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By Vakhrameev S.A.

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23) Coordinate Bethe Ansatz 41 λ12 being integers. The total momentum is ρ(2) (k1 , k2 ) = k1 + k2 = 2π (λ1 + λ2 ). , r. r). kr ) = α 2π kα = N r λα . 29) α The calculation is identical to that in case of the single- and two-particle cases: r r σn+α H|φ(r) = E (r) F | F| α=1 σn+α |φ(r) . 30) α=1 Written explicitly, the above condition gives the following: 1/2{.... ) = 0. 31) The configuration of particles in this case is more complicated than in the two-particle case. Besides, there are other complications associated with the Bethe ansatz for the spin chain, which we do not go into here, but refer the reader to the following reference [32].

2) while the equations of motion are given as usual by ∂ψ = [H, ψ] ∂t ∂q = [H, q] . 3) Note that the commutation and anticommutation relations satisfied by the fields are as follows: {ψ(x), ψ(y)} = {ψ † (x), ψ † (y)} = 0, {ψ(x), ψ † (y)} = δ(x − y), q(x), q † (y) = δ(x − y). 5) Coordinate Bethe Ansatz 45 where |φ = |m, n , with m denoting here the number of bosons and n the number of fermions. To illustrate the calculational procedure, let us consider the state |1, 2 where dx1 f (x1 )q † (x1 )|0 + |1, 2 = dx1 dx2 g(x1 , x2 )ψ † (x1 )ψ † (x2 )|0 .

N+1 αn .... 43) In the sequel we will prove that the S matrix satisfies the Yang-Baxter equation. Until now we have not used any periodic boundary condition. Snn−1 . This equation can be solved by taking recourse to a special trick. 45) L(v) = − 2 n=1 S0n v − σn 2 = ωµ v − σn τ0µ τnµ , 2 the operator L(v) acts in a space of 2N +1 dimension with an additional particle numbered 0. 46) this T (v) is actually Baxter’s transfer matrix [27]. The solution will lead to equations that determine the momenta ki .

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A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems by Vakhrameev S.A.


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