Linear stability and instability of self-interacting spinor field ANDREW COMECH Texas A&M University, College Station, TX, USA October, 2013 Einstein: E2 = p2 + m2, Schrodinger:¨ (i∂ )2ψ = ( i )2ψ + m2ψ t − ∇ 2 E = m2 + p2 m + p , [Schrodinger¨ 26]: i∂ ψ = 1 ( i )2ψ ≈ 2m t 2m − ∇ p [Dirac28]: E = p2 + m2 = α p + βm, · p i∂ ψ = ( iα + βm) ψ, ψ(x,t) C4, x R3 t − ·∇ ∈ ∈ Dm | {z } 2 2 α (1≤j≤3) and β are self-adjoint and such that D = ∆+ m j m − 0 σj I2 0 Standard choice: αj = (Pauli matrices), β = σj 0 0 I2 − 1 Self-interacting spinors Models of self-interacting spinor field: [Ivanenko38, Finkelstein et al.51, Finkelstein et al.56, Heisenberg57] [...] Massive Thirring model [Thirring58] in (n+1)D: k+1 µ µ 2 LMTM = ψ¯(iγ ∂ m)ψ + ψγ¯ ψ ψγ¯ ψ k > 0 (V-V) µ − µ Soler model [Soler70] in (n+1)D: µ k+1 LSoler = ψ¯(iγ ∂ m)ψ +(ψψ¯ ) (S-S) µ − In (1+1)D: massive Gross-Neveu model [Gross & Neveu74, Lee & Gavrielides75] 2 Soler model: NLD with scalar-scalar self-interaction i∂ ψ = ( iα + mβ) ψ (ψψ¯ )kβψ, ψ CN , x Rn t − ·∇ − ∈ ∈ Dm | {z } [Soler70, Cazenave & Vazquez´ 86]: existence of solitary waves in R3, • ψ(x,t) = φ (x)e−iωt, ω (0,m) ω ∈ Attempts at stability: [Bogolubsky79, Alvarez & Soler86, Strauss & Vazquez´ 86] ... • Numerics [Alvarez & Carreras81, Alvarez & Soler83, Berkolaiko & Comech12] suggest that (all?)• solitary waves in 1D cubic Soler model are stable Assuming linear stability, one tries to prove asymptotic stability • [Pelinovsky & Stefanov12, Boussaid & Cuccagna12] 3 Nonrelativistic limit of NLD: ω m → φ(x) Solitary wave: ψ(x,t) = e−iωt; φ, ρ C2 ρ(x) ∈ 0 σ iψ˙ = i ·∇ + mβ ψ (ψψ¯ )kβψ, − σ 0 − ·∇ φ ρ φ φ ω iσ + m φ 2k ρ ≈ − ·∇ φ ρ −| | ρ − − If ω . m: 2mρ iσ φ, φ satisfies NLS: ≈ − ·∇ 1 (m ω)φ = ∆φ φ 2kφ. − − −2m −| | Scaling: φ(x) = ǫ1/kΦ(ǫx), ǫ = √m ω − 1 Φ = ∆Φ Φ 2kΦ − −2m −| | 4 NLD: linearization at a solitary wave −iωt −iωt Given φω(x)e , consider ψ(x,t) = φω(x)+ r(x,t) e Linearized eqn on r(x,t) CN , i∂ r = D r ωr + ... (m + ω)i ∈ t m − Re r 0 Dm ω + ... Re r ∂t = − Im r Dm + ω + ... 0 Im r − (m − ω)i Aω | {z } - σ(D ω) m − σess(Aω) r0 m ω m ω − − − 5 σ(Aω) Linear instability of NLD 2mi (m + ω)i Theorem 1 ([Comech & Guan & Gustafson12]). If NLSk is linearly unstable, then for ω . m, λ σ (A ), (m − ω)i ∃ ± ω ∈ d ω t tt - - Re λω > 0, λω 0 ω−→→m −λω λω 1D, above quintic 2D, above cubic 3D cubic and above −2mi ω < m ω = m Proof: Rescale; use Rayleigh-Schrodinger¨ perturbation theory. 2 6 σ(Aω) Linear stability of NLD 2mi m ω i 12 ( + ) Theorem 2 ([Boussaid & Comech ]). t 2ωi Assume λ σ (A ), ω . m ω ∈ p ω 1. λω 0; 2mi . ω−→→m { ± } (m − ω)i t tt - - 2. If Re λω = 0, then λω 0, ω−→→m −λω λω λω Λ:= lim σp(NLSk) ω→m (m ω) ∈ − 3. Λ = 0 unless critical case: t 2D quintic; 3D cubic −2mi ω < m ω = m Proof: Limiting absorption principle [Agmon75, Berthier & Georgescu87] 2 7 σ(Aω) Linear stability of NLD 2mi (m + ω)i Corollary 3 ([Boussaid & Comech12]). 1D cubic: (m − ω)i φ e−iωt are linearly stable for ω . m t - - ω t −2mi Remark 4. Also true for 1D cubic and 2D quintic ω < m ω m (“charge-critical NLS”) = 8 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 -1 -0.5 0 0.5 1 Figure 1: Upper half of the spectral gap. TOP: 1D cubic Soler BOTTOM: 1D cubic massive Thirring 9 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 -1 -0.5 0 0.5 1 Figure 2: 1D quintic (“charge critical”). TOP: Soler; BOTTOM: massive Thirring 10 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 -1 -0.5 0 0.5 1 Figure 3: 1D, seventh order. Soler and MTM 11 σ(AΩ) Bifurcations from σess (m + Ω)i Let Ω (0,m) λ λΩ ω r r r vr r ∈ r r r r Theorem 5 ([Boussaid & Comech12]). 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