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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] 

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 

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Substitute z = ωx , q = mµB ℏ $^{2}$ω2 and λ = 2 mE ℏ $^{2}$ω2 into equa- tion (8), then this equation becomes 

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 

Since u ( z ) is periodic with period π , it can be expanded in Fourier series: 

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Then substitute (14) into (9), we obtain an infinite symmetric tridiagonal matrix equation, that is, eigen equation. 

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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 

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The total energy of the system is 

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. 

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