By Harry L. Morrison
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Additional resources for Quantum Theory of Many-Particle Systems
19) may be expressed in terms of the Fourier coefficients ¯ ,φ ¯ ) as (φ0,n ), (φ1,n ) of the initial data (φ 0 1 T r C 0 j=1 |∂n φi (t, vj )|2 dt ≥ µn φ20,n + φ21,n . 7. 19) is not true, that is, such that there ¯1 ) ∈ V ′ × H, which are not controllable in time T . 21) n∈N with non-vanishing coefficients cn . 22) is controllable in time T . From that fact, it would hold, in particular, that the system is spectrally controllable (and then approximately controllable) in time T . 22). This would imply that the space Wε is controllable in time T .
Then, (¯ u0 , u ¯1 ) ∈ WT if, and only if, ¯ , −φ ¯ ) (¯ u0 , u ¯1 ), (φ 1 0 (H×V ′ )×(H×V ) ¯ P∗ (φ ¯ , −φ ¯ ) = h, 1 0 T U ¯ ∂n φ| ¯C = h, U. 5. 21) with initial state (φ ¯ ,φ ¯ ). 6. 35) suggests a minimization algorithm for the ¯ If we look for the control in the form h ¯ = construction of the control h. 35) is the Euler equation I ′ (ψ 0 1 quadratic functional I : V × H → R defined by ¯ ,φ ¯ )= 1 I(φ 0 1 2 T 0 r j=1 ¯ − u ¯ . 35) will be verified. Therefore, if (ψ 0 1 The functional I is continuous and convex.
This allows, in particular, to prove observability properties of the solutions of the 1 − d wave equation when measurements are done only on one end-point of the string. 1 D’Alembert Formula Let us assume that the function u(t, x) satisfies the 1 − d wave equation in R × R. Then, for every t∗ ∈ R the function u may be expressed by means of the D’Alembert formula u(t, x) = 1 1 (u(t∗ , x + t − t∗ ) + u(t∗ , x − t + t∗ )) + 2 2 x+t−t∗ ut (t∗ , ξ)dξ. 1) is also valid if we change their role. Thus, if u(t, x) satisfies the 1 − d wave equation in R × [0, ℓ] then, for every a ∈ [0, ℓ], u(t, x) may be expressed by the sidewise formula u(t, x) = 1 1 (u(t + x − a, a) + u(t − x + a, a)) + 2 2 t+x−a ux (τ , a)dτ .