Discrete wavelets and perturbation theory

We show with the help of examples that discrete wavelets can be a useful tool in perturbation theory of finite-dimensional quantum Hamilton systems. PACS numbers: 02.30.Nw, 02.70.−c In perturbation theory the Hamilton operator Ĥ is given by Ĥ = Ĥ 0 + Ĥ 1 where Ĥ 0 and Ĥ 1 are self-adjoint operators in a Hilbert space [1]. It is assumed that the perturbation Ĥ 1 is relatively ‘small’ in comparison to the soluble part Ĥ 0. Quite often Ĥ 0 is the diagonal term. We also quite often have the problem that (for example after a Fourier transform) Ĥ 1 is the soluble part and Ĥ 0 is the perturbation. A typical example is the Hubbard model. Thus it would be quite useful to have a transformation such that Ĥ 0 is always the dominant term independent of the parameters. We assume that the Hamilton operator acts in a finitedimensional Hilbert space. For Hamilton operators acting in a finite-dimensional vector space the discrete wavelet transform [2, 3] can play such a role. In our first example we consider the Hubbard model. For the sake of simplicity we consider the two-point Hubbard model. In Wannier representation we have Ĥ = t(c† 1↑c2↑ + c 1↓c2↓ + c 2↑c1↑ + c 2↓c1↓) + U 2 ∑ j=1 c † j↑cj↑c † j↓cj↓ (1) where the parameters t > 0 and U > 0. After a discrete Fourier transform we find the Bloch representation ĤB = ∑ kσ (k)c † kσ ckσ + U ∑ k1,k2,k3,k4 δ(k1 − k2 + k3 − k4)c k1↑ck2↑c † k3↓ck4↓ (2) where (k) = t cos(k) k = 0, π mod 2π. (3) Thus we would like to consider the cases U t and t U under one approach. The Hubbard operator commutes with the total number operator N̂ and the total spin operator in 0305-4470/03/246807+05$30.00 © 2003 IOP Publishing Ltd Printed in the UK 6807 6808 W-H Steeb et al the z-direction Ŝz. We consider the case with two particles and Sz = 0. Then a basis in Wannier representation is given by c † 1↑c † 1↓|0〉 c 1↑c 1↓|0〉 c 2↑c 1↓|0〉 c 2↑c 2↓|0〉. (4) Thus we find the Hubbard Hamilton operator in Wannier representation has the matrix representation ĤW =   U t t 0 t 0 0 t t 0 0 t 0 t t U   . (5) We see that if t U the non-diagonal elements are dominant. In Bloch representation we have the basis c † 0↑c † 0↓|0〉 c π↑c π↓|0〉 c 0↑c π↓|0〉 c π↑c 0↓|0〉 (6) and the matrix representation ĤB =   U/2 + 2t U/2 0 0 U/2 U/2 − 2t 0 0 0 0 U/2 U/2 0 0 U/2 U/2   . (7) The matrices given by (5) and (7) are related by the unitary transformation ĤB = V ĤWV ∗, where the unitary matrix V is given by V = 1 2   1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 −1   . (8) Now we apply the discrete wavelet transform. The Haar matrices [2] are given by K(k + 1) = ( K(k)⊗ (1 1) 2I2k ⊗ (1 −1) ) k > 1 (9) using the Kronecker product and recursion [2], where K(1) = ( 1 1 1 −1 ) . (10) Thus the 4 × 4 Haar matrix K (after normalizing the columns) is given by K = 1 2   1 1 1 1 1 1 −1 −1 √ 2 −√2 0 0 0 0 √ 2 −√2   . (11)