By Volker Bach, Jean-bernard Bru

ISBN-10: 1470417057

ISBN-13: 9781470417055

The authors examine a non-autonomous, non-linear evolution equation at the area of operators on a fancy Hilbert space.

Abstract

We research a non–autonomous, non-linear evolution equation at the house of operators on a posh Hilbert house. We specify assumptions that make sure the international life of its options and make allowance us to derive its asymptotics at temporal infinity. We display that those assumptions are optimum in an appropriate feel and extra basic than these used ahead of. The evolution equation derives from the Brocket–Wegner stream that was once proposed to diagonalize matrices and operators by way of a strongly non-stop unitary circulation. in reality, the answer of the non–linear circulate equation ends up in a diagonalization of Hamiltonian operators in boson quantum box conception that are quadratic within the box.

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Example text

157) lim (Wt,s − 1) Bs t→s+ 2 2 =0. 159) t − 1) Wt,s Bs (Wt,s 2 2 op t ∗ ¯s ≤ ((Wt,s ) − 1)B Licensed to Tulane Univ. 78. org/publications/ebooks/terms ≤ 1 and the cyclicity 2 2 46 V. TECHNICAL PROOFS ON THE ONE–PARTICLE HILBERT SPACE ∗ ∗ t t and Wt,s is also strongly continuous in t ≥ s ≥ 0 with Ws,s = 1 and t ≤ 1. 150), yield the right continuity in L2 (h) of the Hilbert–Schmidt operator family t (Bt )t∈[0,T+ ) . Furthermore, since the evolution operators Wt,s and Wt,s are also 2 strongly continuous in s ≤ t, the left continuity in L (h) of (Bt )t∈[0,T+ ) is verified in the same way.

9). √ Proof. 19) take ω−1 := 1, ω1 := 2 and b1 ∈ (1/2, 1/ 2), while for k ∈ N\{1}, choose bk := 1/2k , ω−k := ωk := (3/4)k . Similar to the proof of Proposition 2 28, Conditions A1–A3 and Ω−2 0 B0 ∈ L (h) are clearly satisfied for this choice. 22). 33) requires for k = 1 that 1 − b21 > 0 and 1 − 4b21 > 0. 33) does not hold. 6) for k = 1 and k ∈ N\{1} separately. 38) (1) Ω0 = ω−1 0 0 ω1 and (1) B0 = (1) 0 b1 b1 0 . 24) for k = 1 and μ < 1− (2) (2) 2b21 , where b21 < 1/2. 26), that satisfy A1–A3 and A5.

Lemma 34) and prove additional properties on (Wt,s )t≥s≥0 in the next lemma: Lemma 35 (Wt,s as solution of a parabolic evolution equation). 14). 28) ∀s ∈ [0, T0 ), t ∈ (s, T0 ] : Ωt Wt,s op ≤ C0 + D0 (t − s)−1 with C0 , D0 < ∞. Moreover, (Ωt Wt,s )t>s≥0 is strongly continuous in t > s. 29) ∀s ∈ [0, T0 ), t ∈ (s, T0 ] : ∂t Wt,s = −2Ωt Wt,s : ∂s Wt,s = 2Wt,s Ωs ∀t ∈ (0, T0 ], s ∈ [0, t] Licensed to Tulane Univ. 78. org/publications/ebooks/terms , Ws,s := 1 , Wt,t := 1 . 1. WELL–POSEDNESS OF THE FLOW 31 The derivatives with respect to t and s are in the strong sense in h and D (Ω0 ), respectively.

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Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations by Volker Bach, Jean-bernard Bru


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