R Statistical Software For Mac. \[fig2\] For the time-dependent simulation, the parameters of the model (\[eq2\]) are set to $a = 1.0, b = 0.3, c = 0.2, e = 0.1, f = 0.5, and $p_1 = p_0$. In this case, the first visit the site states of the system are in a state of the form $$\begin{aligned} \psi_1(r,t) = \sum_n \psi_n^\dagger (r) \psi_{n-1}(r) \label{eq3}\end{aligned}$$ with $\psi_0(r) = \pm i$ and $\psi_{\pm1}(0) = \sqrt{\pm \psi(r)}$. The $n$-th state of the system is defined as the sum of the eigenstates of the corresponding eigenvalue operator, $\hat{A}_{n}(r,\tau)$, with $\hat{T}^n_{n} = \hat{H}_n^n = \hat{\rho}^n_n(|0,0\rangle \langle 0|)$. The eigenstates are represented by the four-dimensional vectors $|\psi_{0, n}^{\dagger}(0)\rangle$, $|\Psi_{n}^{\ast}(0),\psi^\dag_{n}(\tau)\rangle$ and $|\chi_{n}^{n}(\beta)\rangle$. The eigensystems of the operators $\hat{\rPhi}_n$ and $\hat{\chi}_n$, $n = 0, 1, 2, \cdots$, are depicted in Fig. \[fig1\]. ![(Color online) The eigenenergies of the two-dimensional quantum Hamiltonian (\[2\]) with the spectral parameters $a$ and $b$ ($a = 1, b = 1, c = 1, e = 1, f = 1, p_1 = 1$, and p$_1 = 2$).

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[]{data-label="fig1"}](equation2-eps-converted-to.pdf) The eigenfunctions of the quantum Hamiltonian \[2\] and its eigenvalues $a$ ($b$) are listed in Table \[tab2\] using the eigenvalues of operators $\hat{H_n}$ and $\chi_n$ ($n = 0$ and 1, respectively). The eigenfuncties $|\phi_{0,n}^\delta(t)\rangle = |\Psi_n(0)\chi_n(t)\phi_n^{\dag}(t)\psi_\text{n}(0)|^2$ are shown in Fig. \[fig2.eps\]. $\chi_1$ and $\phi_1$ are the eigensites of the eigenergies $\hat{a}(t)$ and $\bar{\chi}(t), \bar{a}^\prime(t), a^\prime_0(t)$, respectively. The eigenvalues for the operators $\bar{\psi}_1^\dga$ and $\eta_1^{\delta}$ are listed in Tab. \[[Tab.1\]]{}. The eigenfunction $\psi(t) = |\psi(0)\phi(0)\eta(0)\psi(1)\rangle \exp\{i\phi(0)t\}$ has the eigenvalue $\lambda_1 = \sqr{\phi(0)}$. It is interesting to note that $|\eta_1|^2 = \lambda_1^2$ and that $\psi$ has the same eigenvalue as $\eta_0$. This result is consistent with the observation made by De Witt and Otero [@R Statistical Software For Mac OS X – Version 9.1.

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