Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada

Transcription

1 Spectrum of the linearized NLS problem Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaboration: Marina Chugunova (McMaster, Canada) Scipio Cuccagna (Modena and Reggio Emilia, Italy) Vitali Vougalter (Notre Dame, USA) References: Comm. Pure Appl. Math. 58, 1 (2005) Proc. Roy. Soc. Lond. A 461, 783 (2005) Mathematical Physics Seminar, Caltech, April

2 Stability problem in Hamiltonian systems Hamiltonian PDE du dt = J h(u), u(t) X(Rn, R m ) where J + = J and h : X R Linearization at the stationary solution u(t) = u 0 + ve λt, where u 0 X(R n, R m ) and λ C Spectral problem JHv = λv, where H + = H and v X(R n, C m ) 2

3 Main questions Let stationary solutions u 0 decay exponentially as x Let operator J be invertible Let operator H have positive continuous spectrum Let operator H have finitely many isolated eigenvalues Let operator JH have continuous spectrum at the imaginary axis 3

4 Main questions Let stationary solutions u 0 decay exponentially as x Let operator J be invertible Let operator H have positive continuous spectrum Let operator H have finitely many isolated eigenvalues Let operator JH have continuous spectrum at the imaginary axis Is there a relation between isolated and embedded eigenvalues of JH and isolated eigenvalues of H? Is there a relation between unstable eigenvalues of JH with Re(λ) > 0 and negative eigenvalues of H? 3

5 Classes of Hamiltonian evolution equations Nonlinear Schrödinger equation (NLS) iψ t = ψ xx + U(x)ψ + F ( ψ 2 )ψ (J, H) satisfy the main assumptions Korteweg De Vries equation (KdV) u t + x (f(u) + u xx ) = 0 J is not invertible but H satisfy the main assumptions Massive Thirring model (MTM) i(u t + u x ) + v + ūw (u, ū, v, v) = 0, i(v t v x ) + u + v W (u, ū, v, v) = 0 J is invertible but H have positive and negative continuous spectrum 4

6 Review of other results Grillakis, Shatah, Strauss, 1990 If H has no negative eigenvalue, then JH has no unstable eigenvalues. If H has odd number of negative eigenvalues, then JH has at least one real unstable eigenvalue. Number of unstable eigenvalues of JH is bounded by the number of negative eigenvalues of H. 5

7 Review of other results Grillakis, Shatah, Strauss, 1990 If H has no negative eigenvalue, then JH has no unstable eigenvalues. If H has odd number of negative eigenvalues, then JH has at least one real unstable eigenvalue. Number of unstable eigenvalues of JH is bounded by the number of negative eigenvalues of H. Kapitula, Kevrekidis, Sandstede, 2004 Closure relation for negative index N unstable (JH) + N negative Krein (JH) = N negative (H) 5

8 Review of our results: formalism iψ t = ψ + U(x)ψ + F ( ψ 2 )ψ Assume that there exist exponentially decaying C solutions φ + U(x)φ + F (φ 2 )φ + ωφ = 0, where φ : R n R and ω > 0. Assume that U(x) decay exponentially and F (u) C, F (0) = 0 Apply the linearization transformation, ψ(x, t) = e iωt ( φ(x) + ϕ(x)e izt + θ(x)e i zt), where (ϕ, θ) : R n C 2 and z = iλ C. 6

9 Review of our results : formalism The eigenvalue problem becomes where and σ 3 = ( ), H = σ 3 Hψ = zψ, ( + ω + f(x) g(x) g(x) + ω + f(x) f(x) = U(x) + F (φ 2 ) + φ 2 F (φ 2 ), g(x) = φ 2 F (φ 2 ). ), 7

10 Review of our results : formalism The eigenvalue problem becomes where and σ 3 = ( ), H = σ 3 Hψ = zψ, ( + ω + f(x) g(x) g(x) + ω + f(x) f(x) = U(x) + F (φ 2 ) + φ 2 F (φ 2 ), g(x) = φ 2 F (φ 2 ). ), Equivalent form: where and L + u = zw, L w = zu, L ± = + ω + f(x) ± g(x) ψ = (u + w, u w) T. 7

11 Review of our main results : formalism Linearized energy h = 1 2 ψ, Hψ = (u, L +u) + (w, L w) Skew-orthogonal projections ψ, ψ = 1 2 σ 3ψ, ψ = (u, w) + (w, u) Constrained subspace: no kernel of JH X 0 = {ψ L 2 : σ 3 ψ 0, ψ = σ 3 ψ 1, ψ = 0}, where Hψ 0 = 0, σ 3 Hψ 1 = ψ 0 Constrained subspace: no point spectrum of JH X c = {ψ X 0 : { σ 3 ψ j, ψ = 0} zj σ p (JH) }, 8

12 Spectrum of (L +, L ) and JH 9

13 Review of our main results Assuming that dim(ker(h)) = 1, N neg (H) = N neg (H) p(ω), X0 L 2 where p(ω) = 1 for ω φ 2 L 2 > 0 and p(ω) = 0 otherwise. 10

14 Review of our main results Assuming that dim(ker(h)) = 1, N neg (H) = N neg (H) p(ω), X0 L 2 where p(ω) = 1 for ω φ 2 L 2 > 0 and p(ω) = 0 otherwise. Assuming that all eigenvalues of JH are semi-simple, N neg (H) = N neg (H) 2Nreal (JH) N imag(jh) 2N comp (JH) Xc X0 where Nneg(JH) correspond to negative Krein signatures 10

15 Review of our main results Assuming that dim(ker(h)) = 1, N neg (H) = N neg (H) p(ω), X0 L 2 where p(ω) = 1 for ω φ 2 L 2 > 0 and p(ω) = 0 otherwise. Assuming that all eigenvalues of JH are semi-simple, N neg (H) = N neg (H) 2Nreal (JH) N imag(jh) 2N comp (JH) Xc X0 where Nneg(JH) correspond to negative Krein signatures Assuming end-point conditions and simple embedded eigenvalues, ψ X c : ψ, Hψ > 0 10

16 Methods of proofs I. Bounds on the number of isolated eigenvalues: Calculus of constrained Hilbert spaces Sylvester s Inertia Law Rayleigh-Ritz Theorem Pontryagin Krein spaces with sign-indefinite metric 11

17 Methods of proofs I. Bounds on the number of isolated eigenvalues: Calculus of constrained Hilbert spaces Sylvester s Inertia Law Rayleigh-Ritz Theorem Pontryagin Krein spaces with sign-indefinite metric II. Positivity of the essential spectrum: Kato s wave operator formalism Wave function decomposition Smoothing decay estimate on the linearized time evolution 11

18 Quadratic forms and skew-symmetric orthogonality L + u = zw, L w = zu, Let z = z 0 be a simple real eigenvalue with the eigenvector (u R, w R ) (u R, L + u R ) = (w R, L w R ) 0 12

19 Quadratic forms and skew-symmetric orthogonality L + u = zw, L w = zu, Let z = z 0 be a simple real eigenvalue with the eigenvector (u R, w R ) (u R, L + u R ) = (w R, L w R ) 0 Let z = iz 0 be a simple imaginary eigenvalue with the eigenvector (u R, iw I ) (u R, L + u R ) = (w I, L w I ) 0 12

20 Quadratic forms and skew-symmetric orthogonality L + u = zw, L w = zu, Let z = z 0 be a simple real eigenvalue with the eigenvector (u R, w R ) (u R, L + u R ) = (w R, L w R ) 0 Let z = iz 0 be a simple imaginary eigenvalue with the eigenvector (u R, iw I ) (u R, L + u R ) = (w I, L w I ) 0 Let z = z 0 be a simple complex eigenvalue with the eigenvector (u R + iu I, w R + iw I ) where M = (u, L + u) = (w, L w) = (c, Mc) 0, ) = ( (ur, L + u R ) (u R, L + u I ) (u I, L + u R ) (u I, L + u I ) ( (wr, L w R ) (w R, L w I ) (w I, L w R ) (w I, L w I ) ) 12

21 Constrained Hilbert space Let z i and z j be two distinct eigenvalues with eigenvectors (u i, w i ) and (u j, w j ) (u i, w j ) = (w i, u j ) = 0 i j Constrained subspace with no point spectrum of JH X c = {(u, w) L 2 : {(u, w j ) = 0, (w, u j ) = 0} zj σ p (JH) }, Main question: N neg (H) N neg (H) =? L 2 Xc 13

22 Main Theorem I : bounds on isolated eigenvalues Let L be a self-adjoint operator on X L 2 with a finite negative index N neg, empty kernel, and positive essential spectrum. Let X c be the constrained linear subspace on linearly independent vectors: { } X c = v X : {(v, v j ) = 0} N j=1 Let the matrix A be defined by Then, A i,j = (v i, L 1 v j ), N neg (L) = N neg (L) N neg (A), Xc X 1 i, j N 1 i, j N 14

23 Application of the Main Theorem I Consider two matrices A + and A for two operators L + and L constrained with two sets of eigenfunctions {w j } and {u j } 15

24 Application of the Main Theorem I Consider two matrices A + and A for two operators L + and L constrained with two sets of eigenfunctions {w j } and {u j } Due to skew-orthogonality, the matrices A ± are block-diagonal. For real eigenvalue z = z 0 with the eigenvector (u R, w R ) A + j,j = A j,j = 1 z0 2 (u R, L + u R ) = 1 z0(w 2 R, L w R ). For imaginary eigenvalue z = iz 0 with the eigenvector (u R, iw I ) A + j,j = A j,j = 1 z0 2 (u R, L + u R ) = 1 z0(w 2 I, L w I ). 15

25 Application of the Main Theorem I For complex eigenvalue z = z R + iz I with the eigenvector (u R + iu I, w R + iw I ) ( ) A + i,j = A i,j = 1 zr Z2 z M, Z = I zr 2 + z2 z I I z R. 16

26 Application of the Main Theorem I For complex eigenvalue z = z R + iz I with the eigenvector (u R + iu I, w R + iw I ) ( ) A + i,j = A i,j = 1 zr Z2 z M, Z = I zr 2 + z2 z I I z R. Individual counts of eigenvalues N neg (L + ) = N neg (L + ) Nreal (JH) N imag (JH) N comp(jh) Xc X0 and N neg (L ) = N neg (L ) Nreal (JH) N imag + (JH) N comp(jh) Xc X0 End of proof of I 16

27 Projections and decompositions Birman Schwinger representation of the potentials ( ) f(x) g(x) V (x) = = B (x)a(x) g(x) f(x) and of the spectral problem (σ 3 ( + ω) z) ψ = B Aψ, such that (I + Q 0 (z)) Ψ = 0, Q 0 (z) = A (σ 3 ( + ω) z) 1 B, 17

28 Projections and decompositions Birman Schwinger representation of the potentials ( ) f(x) g(x) V (x) = = B (x)a(x) g(x) f(x) and of the spectral problem (σ 3 ( + ω) z) ψ = B Aψ, such that (I + Q 0 (z)) Ψ = 0, Q 0 (z) = A (σ 3 ( + ω) z) 1 B, The set of isolated eigenvalues of σ 3 H is finite No resonances occur in the interior points There exists a spectrum-invariant Jordan-block decomposition L 2 = N g (JH z) X c z σ p (JH) 17

29 Main Theorem II : positivity of the essential spectrum Assume that no resonance exist at the endpoints of σ e (JH) Assume that no multiple embedded eigenvalues exist in σ e (JH) R n = R 3 18

30 Main Theorem II : positivity of the essential spectrum Assume that no resonance exist at the endpoints of σ e (JH) Assume that no multiple embedded eigenvalues exist in σ e (JH) R n = R 3 There exists isomorphisms between Hilbert spaces W : L 2 X c, Z : X c L 2 W and Z are inverse of each other, where 1 + lim ɛ 0 + 2πi + u X c, v L 2 : Zu, v = u, v A(σ 3 H λ iɛ) 1 u, B(σ 3 ( +ω) λ iɛ) 1 v dλ, 18

31 Application of the Main Theorem II It follows from the wave operators that W σ 3 = σ 3 Z, Z σ 3 = σ 3 W, Zσ 3 H = σ 3 ( + ω)z. 19

32 Application of the Main Theorem II It follows from the wave operators that W σ 3 = σ 3 Z, Z σ 3 = σ 3 W, Zσ 3 H = σ 3 ( + ω)z. If ψ X c, there exists ˆψ L 2, such that ψ = W ˆψ. Then ψ, Hψ = W ˆψ, σ 3 σ 3 HW ˆψ = W ˆψ, (σ 3 H) Z σ 3 ˆψ = σ 3 ( + ω)zw ˆψ, σ 3 ˆψ = ( + ω)i ˆψ, ˆψ > 0. End of proof of II. 19

33 Extensions of the main results Fermi Golden Rule for an embedded real eigenvalue It disappears if it has positive energy It becomes complex eigenvalue if it has negative energy 20

34 Extensions of the main results Fermi Golden Rule for an embedded real eigenvalue It disappears if it has positive energy It becomes complex eigenvalue if it has negative energy Jordan blocks for multiple real eigenvalues of zero energy N real + = N real N real + = N real for even multiplicity ± 1 for odd multiplicity 20

35 Extensions of the main results Fermi Golden Rule for an embedded real eigenvalue It disappears if it has positive energy It becomes complex eigenvalue if it has negative energy Jordan blocks for multiple real eigenvalues of zero energy N real + = N real N real + = N real for even multiplicity ± 1 for odd multiplicity Weighted spaces for endpoint resonance and eigenvalue Resonance results in real eigenvalue of positive energy Eigenvalue repeats the scenario of embedded eigenvalue 20

36 Example: two coupled NLS equations iψ 1t + ψ 1xx + iψ 2t + ψ 2xx + ( ψ χ ψ 2 2) ψ 1 = 0 ( χ ψ ψ 2 2) ψ 2 = 0 21

37 Example: two coupled NLS equations Lyapunov-Schmidt ( reductions near local bifurcation boundary ψ 1 = e it ) ( ) 2 sechx + O(ɛ 2 ), ψ 2 = e iωt ɛφ n (x) + O(ɛ 3 ), where ( ) x 2 + ω n (χ) 2χ sech 2 (x) φ n (x) = 0 By continuity of eigenvalues, we count isolated eigenvalues N neg (H) = 2n, where n is the number of zeros of φ n (x). Therefore, 2N real (JH) + N imag(jh) + 2N comp (JH) = 2n 22

38 Example: two coupled NLS equations n = 1 23

39 Example: two coupled NLS equations n = 2 24

40 Work in progress Bounds on N real + (JH) in terms of positive eigenvalues of H Relation between bifurcations of resonances in JH and H What if continuous spectrum of H is sign-indefinite? What if J is not invertible? Dynamics of nonlinear waves beyond the linearized system 25