SYMPLECTIC DIMENSIONAL EXTENSIONS OF A PSEUDO-DIFFERENTIAL OPERATOR M.A. de Gosson (joint work with N.C. Dias and J.N. Prata) University of Vienna, Faculty of Mathematics, NuHAG September 2012 (NuHAG) Dimensional extensions September 2012 1 / 19
(NuHAG) Dimensional extensions September 2012 2 / 19
References N. C. Dias, M.A. de Gosson, and J.N. Prata. Prata. Dimensional Extension of Pseudo-Di¤erential Operators: Properties and Spectral Results. Preprint 2012 (submitted) N.C. Dias, M. de Gosson, F. Luef, and J.N. Prata. A pseudo–di¤erential calculus on non–standard symplectic space; spectral and regularity results in modulation spaces. J. Math. Pure Appl. 96 (2011) 423–445 M. de Gosson. Spectral Properties of a Class of Generalized Landau Operators, Comm. Partial Di¤erential Equations, 33(11) (2008), 2096–2104 M. de Gosson. Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Birkhäuser; Springer Basel (2011) (NuHAG) Dimensional extensions September 2012 3 / 19
Introductory example We begin by making a simple observation: in phase space QM one often ! ( b x , b replaces the usual "canonical quantization rules" ( x , p ) � p ) with b x ψ = x ψ , b p ψ = � i � h ∂ x ψ by the "Bopp quantization rules" ( x , p ) � ! ( e x , e p ) where x = x + 1 p = p � 1 e 2 i � h ∂ p , e 2 i � h ∂ p . (1) This is an example of what we call a "symplectic dimensional extension". ! e p ) . The operator e A = a ( e x , e Suppose indeed that a function a ( x , p ) � A can be viewed as the Weyl quantization a ( x , y ; p , q ) = a � s ( x , y ; p , q ) e (2) where s is the symplectic matrix � I � � 1 � � 0 � � 1 2 D 0 1 s = , I = , D = . 1 D 2 I 0 1 1 0 (NuHAG) Dimensional extensions September 2012 4 / 19
Symplectic and metaplectic The symplectic group � 0 � I A 2 n � 2 n matrix s is symplectic if s T Js = J where J = is the � I 0 "standard symplectic matrix". More abstract de…nition: � s linear automorphism of R 2 n s 2 Sp ( 2 n ) ( ) σ 2 n ( sz , sz 0 ) = σ 2 n ( z , z 0 ) Symplectic automorphisms/matrices form a group: Sp ( 2 n ) . Note that s T Js = J = ) det s = � 1. In fact: s 2 Sp ( 2 n ) = ) det s = 1. The importance of symplectic automorphisms in mathematical physics comes from the fact that Sp ( 2 n ) is the main group of symmetries in Hamiltonian mechanics. In addition, Hamiltonian ‡ows consist of symplectomorphisms, i.e. of di¤eomorphisms whose Jacobian matrix is, at every point, a symplectic matrix. The symplectic group Sp ( 2 n ) is a connected Lie group with dimension n ( 2 n + 1 ) . (NuHAG) Dimensional extensions September 2012 5 / 19
Symplectic and metaplectic The metaplectic group We have π 1 ( Sp ( 2 n )) = Z , hence Sp ( 2 n ) has covering groups of all orders 2 , 3 , ..., ∞ . One of these plays an essential role, both in theoretical questions and in applications: the twofold covering group Sp 2 ( 2 n ) has a faithful representation by a group of unitary operators acting on L 2 ( R n ) : the metaplectic group Mp ( 2 n ) . It is the group generated by the generalized Fourier transforms Z � 1 � n / 2 ∆ m ( W ) R n e iW ( x , x 0 ) ψ ( x 0 ) dx 0 S W ,. m ψ ( x ) = (3) 2 π i 2 Px � x � Lx � x 0 + 1 2 Qx 0 � x 0 W ( x , x 0 ) 1 = (4) with P = P T , Q = Q T and det L 6 = 0, and ∆ m ( W ) = i m p j det L j where m is an integer ("Maslov index") corresponding to a choice of arg det L . The projection of S W ,. m on Sp ( 2 n ) is de…ned by � p = ∂ x W ( x , x 0 ) ( x , p ) = s ( x 0 , p 0 ) ( ) p 0 = � ∂ x 0 W ( x , x 0 ) . (5) (NuHAG) Dimensional extensions September 2012 6 / 19
Symplectic and metaplectic Covariance Let b A be the pseudo-di¤erential operator with Weyl symbol a : Weyl b ! a . A AS � 1 has Weyl Let S 2 Mp ( 2 n ) have projection s 2 Sp ( 2 n ) . Then S b symbol a � s � 1 : AS � 1 Weyl S b ! a � s � 1 . (NuHAG) Dimensional extensions September 2012 7 / 19
Symplectic and metaplectic Covariance Let b A be the pseudo-di¤erential operator with Weyl symbol a : Weyl b ! a . A AS � 1 has Weyl Let S 2 Mp ( 2 n ) have projection s 2 Sp ( 2 n ) . Then S b symbol a � s � 1 : AS � 1 Weyl S b ! a � s � 1 . Let � n Z � 1 R n e � ipy ψ ( x + 1 2 y ) φ ( x � 1 W ( ψ , φ )( z ) = 2 y ) dy (6) 2 π be the cross-Wigner transform; then W ( S ψ , S φ )( z ) = W ( ψ , φ )( s � 1 z ) . This follows from the fundamental relation between Wigner formalism and Weyl calculus: ( b A ψ j φ ) L 2 = h a , W ( ψ , φ ) i . (7) (NuHAG) Dimensional extensions September 2012 7 / 19
Symplectic dimensional extensions Let us then make the identi…cations R 2 n = R n x � R n p and R 2 ( n + k ) = R ( n + k ) � R ( n + k ) and let the two spaces R 2 n and R 2 ( n + k ) be x , y p , q equipped with the standard symplectic forms. Consider E s : S 0 ( R 2 n ) � ! S 0 ( R 2 ( n + k ) ) (8) ! e a 7� a s = E s [ a ] = ( a � 1 2 k ) � s where 1 2 k : R 2 k � ! R ; 1 2 k ( y , η ) = 1 and s 2 Sp ( 2 ( n + k ) , R ) . We call E s a symplectic dimensional extension map . In the simplest case s = I we write E = E I and the function (or distribution) e a = E [ a ] is given explicitly by: e a ( x , y ; p , q ) = a ( x , p ) . (9) We note that e a can also be de…ned in terms of the orthogonal projection operator Π : R 2 ( n + k ) � ! R 2 n : under the identi…cation Π ( x , y ; p , q ) � ( x , p ) we have e a s = a � Π � s . (NuHAG) Dimensional extensions September 2012 8 / 19
De…nition Let us write A = Op w ( a ) and e A s = Op w ( e a s ) (Weyl operators with symbols a and e a s , respectively). The commutative diagram: W a � � � � � � � � �� > A j j E s j j E s (10) # # f W e a s e � � � � � � � � �� > A s where W and f W are the Weyl correspondences, de…nes, for each E s , a linear embedding map E s E s : L ( S ( R n ) , S 0 ( R n )) � ! L ( S ( R n + k ) , S 0 ( R n + k )) (11) ! e A 7� A s = E s [ A ] which we call the symplectic dimensional extension map for Weyl operators. (NuHAG) Dimensional extensions September 2012 9 / 19
Let denote by e S the elements of Mp ( 2 ( n + k ) , R ) ; they are unitary on L 2 ( R n + k ) . Theorem Let s 2 Sp ( 2 ( n + k ) , R ) and a 2 S 0 ( R 2 n ) . Let e A = E [ A ] and e A s = E s [ A ] be the corresponding dimensional extensions of b A. We have A s = e e S � 1 e A e S (12) where e S 2 Mp ( 2 ( n + k ) , R ) is any of the two metaplectic operators with projection s. Proof. Formula (12) is equivalent to S � 1 \ ( a � 1 2 k ) � s = e \ a � 1 2 k e S (13) which follows from the symplectic covariance of Weyl operators. (NuHAG) Dimensional extensions September 2012 10 / 19
Intertwiners Let us shortly return to the example in the Introduction, the Bopp operators A = a ( x + 1 e h ∂ p , p � 1 h ∂ p ) . 2 i � 2 i � We have shown that e A is related to the usual Weyl operator b A = a ( b x , b p ) by the intertwining relations e AW χ = W χ b A , χ 2 S ( R n ) , jj χ jj L 2 = 1 where W φ is a partial isometry L 2 ( R n ) � ! L 2 ( R 2 n ) de…ned by W χ ψ = ( 2 π ) n / 2 W ( ψ , χ ) (14) where � n Z � 1 R n e � ipy f ( x + 1 2 y ) χ ( x � 1 W ( f , χ )( z ) = 2 y ) dy 2 π is the cross-Wigner transform. This allows, among other things, to study the spectral properties of the Bopp operator e A knowing those of b A . We are going to see that similar intertwiners exist for general symplectic dimensional extensions e A s . (NuHAG) Dimensional extensions September 2012 11 / 19
Intertwiners: general case For χ 2 S ( R k ) , jj χ jj L 2 = 1, and e S 2 Mp ( 2 ( n + k ) , R ) de…ne S , χ : S ( R n ) � ! S ( R n + k ) by T e S , χ ψ = e S � 1 ( ψ � χ ) . T e (15) S , χ are partial isometries L 2 ( R n ) � ! L 2 ( R n + k ) and: These operators T e For a 2 S 0 ( R 2 n ) we have e S , χ b A , T � S , χ e A s = b AT � A s T e S , χ = T e (16) e e S , χ The operator T e S , χ extends into a continuous operator S 0 ( R n ) � ! S 0 ( R n + k ) and we have Z R k χ ( y ) e T � S , χ Φ ( x ) = S Φ ( x , y ) dy . (17) e The map T � S , χ extends into the continuous operator e S 0 ( R n + k ) � ! S 0 ( R n ) de…ned for ψ 2 S ( R n ) by S , χ Φ , ψ i = h e h T � S Φ , χ � ψ i . (18) e (NuHAG) Dimensional extensions September 2012 12 / 19
Proof. Let us …rst prove that e S , χ ψ 2 S ( R n + k ) A s T e S , χ = T e S , χ A . It is clear that T e if ψ 2 S ( R n ) and χ 2 S ( R k ) . Let us …rst consider the case e S = e I , and I , χ . We have to show that e AT χ = T χ A . Let ψ 2 S ( R n ) ; we set T χ = T e have A ( T χ ψ )( x , y ) = b e A ( T χ ψ ) y ( x ) = b A ψ ( x ) χ ( y ) = T χ ( b A ψ )( x , y ) . The general cases follows using the symplectic covariance of Weyl calculus e S , χ = e S � 1 e A e S ( e S � 1 T χ ) = e S � 1 e A s T e AT χ = e S � 1 T χ b S , χ b A = T e A . The second formula (16) is deduced from the …rst: since e S , χ b A � A � s T e S , χ = T e S , χ ) � = ( T e A � ) � that is T � we have ( e S , χ b S , χ e A s = b A � AT � s T e S , χ . e e (NuHAG) Dimensional extensions September 2012 13 / 19
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