2 As the ground state of a bose system is non-negative [19], mapping (3) for the stationary ground state of a system reduces to a simplified form [2] Then substitute (14) into (9), we obtain an infinite symmetric tridiagonal matrix equation, that is, eigen equation. To illustrate our general ideas, we now study the case that magnetized bosons in an external magnetic field B ( x ) = − B cos(2 ωx ). For simplicity, we take B ( x ) inde- pendent of t . therefore, in which We suppose L = Mπ , where M is an integer and µ is the magnetic moment of atom. For the sake of simplicity, we assume that both N and M are odd (As a result of the introduction of A ( x$_{1}$, · · · , x$_{N}$ ), ψ B is periodic if N is odd and antiperiodic if N is even [2, 12]). From Bloch theory, the corresponding energy spectrum of system in periodic potentials has the energy band structure. We use the notation n as the band index and k as the Bloch wave vector in the first Brillouin zone. For convenience, we use α = { n, k } to denote both the band index and the Bloch wave vector. Not considering the interparticle interaction, every particle in x ∈ [0 , L/ω ] is governed by By truncating the matrix in each direction (centered at the smallest diagonal element ν $^{2}$) at sufficiently large di- mensions, approximations to the desired eigenvalues can be obtained to any desired precision. We could obtain the eigenvalue and eigenfunction easily using Mathemat- ica or Matlab. In practice, the eigenvalue and eigenfunc- tion converge quickly as the dimension increases when q is not too large. For example, if we choose ν = 6 / 7 and q = 1, the relative difference of the lowest eigenvalue between truncating the matrix at 21-D and 2001-D is only 10 − $^{38}$. For large values of q and for high orders, we can use asymptotic expansions instead [21, 22]. since the matrix dimension that needed to get accurate eigenvalues and eigenfunctions becomes large. With Girardeau’s Fermi-Bose mapping (3), the eigen- states of the hard-core Bose system is given by the Slater determinant It is called Mathieu’s Differential Equation, which can be solved with Randall B. Shirts’s [18] method. Using Bloch’s theorem, we first set Substitute z = ωx , q = mµB ℏ $^{2}$ω2 and λ = 2 mE ℏ $^{2}$ω2 into equa- tion (8), then this equation becomes Since u ( z ) is periodic with period π , it can be expanded in Fourier series: As a result of the Bose-Fermi mapping, the energy spec- trum of the Bose and corresponding Fermi system are identical. When the temperature is 0, the system will be on the ground state — N particles on the N lowest eigenstates, respectively. The total energy of the system is ψ $^{B}$( x$_{1}$, · · · , x$_{N}$ ) = | ψ $^{F}$( x$_{1}$, · · · , x$_{N}$ ) | (5)    $^{.}$$^{.}$$^{.}$ · · · · · · · · · · · · · · · · · · · · · ( ν − 4) 2 q 0 0 0 q ( ν − 2) 2 q 0 0 · · · · · · 0 q ν 2 q 0 0 0 q ( ν + 2) 2 q · · · · · · 0 0 0 q ( ν + 4) 2 · · · · · · · · · · · · · · · · · · . . .     ×     . c$_{−}$$_{2}$ c$_{−}$$_{1}$ c$_{0}$ c$_{1}$ c$_{2}$ .     = λ     . c$_{−}$$_{2}$ c$_{−}$$_{1}$ c$_{0}$ c$_{1}$ c$_{2}$ .    $^{.}$ (15) ˆ H = N ∑ i =1 [ − ℏ 2 2 m ∂ 2 ∂x 2 i + V ( x$_{i}$ ) ] . (6) V ( x ) = { µB cos(2 ωx ) , 0 ⩽ x$_{i}$ ⩽ L $_{ω}$, 0 , else . (7) [ − ℏ 2 2 m ∂ 2 ∂x 2 + µB cos(2 ωx ) ] ϕ$_{α}$ ( x ) = E$_{α}$ϕ$_{α}$ ( x ) . (8) d $^{2}$ϕ ( z ) dz 2 + [ λ − 2 q cos(2 z )] ϕ ( z ) = 0 , (9) z ∈ [0 , L ] . ψ $^{B}$( x$_{1}$, · · · , x$_{N}$ ) = A ( x$_{1}$, · · · , x$_{N}$ ) √$_{N}$$_{!}$ × ∣ ∣ ϕ$_{α}$$_{2}$ ( x$_{1}$ ) · · · ϕ$_{α}$$_{N}$ ( x$_{1}$ ) ϕ$_{α}$$_{1}$ ( x$_{2}$ ) ϕ$_{α}$$_{2}$ ( x$_{2}$ ) ϕ$_{α}$$_{N}$ ( x$_{2}$ ) . . . . ϕ$_{α}$$_{1}$ ( x$_{N}$ ) ϕ$_{α}$$_{2}$ ( x$_{N}$ ) · · · ϕ$_{α}$$_{N}$ ( x$_{N}$ ) ∣ ∣ . (16) E = N ∑ i =1 E$_{α}$$_{i}$. (17) ϕ ( z ) = exp( iνz ) u ( z ) , (10) ϕ ( z ) = ϕ ( z + π ) , (11) u ( z ) = u ( z + L ) , (12) ν = 2 l M , ( l ∈ 0 , ± 1 , · · · , ± M − 1 2 ) . (13) ϕ ( z ) = exp( iνz ) ∑ n c$_{n}$ exp( i 2 nz ) . (14)