By Chae Manjunath
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Extra info for A Robust Embedded Data from Wavelet Coefficients
7) where f1 , f0 are the respective wave functions. 8) then we get 1 ∂ 2 f1 1 ∂ 2 f2 = Ef , = Ef2 . 1 2 ∂x2 2 ∂x2 For the two-particle state the state vector is written in the form, |2, 11, 2 = dx1 dx2 g1 (x1 , x2 )q1 (x)q1 (x2 )|0 + dx1 dx2 g2 (x1 , x2 )q1 (x)q0 (x2 )|0 + dx1 dx2 g3 (x1 , x2 )q0 (x1 )q0 (x2 )|0 . 11) + ∂ 2 g3 ∂x22 − 2g2 (x1 , x2 )δ(x1 − x2 ) = Eg3 (x1 , x2 ). 12) In a similar way the corresponding three-particle case can be expressed as follows: |3, 12, 21, 3 = + dx1 dx2 dx3 h1 (x1 , x2 , x3 )q1 (x1 )q1 (x2 )q1 (x3 )|0 + dx1 dx2 dx3 h2 (x1 , x2 , x3 )q0 (x1 )q1 (x2 )q1 (x3 )|0 + dx1 dx2 dx3 h3 (x1 , x2 , x3 )q0 (x1 )q0 (x2 )q1 (x3 )|0 dx1 dx2 dx3 h4 (x1 , x2 , x3 )q0 (x1 )q0 (x2 )q0 (x3 )|0 .
30) Coordinate Bethe Ansatz 23 where x12 = x1 − x2 . 31) where i 1 k1 . 32) with P denoting terms obtained by permutation of (1, 2, 3). Note that in this manner it is not diﬃcult to formulate the n-particle problem. Moreover, the solution may be obtained by following the previous strategy. Hence, even though the commutation relation is not canonical, it is in principle possible to solve the quantum mechanical problem. Formulation of the n-particle case essentially represents a combinatorial problem.
1 Introduction In the previous chapter we introduced the coordinate Bethe ansatz, which is a fundamental tool for the exact solution of a large number of quantum mechanical many-body problems. We also introduced the notion of the quantum R matrix, a central object in the modern approach to studies of quantum integrable systems deriving in particular a basic equation satisﬁed by it, namely the Yang-Baxter equation (YBE). In this chapter we will analyze the Yang-Baxter equation in greater detail, discussing its origin and structure, the nature of its solutions, their properties and also its connection with exactly solvable models of statistical mechanics and the quantum inverse problem in general.