Semiclassical estimates for non-selfadjoint operators with double - PowerPoint PPT Presentation

Semiclassical estimates for non-selfadjoint operators with double characteristics Michael Hitrik Department of Mathematics, University of California, Los Angeles Joint work with Karel Pravda-Starov M. Hitrik (UCLA) 1 / 24

Introduction The study of operators with double characteristics has a long tradition in the analysis of linear PDE. Boutet de Monvel, Grigis, Helffer, H¨ ormander, Ivrii, Petkov, Sj¨ ostrand... (classical works on hypoellipticity from the 1970’s). Recent works on Kramers–Fokker–Planck type operators have brought about a renewed interest in this subject. H´ erau – Nier, Helffer – Nier, H´ erau – Sj¨ ostrand – Stolk (2004 – 2006). In a recent work with K. Pravda–Starov we have investigated spectral and semigroup properties for a class of non-selfadjoint quadratic operators that are also non-elliptic. M. Hitrik (UCLA) 2 / 24

Specifically, let q : T ∗ R n → C be a quadratic form such that Re q ( x , ξ ) ≥ 0, ( x , ξ ) ∈ T ∗ R n . Associated to q is the Hamilton map F : T ∗ C n → T ∗ C n defined by X , Y ∈ T ∗ C n , q ( X , Y ) = σ ( X , FY ) , where σ is the canonical symplectic form on T ∗ R n . M. Hitrik (UCLA) 3 / 24

It turns out that in order to understand the quadratic operator Q = q w ( x , D x ) , defined as the Weyl quantization of q , it is both helpful and natural to introduce the singular space S defined as follows :   � Re F ( Im F ) j � ∞ �   ∩ R 2 n . S = Ker j =0 Notice that Re F ( S ) = { 0 } and ( Im F ) S ⊂ S . M. Hitrik (UCLA) 4 / 24

Example. The one-dimensional quadratic Kramers–Fokker–Planck operator is given by K = q w ( x , y , D x , D y ) − 1 , where q ( x , y , ξ, η ) = η 2 + y 2 + i ( y ξ − ax η ) , a ∈ R \{ 0 } . (1.1) In this case, S = Ker ( Re F ) ∩ Ker ( Re F Im F ) ∩ R 4 = { 0 } . M. Hitrik (UCLA) 5 / 24

Theorem (Pravda-Starov – H., 2008). Assume that the quadratic form q is such that Re q ≥ 0 and that the restriction of q to S is elliptic , X ∈ S , q ( X ) = 0 ⇒ X = 0 . Then the singular space S is symplectic and the spectrum of q w ( x , D x ) on L 2 is discrete. The eigenvalues are of the form � � � r λ + 2 k λ ( − i λ ) , k λ ∈ N , λ ∈ Spec ( F ) , − i λ ∈ C + ∪ (Σ( q | S ) \{ 0 } ) where r λ is the dimension of the space of generalized eigenvectors of F in T ∗ C n associated to the eigenvalue λ ∈ C , Σ( q | S ) = q ( S ) and C + = { z ∈ C ; Re z > 0 } . M. Hitrik (UCLA) 6 / 24

Remarks . The structure of the spectrum of q w ( x , D x ) in the globally elliptic case is known since the work of J. Sj¨ ostrand (1974). In the quadratic Kramers–Fokker–Planck case, this result is known (Helffer – Nier, H´ erau – Sj¨ ostrand – Stolk, Risken). This talk : work in progress on non-selfadjoint semiclassical operators with double characteristics, when the quadratic approximations at doubly characteristic points are merely partially elliptic along the singular space. M. Hitrik (UCLA) 7 / 24

In addition to the classical PDE works, our main source of inspiration is the work by H´ erau – Sj¨ ostrand – Stolk (2004) and also the recent work by H´ erau – Sj¨ ostrand – H. (2007) on second order differential operators of Kramers–Fokker–Planck type. M. Hitrik (UCLA) 8 / 24

Statement of the results Let m ≥ 1 be an order function on R 2 n , satisfying for some C 0 > 0, N 0 > 0, m ( X ) ≤ C 0 � X − Y � N 0 m ( Y ) , X , Y ∈ R 2 n . Associated to m is the symbol space S ( m ) defined by a ∈ S ( m ) ⇔ ∂ α a ( X ) = O α (1) m ( x ) , α ∈ N 2 n . Let P ( x , ξ ; h ) ∼ p ( x , ξ ) + hp 1 ( x , ξ ) + . . . in S ( m ) be such that X = ( x , ξ ) ∈ R 2 n . Re p ( X ) ≥ 0 , M. Hitrik (UCLA) 9 / 24

Assume that for some C > 0, Re p ( X ) ≥ m ( X ) , | X | ≥ C . C For h > 0 small enough, we introduce the h – Weyl quantization of P ( x , ξ ; h ), P = P w ( x , hD x ; h ) . The spectrum of P in a fixed neighborhood of 0 ∈ C is discrete. Assume that the set { X ∈ R 2 n ; Re p ( X ) = 0 } is finite = { X 1 , . . . X N } . M. Hitrik (UCLA) 10 / 24

Then necessarily � ( X − X j ) 2 � Re p ( X ) = O , X → X j , 1 ≤ j ≤ N , and assume that the same holds for Im p , � ( X − X j ) 2 � Im p ( X ) = O , X → X j , 1 ≤ j ≤ N . Write p ( X j + Y ) = q j ( Y ) + O ( Y 3 ) , Y → 0 , where q j is quadratic with Re q j ≥ 0. Let S j stand for the singular space of q j , 1 ≤ j ≤ N . M. Hitrik (UCLA) 11 / 24

Theorem (Pravda-Starov – H., 2008) Assume that q j is elliptic along S j , for each 1 ≤ j ≤ N, X ∈ S j , q j ( X ) = 0 ⇒ X = 0 . Then for each B > 1 and for every fixed neighborhood Ω ⊂ C of N � � � p 1 ( X j ) + Spec ( q w j ( x , D x )) j =1 there exists h 0 > 0 and C > 0 such that for | z | ≤ B, z / ∈ Ω , and h ∈ (0 , h 0 ] , we have || ( P − hz ) u || L 2 ≥ h u ∈ S ( R n ) . C || u || L 2 , M. Hitrik (UCLA) 12 / 24

Remarks . We get the same estimate as in the quadratic case, when P = q w ( x , hD x ), with q quadratic, elliptic along S . In the case when q j are globally elliptic, 1 ≤ j ≤ N , this result is essentially well-known (J. Sj¨ ostrand). For Kramers–Fokker–Planck type operators, this result was established by H´ erau – Sj¨ ostrand – Stolk. For m = 1, say, the result implies that for z ∈ C with | z | ≤ B as in the theorem, � 1 � ( P − hz ) − 1 = O : L 2 → L 2 . h Following the methods of H´ erau – Sj¨ ostrand – Stolk, one can go further and compute Spec ( P ) for | z | < Bh , modulo O ( h ∞ ). M. Hitrik (UCLA) 13 / 24

Example. Let q = q ( x ′ , ξ ′ ) be the quadratic form defined in (1.1) and let q ( x ′′ , ξ ′′ ) be a real-valued elliptic quadratic form in another group of � q = � symplectic variables ( x ′′ , ξ ′′ ). Then the quadratic form Q ( x ′ , x ′′ , ξ ′ , ξ ′′ ) = q ( x ′ , ξ ′ ) + i � q ( x ′′ , ξ ′′ ) is elliptic along the associated singular space S = { ( x ′ , x ′′ , ξ ′ , ξ ′′ ); x ′ = ξ ′ = 0 } . M. Hitrik (UCLA) 14 / 24

Ideas of the proof Take m = 1 and assume for simplicity that N = 1 and that X 1 = (0 , 0) ∈ T ∗ R n . Write p ( X ) = q ( X ) + O ( X 3 ) , X → 0 , where q is elliptic when restricted to S . It follows then that S is symplectic. We have the F – invariant decomposition T ∗ R n = S σ ⊕ S , with linear symplectic coordinates ( x ′ , ξ ′ ) ∈ S σ , ( x ′′ , ξ ′′ ) ∈ S , so that q ( x , ξ ) = q 1 ( x ′ , ξ ′ ) + iq 2 ( x ′′ , ξ ′′ ) , X = ( x , ξ ) = ( x ′ , x ′′ , ξ ′ , ξ ′′ ) . M. Hitrik (UCLA) 15 / 24

Here q 2 is elliptic and real-valued. The quadratic form q 1 enjoys the following dynamical property : for each T > 0, � T 1 Re q 1 ◦ exp( tH Im q 1 ) dt > 0 . T 0 It follows that the flow average � T � Re p � T , Im p = 1 Re p ◦ exp( tH Im p ) dt T 0 satisfies q ( x ′ , ξ ′ ) + O ( X 3 ) , � Re p � T , Im p ( X ) = � where the quadratic form q ( x ′ , ξ ′ ) > 0 . � M. Hitrik (UCLA) 16 / 24

We shall introduce a weight corresponding to the procedure of averaging along the H Im p – flow in a small neighborhood of 0. Let g ∈ C ∞ ([0 , ∞ ); [0 , 1]) be decreasing and such that g ( t ) = t − 1 , g ( t ) = 1 , t ∈ [0 , 1] , t ≥ 2 . Let � � | X | 2 ( Re p ) ε ( X ) = g Re p ( X ) , ε > 0 . ε Then ( Re p ) ε ( X ) = O ( ε ) . M. Hitrik (UCLA) 17 / 24

Set, for T > 0 � � � − t G ε = − J ( Re p ) ε ◦ exp( tH Im p ) dt . T Here J is the compactly supported piecewise affine function solving J ′ ( t ) = δ ( t ) − 1 [ − 1 , 0] ( t ) . Then G ε = O ( ε ) satisfies H Im p G ε = � ( Re p ) ε � T , Im p − ( Re p ) ε M. Hitrik (UCLA) 18 / 24

Associated to G ε we have the IR–manifold Λ δ,ε = { X + i δ H G ε ( X ); X ∈ T ∗ R n } , 0 < δ ≪ 1 . The distorted symbol p | Λ δ,ε = p ( X + i δ H G ε ( X )) satisfies � � = Re p + δ H Im p G ε + O ( δ 2 |∇ G ε | 2 ) , Re p | Λ δ,ε and � � p | Λ δ,ε = Im p + O ( δ |∇ G ε | ) . Im M. Hitrik (UCLA) 19 / 24

Using the ellipticity of � � S σ Re p | Λ δ,ε along and the ellipticity of � � Im p | Λ δ,ε along S , we obtain that for X ∈ T ∗ R n in a small but fixed neighborhood of 0, � � | p ( X + i δ H G ε ( X )) | ≥ δ | X | 2 , ε C min , C > 1 , M. Hitrik (UCLA) 20 / 24

for δ ∈ (0 , δ 0 ] and ε ∈ (0 , ε 0 ], with δ 0 > 0 and ε 0 > 0 sufficiently small. More precisely, in the entire region | X | ≥ √ ε , we have for some C > 1, �� � � δε ≥ δε Re 1 − p ( X + i δ H G ε ( X )) C . C | X | 2 M. Hitrik (UCLA) 21 / 24

Associated to Λ δ,ε is the Hilbert space H (Λ δ,ε ) of functions that are microlocally � � G ε �� O exp h in the L 2 –sense. We shall take ε = Ah , where A is a constant. Then we have || · || H (Λ δ,ε ) ∼ || · || L 2 , uniformly as h → 0, for each fixed A > 1. M. Hitrik (UCLA) 22 / 24