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Remarks on the quantum adiabatic theorem. (English. Russian original) Zbl 1081.81040

St. Petersbg. Math. J. 16, No. 4, 639-648 (2005); translation from Algebra Anal. 16, No. 4, 41-53 (2004).
To recall the quantum adiabatic theorem, let \(t\in\Delta=[0,a]\subset{R}\), and let \(H_0(t)\) be a \(t\)-dependent linear operator on a complex Banach space \(X\). Suppose the spectrum \(\sigma(t)\) of \(H_0(t)\) consists of two separate components \(\sigma_1(t)\), \(\sigma_2(t)\) such that \(\text{dist} (\sigma_1(t), \sigma_2(t))\geq a >0\) for \(t\in\Delta\). Let \(P_1(t)\) and \(P_1(t)\) denote the spectral projections of \(H_0(t)\) corresponding to the components \(\sigma_1(t)\) and \(\sigma_2(t)\). Consider equation \[ i\varepsilon \frac{\partial \psi}{\partial t} = H(t, \varepsilon)\psi \quad (1) \] with the Hamiltonian \[ H(t,\varepsilon) = H_0(t)+\varepsilon H_1(t,\varepsilon), \quad \varepsilon > 0. \] Then, for the resolving operator \(U(t,s,\varepsilon): X \rightarrow X\) of the equation, so that \[ i\varepsilon \frac{\partial U(t,s,\varepsilon)}{\partial t} = H(t,\varepsilon)\,U(t,s,\varepsilon), \quad U(s,s,\varepsilon) = I, \] the following statement \[ (I-P_j(t))\,U(t,s,\varepsilon)P_j(s)= O(\varepsilon), \quad \varepsilon \rightarrow 0. \] is valid. In the paper, generalized form of this statement is considered. \[ (I-P_{j,k}(t,\varepsilon))\,U(t,s,\varepsilon)P_{j,k}(s,\varepsilon)= O(\varepsilon^k), \quad k=1,2,\ldots. \tag{2} \] This formula means that the solution of the equation (1) with initial condition \( \psi(s) \in P_{j,k}(s,\varepsilon)X \) stays with high accuracy in the subspace \(P_{j,k}(t,\varepsilon)X\). Explicit asymptotic expressions of the projections \(P_{j,k}(t,\varepsilon)\) for small \(\varepsilon\) are rarely provided in the literature. However, the asymptotic representations for the solutions of equation (1) were constructed in M. G. Krein [Linear differential equations in Banach space. American Mathematical Society (AMS) (1972; Zbl 0229.34050)] and Yu. L. Daletskii, M. G. Krein [Stability of solutions of differential equations in Banach space. American Mathematical Society (AMS) (1974; Zbl 0286.34094)]. It was shown there that the solutions split into noninteracting branches in the sense of formula of type (2).
In the present paper, similar result is provided. It is shown that, under certain conditions, initial equation (1) can be transformed into \[ i\varepsilon \frac{\partial z}{\partial t} = \begin{pmatrix} M_{11}^{(N)}(t,\varepsilon) & \varepsilon^N M_{12}^{(N)}(t,\varepsilon) \\ \varepsilon^N M_{21}^{(N)}(t,\varepsilon) & M_{22}^{(N)}(t,\varepsilon) \end{pmatrix} z. \] The spectra of the leading terms \(M_{11}^{(N)}(t,\varepsilon) \sim M_{1}^{(N)}(t)\), \(M_{22}^{(N)}(t,\varepsilon) \sim M_{2}^{(N)}(t)\) coincide with the \(\sigma_1(t)\) and \(\sigma_1(t)\) respectively. The components of the vector \(z = (z_1, z_2)^T\) belong to the fixed, not ‘moving’, that is independent of \(t\) and \(\varepsilon\), subspaces \(Z_1=P_1(0)X\), \(Z_2=P_2(0)X\). The statement (2) readily follows from this result, and explicit formulas for the \(P_{j,k}\) are provided. In the authors’ opinion, the approach proposed in the present paper possesses some advantages over the approach of the works cited above.

MSC:

81Q15 Perturbation theories for operators and differential equations in quantum theory
47N50 Applications of operator theory in the physical sciences
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References:

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