Fermionic renormalization group method based on the smooth Feshbach map
Itaru Sasaki, Akito Suzuki
For a fermion system, an operator theoretic renormalization group method based on the smooth Feshbach map is constructed. By using the fermionic renormalization group method, the closed operator of the form: Hg(θ) = HS ⊗ 1 + eθν1 ⊗ Hf + Wg(θ) is analyzed, where HS is a selfadjoint operator on a separable Hilbert space and bounded from below, Hf denotes the fermionic quantization of the one fermion kinetic energy c|k|ν, k ∈ Rd (c, ν > 0), Wg(θ) is a small perturbation with respect to HS ⊗ 1 + eθν1 ⊗ Hf and θ ∈ C is a complex scaling parameter. The constant g ∈ R denotes a coupling constant such that Wg(θ) → 0(g → 0) in some sense. It is assumed that HS has a discrete simple eigenvalue E ∈ σd(HS), and proved that Hg(θ) has an eigenvalue Eg(θ) close to E for a small coupling constant g. Moreover, the eigenvalue Eg(θ) and the corresponding eigenvector Ψ(θ) is constructed by the process of the operator theoretic renormalization group method.
Preprint Series of Department of Mathematics, Hokkaido University•May 9, 2007•0 citations