3 and then rewrite Eq. (2) as where ¯ E = Ee λ/ $^{2}$. Now let us consider the effect of the lattice translation. By shifting n = n ′ + m and rewriting φ ( n ′ + m ) → φ ( n $^{′}$), we obtain which becomes identical to Eq. (4) by redefining ¯ E ′ = ¯ Ee − $^{λm}$. In other word, the lattice translation of the wavefunction has one-to-one correspondence to the eigen- state of the e − $^{λm}$-scaled energy eigenvalue, which has been mentioned in Ref. 19 in the operator level. Thus, once a certain energy eigenvalue is obtained, the other eigenvalues and the corresponding eigenvectors can be obtained by the lattice translation combined with E ′ = e − $^{λm}$E . In addition, we note that, if λ is sufficiently small, we can always find a certain m such that E ′ is unity, except for E being exactly zero. This implies that the Wilson Hamiltonian is energy-scale-free theory and thus the problem can be physically well-defined, although the Hamiltonian includes very wide range of the energy scale, Λ ∼ Λ $^{N}$. We analyze features of the bulk wavepacket itself. For this purpose, we concentrate on the positive energy solu- tion of Eq. (4) and consider its “continuum limit” λ → 0. Writing x = λn , we have III. WAVEPACKET BASIS where we have omitted O ( λ $^{3}$). Note that the translation x → x + a combined with ¯ Ee − a → ¯ E keeps Eq. (6) invari- ant. Thus we can set ¯ E = 1 in the following analysis. It should be also remarked that the continuum limit λ ≪ 1 can be taken with no reference to the lattice space. Introducing y ≡ exp( − x/ 2), we can convert Eq. (6) to the modified Bessel type equation, Assuming that the boundary is at infinity with respect to x , and taking account of the normalizability of the wavefunction in x ≪ − 1( y ≫ 1), we obtain φ ( y ) = const × K$_{i}$$_{2}$$_{c}$ ( cy ) with c ≡ √$_{2}$$_{/λ}$$_{,}$ 20 where K$_{iν}$ ( z ) is the modified Bessel function of the imaginary order ν . Using the formula ∫ ∞ 0 dzzK$_{−}$$_{iν}$ ( z ) K$_{iν}$ ( z ) = πν 2 sinh $_{πν}$, we obtain the eigenfunction of Eq. (2) as The behavior of this solution can be resolved in special limits. When x ≫ 1( y ≪ 1), we can easily see that Eq. (6) reduces to the uniform tight binding model, which yields This is consistent with the asymptotic behavior of the modified Bessel function K$_{iν}$ ( z ) ∼ 2 iν − $^{1}$Γ( iν ) z − iν in the z → 0 limit. Thus, the wavefunction ψ ( x ) exponentially decays in x ≫ 1 with the oscillation. This behavior is consistent with the numerical result in Fig. 2, where we have checked that the tail of the wavepacket decays as e − λn/ $^{2}$. In the limit z ≫ 1, the asymptotic form is given by K$_{iν}$ ( z ) ∼ √ π 2 z exp( − z ). For x ≪ 0( y ≫ 1), thus, we can see which decays very rapidly in x < 0. This is also in good agreement with the numerical result in Fig. 2. From these, we can see that the localization of the wavepacket is basically good and the width of the packet is ex- actly described by the decay rate of the high-energy tail, ξ ≡ 2 /λ (= 2 / ln Λ), because of Eq. (3). Also this good localization of the packet justifies that the effect of the boundary can be negligible in the bulk region. Since the Kondo impurity is located at the right edge of the chain, the amplitude at n = N is of particular importance. Here we return to the lattice problem and write the index of the wavepacket as j . Figure 3 shows the absolute value of ψ$_{j}$ ( N ), for which we can see the ex- ponential dependence of ψ$_{j}$ ( N ) with respect to the index j . Because of the resolution of the figure, we plotted the case of N = 50. Note that the particle-hole symmetry of (1) gives α$_{j}$ = − α$_{N}$$_{−}$$_{j}$$_{+1}$ with j ≤ N/ 2( N =even), which can be also confirmed in Fig. 3. According to Eq. FIG. 3: (Color online)Amplitude of the wavefunction at the impurity site ψ$_{j}$ ( N ) of N = 50 for λ = 0 . 05, 0.1, 0.2 and 0.5, where j indicates the label of an eigenstate. We can see the clear exponential dependence on j for the bulk wavefunction. The small hump around Fermi surface( j ∼ 25) corresponds to the edge states of the lowest energy scale. The hump region of the edge states extends, as λ becomes small. 0 25 50 10 −6 10 −4 10 −2 10 0 j | j (N) | =0.1 N=50 =0.05 =0.2 =0.5 φ ( n − 1) + φ ( n + 1) = ¯ Ee − $^{λn}$φ ( n ) , (4) φ ( n ′ − 1) + φ ( n ′ + 1) = ¯ Ee − $^{λm}$e − λn $^{′}$φ ( n $^{′}$) , (5) λ $^{2}$φ $^{′′}$( x ) + (2 − ¯ Ee − $^{x}$) φ ( x ) = 0 , (6) [ d 2 dy 2 + 1 y d dy + 4 λ $_{2}$( 2 y 2 − 1) ] φ ( y ) = 0 . (7) ψ ( x ) = √ sinh 2 πc 2 πc ce − x/ $^{2}$K$_{i}$$_{2}$$_{c}$ ( ce − x/ $^{2}$) . (8) φ ( x ) ∼ e x/ $^{4}$exp( − ce − x/ $^{2}$) , (10) φ ( x ) ∼ exp( − icx ) . (9)