Estimates for non-elliptic operators Fabian Portmann, KTH Stockholm - PowerPoint PPT Presentation

Estimates for non-elliptic operators Estimates for non-elliptic operators Fabian Portmann, KTH Stockholm July 14, 2010

Estimates for non-elliptic operators Outline Introduction Recent Improvements Results for Sub-elliptic operators

Estimates for non-elliptic operators Introduction This is joint work with A. Laptev.

Estimates for non-elliptic operators Introduction LT Inequalities vs. Sobolev Inequalities Well known Lieb-Thirring inequalities for a Schr¨ odinger operator − ∆ − V , V ∈ L γ + d / 2 ( R d ), state that for the γ - moments of its negative eigenvalues {− λ k } the estimate � λ γ R d V γ + d / 2 � k ≤ L γ, d ( x ) dx (2.1) + k holds, where V + = ( | V | + V ) / 2 is the positive part of V . The constants L γ, d in this inequality are finite if γ ≥ 1 / 2 ( d = 1), γ > 0 ( d = 2) and γ ≥ 0 ( d ≥ 3).

Estimates for non-elliptic operators Introduction LT Inequalities vs. Sobolev Inequalities If γ = 1, E.H. Lieb and W. Thirring proved that (2.1) is equivalent to a so-called generalised Sobolev inequality for an orthonormal system of functions { ϕ k } N k =1 in L 2 ( R d ), N � � R d [ ρ N ( x )] (2+ d ) / d dx ≤ C d � R d |∇ ϕ k ( x ) | dx , (2.2) k =1 where ρ N ( x ) = � N k =1 | ϕ k ( x ) | 2 . With the help of the Fourier transform (2.2) can be rewritten as N � � d +2 ϕ k ( ξ ) | 2 d ξ, � R d | ξ | 2 | ˆ d dx ≤ C d (2 π ) d x , ξ ∈ R d . R d ( ρ N ( x )) k =1

Estimates for non-elliptic operators Recent Improvements Barsegyan’s Results Recently, D.S. Barsegyan has obtained L-T type inequalities in R 2 , where the Laplace operator (whose symbol equals | ξ | 2 ) has been substituted by the product | D x D y | , D x = − i ∂ x . In this case the latter inequality takes the form N � � R 2 ( ρ N ( x , y )) 2 dxdy ≤ C (log N + 1) ϕ k ( ξ, η ) | 2 d ξ d η, � R 2 | ξη || ˆ k =1 (3.1) where the constant C is independent of N .

Estimates for non-elliptic operators Recent Improvements Reformulation in Terms of an Operator This inequality could be rewritten as an inequality for the negative eigenvalues {− λ k } of the operator | D x D y | − V (3.2) acting in L 2 ( R 2 ). Let − λ 1 ≤ − λ 2 ≤ · · · ≤ − λ N ≤ . . . be the sequence of negative eigenvalues, then (3.1) implies that for any N , N � � R 2 V 2 λ k ≤ C (log N + 1) + ( x , y ) dxdy . (3.3) k =1

Estimates for non-elliptic operators Recent Improvements Proof. Indeed, if { ϕ k } is an orthonormal system of eigenfunctions of the operator (3.2), then by (3.1) and the Cauchy-Schwartz inequality we have N N N � � ϕ k ( ξ, η ) | 2 d ξ d η − | ϕ k ( x , y ) | 2 dxdy � � � − λ k = R 2 | ξη | | ˆ R 2 V k =1 k =1 k =1 � R 2 [ ρ N ( x , y )] 2 dxdy ≥ C (log N + 1) − 1 � 1 / 2 �� � 1 / 2 �� R 2 V 2 dxdy R 2 [ ρ N ( x , y )] 2 dxdy − . (3.3) follows when minimizing the right hand side with respect to � 1 / 2 �� R 2 [ ρ N ( x , y )] 2 dxdy X = .

Estimates for non-elliptic operators Recent Improvements Incompleteness Although the inequality (3.1) is sharp, it does not give a satisfactory inequality for the sum of all negative eigenvalues, because the right hand side of (3.3) depends on log N + 1. When d = 2, estimates for the number of negative eigenvalues even for Schr¨ odinger operators is a delicate problem. Necessary and sufficient conditions for the finiteness of the negative spectrum are so far not known.

Estimates for non-elliptic operators Results for Sub-elliptic operators Main Result We consider a related problem and obtain spectral inequalities for the operator D 2 x D 2 y u − Vu = − λ u , u ( x , 0) = u (0 , y ) = 0 . (4.1) in L 2 ( R 2 ++ ), where R 2 ++ = R + × R + . Theorem Let γ ≥ 1 / 2 . Then for the negative eigenvalues {− λ k } of the operator (4.1) we have k ≤ ( R γ, 1 ) 2 � V 1 / 2+ γ � λ γ � 4 γ (2 π ) 2 B (1 / 2 , γ + 1) log(1 + 4 xy V + ) dxdy . + R 2 k ++ (4.2)

Estimates for non-elliptic operators Results for Sub-elliptic operators Important Remarks For both (3.2) and (4.1) the phase volume type estimates do not exist, because the classical phase volume is infinite. Differential parts of these operators are highly non-elliptic. Some examples of operators with infinite classical phase volume were previously considered by B. Simon, M.Z. Solomyak and M.Z. Solomyak & I.L. Vulis.

Estimates for non-elliptic operators Results for Sub-elliptic operators Sharpness of the Result The sharpness of the inequality (4.2) in terms of large potentials could be confirmed by the following argument. Simon showed that for the number N ( λ ) of the eigenvalues { λ k } below λ of the y + x 2 y 2 there is the following asymptotic formula operator D 2 x + D 2 N ( λ ) = π − 1 λ 3 / 2 log λ + o ( λ 3 / 2 log λ ) , λ → ∞ . This formula immediately implies that 1 ( γ + 3 / 2) π λ γ +3 / 2 log λ + o ( λ γ +3 / 2 log λ ) , � ( λ − λ k ) γ + = λ → ∞ . k

Estimates for non-elliptic operators Results for Sub-elliptic operators Sharpness of the Result Using the duality of the Fourier transform it is equivalent to study y + x 2 + y 2 . the spectrum below λ of the operator D 2 x D 2 We now reduce this problem to studying of the negative spectrum y − ( λ − x 2 − y 2 ) − . By Theorem 4.1 we find of the operator D 2 x D 2 that for γ ≥ 1 / 2 + ≤ 1 4 γ (2 π ) − 2 ( R γ, 1 ) 2 B (1 / 2 , γ + 1) × � ( λ − λ k ) γ k � � ( λ − x 2 − y 2 ) 1 / 2+ γ ( λ − x 2 − y 2 ) + ) dxdy × log(1 + 4 xy + R 2 ++ ≤ C λ γ +3 / 2 (1 + log( λ + 1)) , where C is independent of λ .