By Volker Bach, Jean-bernard Bru
The authors research a non-autonomous, non-linear evolution equation at the area of operators on a posh Hilbert space.
We examine a non–autonomous, non-linear evolution equation at the house of operators on a fancy Hilbert area. We specify assumptions that make sure the international life of its strategies and make allowance us to derive its asymptotics at temporal infinity. We reveal that those assumptions are optimum in an appropriate feel and extra common than these used prior to. The evolution equation derives from the Brocket–Wegner stream that used to be proposed to diagonalize matrices and operators by way of a strongly non-stop unitary stream. actually, the answer of the non–linear circulate equation results in a diagonalization of Hamiltonian operators in boson quantum box idea that are quadratic within the box.
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Extra info for Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations
114) sup Ωt,μ − Ωt t∈[0,T0 ] op ≤ 2μ + 4 1 − e−2T0 μ B0 2 . 116) ∀t ∈ [0, T0 ] : (ϕ, Ωt ϕ) = lim+ (ϕ, Ωt,μ ϕ) ≥ 0 , μ→0 because of Lemma 38 applied to the operator family (Ωt,μ )t∈[0,T0 ] . In other words, the operator Ωt is positive for any t ∈ [0, T0 ]. Licensed to Tulane Univ. 78. 1. 3) ∈ (0, ∞] . 118) Tmax ≥ T0 := (128 B0 2 )−1 > 0 on this maximal time. Then, we can clearly extend Lemma 34 and Corollary 37 to all times t ∈ [0, Tmax ): Theorem 40 (Local existence of (Ωt , Bt )). Assume Ω0 = Ω∗0 ≥ 0 and B0 = B0t ∈ L2 (h).
5), is a family of Hilbert–Schmidt operators. 120) Bt = B0 − 2 ∀t ∈ [0, Tmax ) : Ωτ Bτ + Bτ Ωtτ dτ , 0 with Bt>0 h ⊆ D(Ω0 ). Furthermore, (Bt )t∈(0,Tmax ) is locally Lipschitz norm continuous. Proof. 117) of Tmax as Ωt := Ω0 − Δt for all t ∈ [0, Tmax ). Only the Lipschitz continuity of (Ωt )t∈[0,T0 ] must be proven. 121) r= Δ ∞ := sup Δt t∈[0,T ] op <∞ is obviously ﬁnite as (Δt )t∈[0,T ] ∈ C. , the operator family (Ωt )t∈[0,Tmax ) is Lipschitz continuous on [0, T ]. 122). 121). This theorem says nothing about the uniqueness of the solutions (Ωt )t∈[0,Tmax ) and (Bt )t∈[0,Tmax ) .
In the Hilbert–Schmidt topology. Proof. 167) ≤ 2 t+δ 2δ −1 Ωt Bt − Ωτ Bτ t 2 + Bt Ωtt − Bτ Ωtτ 2 dτ . 167) in the limit δ → 0, we conclude that (Ωt − Ω0 , Bt )t∈[0,T+ ) is a solution of the diﬀerential equations stated in the corollary. 2), the operator Ωt is unique, see Lemma 41. 168) Qt = −16 ¯τ dτ = Q∗t . 169) Qt 2 ≤ 16t B0 2 <∞ for all t ∈ [0, T+ ). Note that the uniqueness can also be directly deduced from Corollary 37 extended to [0, T+ ). 165). 2) with Ω 2 ˜ by Lemma 99. 3) ˜t = Bt for all t ∈ [0, T+ ).