), and the transport equation (A 2 ) t + x (A 2 x S) = 0, A(x, 0) = A 0 (x)

7 : Rays For integrating the eikonal and the transport equations we consider the rays x(q, t) = k, k(q, t) = S x (x(q, t), t) solving the Hamiltonian system d dt x = k, with initial data d k = V (x), dt x(0) = q, k(0) = S 0 (q) Along the rays the eikonal is written as an ODE ds dt = S t + x S 2 = k 2 2 V, S(x, 0) = S 0(q)

8 : Caustics Remark: Since the eikonal is a nonlinear equation, it has, in general, a smooth solution only up to some finite time t c, when the rays cross each other and singularities are developed that are called caustics. At a caustic point the Jacobian J(q, t) of the ray map q x(q, t) is zero, and the amplitude becomes infinite since A(x, t) = A 0 (q)j 1/2 (q, t)

9 The Wigner transform Consider the Wigner transform W ɛ (x, k) := W ɛ [ψ ɛ (x)] (Wigner-1932, Gerard et. al-1997) W ɛ (x, k) = 1 (2π) N e ik y ψ ɛ (x+ ɛy R N 2 )ψɛ (x ɛy 2 )dy, (x, k) RN R N which has the basic properties R N W ɛ (x, k)dk = ψ ɛ (x) 2 and for ψ ɛ (x) = A(x) exp ( is(x)/ɛ ) (Lions & Paul-1993) W ɛ (x, k) A 2 (x)δ(k S(x)), weakly as ɛ 0

10 The -1 The Wigner transform f ɛ (x, k, t) = W ɛ [ψ ɛ (x, t)] of the solution ψ ɛ of the Schrödinger equation satisfies the f ɛ t (x, k, t) + k x f ɛ (x, k, t) + Z ɛ f ɛ (x, k, t) = 0, where the operator Z ɛ is defined by the convolution with respect to the momentum k, i Z ɛ f (x, k, t) = f (x, k, t) (2π) N ɛ exp( ik y) (V (x+ ɛ 2 y) V (x ɛ ) 2 y) d

11 The -2 Assuming that the potential V is smooth enough, we can expand V (x ± ɛ 2y) into Taylor series, so that we can rewrite the Wigner equation as a transport dispersive equation ft ɛ + k x f ɛ V k f ɛ = c m ɛ 2m D α VDk α f ɛ, m 1 α =2m+1 which sheds some light to the combined (transport-dispersive) mechanism of the evolution of f ɛ near the Lagransian manifold Λ t = {k = S(x, t)}.

12 The limit In the limit ɛ 0, the reduces formally to the Liouville equation for f 0 (dispersion disappears!!) f 0 t (x, k, t) + k x f 0 (x, k, t) V (x) k f 0 (x, k, t) = 0 with initial data f 0 0 (x, k, t = 0) = A 2 0(x)δ(k S 0 (x)) For 0 < t < T, T < t c = time up to which the solution of the eikonal is single-valued, f 0 (x, k, t) = A 2 (x, t)δ(k x S(x, t))

13 Semiclassical Wigner function We rewrite the Wigner integral W ɛ (x, k) = 1 (ɛπ) N in the form e 2i R N ɛ k σ ψ ɛ (x + σ)ψ ɛ (x σ)dσ W ɛ (x, k) = 1 (ɛπ) N D(x, σ)e i F (x,k;σ) ɛ dσ R N D(x; σ) := A(x + σ)a(x σ) F (x, k; σ) := S(x + σ) S(x σ) 2k σ (Wigner phase)

14 Critical points of the Wigner phase-1 The stationary points of the Wigner phase are found from σ F = x S(x + σ) + x S(x σ) 2k 1. The stationary points of F always come in pairs ±σ 0 (x, k), and as k S(x), there always exist stationary points which coalesce to σ 0 (x, k) In particular, when k = S(x), then σ 0 = 0 is always a degenerate critical point.

15 Critical points of the Wigner phase-2 Remark 1: The structure of the critical set of F can be quite complicated (even in the case N = 1), but we expect that, modulo highly oscillatory terms that tend weakly to zero as ɛ 0, the main contribution to the asymptotics of W ɛ comes from points close to the Lagrangian manifold, that is, from k S(x). This is in agreement with the fact that as ɛ 0 then W ɛ W 0 = A 2 (x)δ(k S(x)), is concentrated on the Lagrangian manifold.

16 Critical points of the Wigner phase-3 Remark 2: Let us consider the projection identity ψ ɛ (x) 2 = W ɛ (x, k)dk = 1 R N (ɛπ) N R N R N D(x, σ)e i F (x,k;σ) ɛ dσ dk The main contribution in the calculation of the last integral (over R N σ R N k ) comes from the points (σ, k) at which σ,kf = 0. These are easily is seen to be σ = 0, k = S(x). At each x these critical points are always non-degenerate. Thus in the calculation of ψ ɛ (x), the main contribution also comes from the points near the Lagrangian manifold.

17 The semiclassical approximation-1 Once we restrict attention to points k S(x), it is natural to approximate D and F by their Taylor expansion about σ = 0 D(x; σ) = A 2 (x) + O( σ 2 ), F (x, k; σ) = 2( S(x) k) σ+ 1 3 N i,j,k=1 3 S(x) x i x j x k σ i σ j σ k +O( σ 5 )

18 The semiclassical approximation-2 Keeping only the linear term in the right hand side of F, we get the (limit) Wigner measure: A 2 (x) (ɛπ) N e i ɛ 2(k S(x)) σ dσ = A 2 (x)δ(k S(x)) =: W 0 (x, k), R N while keeping the cubic terms we are led to define the semi-classical Wigner function W ɛ, ( 2 W ɛ (x, k) = ɛ 2 3 ) N A 2 (x)p N ( 2 ɛ 2 3 (k S(x)), 3 ) S(x) x i x j x k

19 The semiclassical approximation-3 where P N is the oscillatory integral P N (z, C k ) := 1 (2π) N with Note that C k = {c ijk } = S xi,x j,x k e i[zt 1 σ+ R N 3 σt C k σσ k ] dσ i, j, k = 1, 2... N P 1 (z, c) = c 1/3 Ai(zc 1/3 ) and P N (z, 0) = δ(z) for C k = 0 Remark: The integral P N is defined, in general, as a distribution, but it is a smooth function if N (σ T C i σ) 2 0, σ R N \ {0} i=1

20 The semiclassical approximation-4 Degenerate points Failure of the last condition is equivalent to the fact that there exists a unit vector ν R N such that ν T C i ν = 0 for all i = 1,... N (degenerate points). Since C i is the second derivative matrix of S(x) x i, this is equivalent to the fact that, for all i s, the surfaces k i = S(x) x i have zero normal curvature in the direction of ν R N x.

21 The semiclassical approximation-5 It is important to observe that W ɛ obeys the basic properties and W ɛ (x, k) A 2 (x)δ(k S(x)), as ɛ 0, R N W ɛ (x, k)dk = A 2 (x). Moreover, near non degenerate points the semi-classical Wigner W ɛ is a smooth function that approximates the scaled Wigner W ɛ, as ɛ tends to zero. At degenerate points either the semi-classical Wigner W ɛ fails to approximate W ɛ, or else W ɛ itself is a distribution.

22 Evolution near the Lagrangian manifold-1 Consider the Hamiltonian system d dt ˆx(q, p, t) = ˆk(q, d p, t), dt ˆk(q, p, t) = x V (ˆx(q, p, t)) with initial data ˆx(q, p, t = 0) = q ˆk(q, p, t = 0)p The distance from the manifold a(t) = a(q, p, t) = ˆk(q, p, t) x S(ˆx(q, p, t), t) evolves from its initial value a(0) = p S 0 (q), according to the ODE d dt a(t) = B(t)a(t), B(t) = ( b ij (t) = S xi x j (ˆx(q, p, t), t) )

23 Evolution near the Lagrangian manifold-2 If Φ(t) = {φ ij (t)} i,j=1,...,n Φ(0) = I N, is the fundamental solution, then a(t) = Φ(t)a(0). Similarly, the matrix C k (t) = ( c ijk (t) = S xi x j x k (ˆx(q, p, t), t) ) with initial data C k (0) = ( c ijk (0) = S 0,qi q j q k (q) ) evolves according to the ODE d dt C k(t) = B(t)C k (t) C k (t)b(t) b kl (t)c l (t) V k (t)+o(t a(0) )

24 Evolution near the Lagrangian manifold-3 and it is given by C k (t) = Φ(t) ( C l (0) + U l (t) + O(t a(0) ) ) Φ T (t)φ kl (t) where V k (t) = ( V xi x j x k (ˆx(q, p, t), t) ), and φ kl d dt U l = Φ 1 V k Φ T, U k (0) = 0, k, l = 1, 2,..., N

25 Identification of the evolution law -1 Key observation: The semiclassical Wigner W ɛ (x, k, t) = ( ) 2ɛ 2 N ( ) 3 A 2 (x, t)p N 2ɛ 2 3 â(t), Ĉk (t) where Ĉk(t) := C k (ˆq, ˆp, t) = C k (ˆq(x, k, t), ˆp(x, k, t), t) obeys the evolution law W ɛ (x, k, t) = W ) 0 ɛ (ˆq, p) p P N ( 2ɛ 2 3 p, Ûk (t) + O(t â(0) ) p=ˆp ( ) (1 + O(t 2 â(0) ) 2ɛ 2 N 3

26 Identification of the evolution law -2 Evolution law : at the exception of error terms O(t 2 â(0), the semiclassical Wigner W ɛ (x, k, t) is equal to the convolution of W 0 ɛ (ˆq, p) with a suitable phase integral, the convolution being evaluated at the point p = ˆp(x, k, t)

27 Approximate parametrix & evolution formula We set G ɛ (p, Û k (t)) := ( 2ɛ 2 3 ) N PN ( 2ɛ 2 3 p, Ûk (t)) and we then define f ɛ (x, k, t) as f ɛ (x, k, t) := G ɛ ɛ (p, Û k (t)) p W 0 (ˆq, p) = G ɛ ɛ (p, Û k (t)) p f 0 (ˆq, p). p=ˆp p=ˆp

28 Remarks on the structure of f ɛ 1. f ɛ (x, k, t) is near to W ɛ (x, k, t), for â(0) small (i.e, when phase space dynamics remains locally near the Lagrangian manifold) 2. The evolved semiclassical Wigner is in fact a P N phase integral, given by f ɛ (x, k, t) = and it follows easily that ( ) 2ɛ 2 N ( ) 3 A 2 0 (ˆq)P N 2ɛ 2 3 â(0), Ĉk (0) + Û k (t), f ɛ (x, k, t) f 0 (x, k, t) = A 2 (x, t)δ(k x S(x, t)), as ɛ 0.

29 Remarks on the structure of f ɛ 3. f ɛ (x, k, t) is a smooth function iff the point (x, S(x, t)) on Λ t is a non degenerate point. In particular, for any 0 < t < t c, the approximation f ɛ provides a P N regularization of f 0 away from degenerate points. 4. Combined evolution mechanism: the semiclassical Wigner f ɛ is not only transported by the Hamiltonian flow but also dispersed in the k direction, through its convolution with the kernel G ɛ. Such a behavior is in complete agreement with the transport dispersive character of the

30 Remarks on the structure of f ɛ 5. In the ɛ = 0 limit and W ɛ 0 (ˆq, ˆp) f 0 0 (ˆq, ˆp) G ɛ (p, Ûk(t)) δ(p), ɛ 0 Consequently, as ɛ 0, we recover Liouvillian dynamics f ɛ (x, k, t) = G ɛ (p, Ûk(t)) ɛ p W 0 (ˆq, p) p=ˆp δ(p) p f0 0 (ˆq, p) = W0 0 (ˆq, ˆp) p=ˆp where W 0 0 (ˆq, ˆp) = A 0 (ˆq)δ(ˆp S 0 (ˆq))

31 What happens at a caustic? In order to arrive at the definition of f ɛ we required the evolved semiclassical Wigner to be in agreement with W ɛ. In particular all the underlying analysis was restricted in the time interval (0, T ), T < t c. Once however we define f ɛ, it follows that f ɛ is well defined even at a caustic point (x c, t c ), if this point is a non degenerate point. At such a point, the integration R f ɛ (x N c, k, t c )dk is now k meaningful

32 Example with caustic: elliptic umbilic-1 The initial WKB function A 0 (q) 1 S 0 (q 1, q 2 ) = 1 3 q3 1 q 1 q 2 2 a(q q 2 2), a > 0 Assuming zero potential,the bicharacteristics are given by q = ˆq(x, k, t) = x kt, and the caustic in (x 1, x 2, t) space is given by p = ˆp(x, k, t) = k x 1 q 1 x 2 q 2 S 0 (q 1, q 2 ) = 0, S 0 (q 1, q 2 ) = 0, t q 1 t q 2 ( 1 + t 2 S )( t 2 S ) 0 q 1 2 t 2( 2 S ) 0 2 = 0 q 2 q 1 q 2

33 elliptic umbilic-2 Geometry of the caustic: The t = const. sections are hypocycloids with cusps A, B, C x2 B A x1 C

34 elliptic umbilic-3 For u 1 := x 1 t, u 2 := x 2 t, v := 1 2t a, the equations of the caustic are u 1 = q1 2 q q 1 v u 2 = 2q 1 q 2 + 2q 2 v v 2 = q1 2 + q2. 2 For v > 0, we set q 1 = v cos θ, q 2 = v sin θ, and u 1 = v 2 (cos 2θ + 2 cos θ) u 2 = v 2 (sin 2θ 2 sin θ)

35 elliptic umbilic-4 For fixed v (that is, fixed t) this describes a hypocycloid with three cusps. A similar analysis shows that the picture remains the same for v < 0. All three points move to infinity as t tends to either 0 or infinity. Also, at t = 1/2a, the hypocycloid reduces to a point (the origin) which is called the focus of the caustic.

36 The initial Wigner function f ɛ 0 The initial scaled Wigner function is f0 ɛ (q, p) = 1 (ɛπ) 2 e i ɛ F (q,p;σ) dσ R 2 where with F (q, x q ; σ) = 2ẑ σ + 1 t 3 (2σ3 1 6σ 1 σ2), 2 ẑ = (ẑ 1, ẑ 2 ) ẑ 1 (u, v; q) := u 1 2vq 1 + q 2 2 q 2 1, ẑ 2 (u, v; q) := u 2 2vq 2 + 2q 1 q 2,

37 The integral P 2 Let P 2 (z) = 1 (2π) 2 and define the new phase integral P 2 (z, w) := 1 (2π) 2 so that e i[z σ+ 1 3 (2σ3 1 6σ 1σ2 2)] dσ R 2 e i[z σ+w σ (2σ3 1 6σ 1σ2 2)] dσ R 2 P 2 (z, 0) = P 2 (z). Then, the following projection identity holds R2 P 2 ( γẑ(u, v; q))dq = 21/3 4π 2 P 2 ( γu, 2 1/6 γ 1/2 v) 2 γ

38 The evolved Wigner function f ɛ -1 f ɛ (x, k, t) = 1 (ɛπ) 2 e i ɛ ( 2ẑ 1 1σ 1 2ẑ 2 σ (2σ3 1 6σ 1σ2 2) dσ R 2 = (2ɛ 2 3 ) 2 P 2 ( 2ɛ 2 3 ẑ(u, v; q)) and integrating wrt. k ψ ɛ (x, t) 2 = 1 (2πɛt) 2 R 2 e i ɛ ( 1 3 ξ3 1 ξ 1ξ2 2+v(ξ2 1 +ξ2 2 ) u 1ξ 1 u 2 ξ 2 2 ) dξ 1 dξ 2 where u = x t, v = 1 2t a.

39 The evolved Wigner function f ɛ -2 This formula is valid at any point including the points on the caustic. For instance, at the focus of the caustic (x 1, x 2, t) = (0, 0, 1 2a ), that is (u 1 = u 2 = v = 0), we get ψ ɛ (0, 0, 1 2a ) = O(ɛ 1/3 ).

40 The projection identity (Berry & Wright-1980)-1 Let with ψ E (C 1, C 2, C 3 ) = 1 2π e iφ E (S 1,S 2 ;C 1,C 2,C 3 ) ds 1 ds 2 φ E (S 1, S 2 ; C 1, C 2, C 3 ) := C 1 S 1 C 2 S 2 C 3 (S 2 1 +S 2 2 )+S 3 1 3S 1 S 2 2 Put C 1 := 2 2/3 (C 1 + 2C 3 U 1 + 3(U 2 2 U 2 1 )) C 2 := 2 2/3 (C 2 + 2C 3 U 2 + 6U 1 U 2 ) Then ψ E (C 1, C 2, C 3 ) 2 = 21/3 π R 2 ψ E ( C 1, C 2, 0)dU 1 du 2

41 The projection identity -2 The important fact here is that the zero section of ψ E is expressed in terms of the Airy functions Ai&Bi, and in this sense the wave function is expressed in terms of lower order singularities.

42 -1 A.We have introduced the semiclassical Wigner function f ɛ as a formal approximation of the scaled Wigner trasform of the WKB solution to the problem. This approximation is valid near the manifold k = x S(x, t). f ɛ is an object that is richer than the limit Wigner function f 0 which is a Dirac mass concentrated on k = x S(x, t). In particular, f ɛ is an ɛ dependent oscillatory integral that tends to f 0 as ɛ tends to zero. Moreover, it obeys an evolution law which is in agreement with the transport dispersive character of the.

43 -2 B. If (x c, t c ) is a caustic point, the restriction of f 0 at this point, that is f 0 (x c, k, t c ), is not a well defined distribution in R N k, and as a consequence the amplitude at (x c, t c ) cannot be computed via the projection identity. On the contrary, f ɛ (x c, k, t c ) is a well defined function in R N k, and therefore the integral of f ɛ with respect to k is meaningful and is expected to approximate the (ɛ dependent) exact amplitude, at least on simple caustics. However, an approximation result in this direction is still missing.

44 Open problem More generally, the asymptotic nearness of f ɛ and f ɛ is an open question. The difficulty in settling these questions stems from the fact that one deals with complicated multidimensional oscillatory integrals.