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Adiabatic expansions near eigenvalue crossings. (English) Zbl 0875.47002

Summary: Quantum mechanical adiabatic approximations describe the small \(\varepsilon\) behavior of solutions of the Schrödinger equation \(i\varepsilon (\partial \psi/\partial t)= H(t) \psi\) for \(-T\leq t\leq T\). It is well known that if \(H(t)\) is analytic in \(t\), selfadjoint for real \(t\), and has an isolated eigenvalue \(E(t)\) of multiplicity one for \(-T\leq t\leq T\), then there is a solution that is asymptotic to \[ e^{-i \int^t_0 E(r) dr/ \varepsilon} (\Phi (t)+ \varepsilon \psi_1 (t)+ \varepsilon^2 \psi_2 (t)+ \dots). \] The leading term, \(\Phi (t)\), is a unit vector in the spectral subspace for \(H(t)\) that corresponds to \(E(t)\). In this paper we investigate eigenvalue crossings. We assume \(H(t)\) has two analytic eigenvalues \(E_A (t)\) and \(E_B (t)\) that are isolated from the rest of the spectrum of \(H(t)\). We further assume that \(E_A (t)\) and \(E_B (t)\) are isolated from one another and are of multiplicity one, except at \(t=0\), where they are equal. We prove that in this situation, the Schrödinger equation has solutions with asymptotic expansions of the form \[ e^{-i \int^t_0 E_j (r) dr/ \varepsilon} \Phi_j (t)+ \nu_1 (\varepsilon) \psi^j_1 (t, \varepsilon)+ \nu_2 (\varepsilon) \psi^j_2 (t, \varepsilon)+ \cdots \qquad (j=A, B). \] The \(\psi^j_k (t, \varepsilon)\) are uniformly bounded for \(-T\leq t\leq T\), and \(\Phi_j (t)\) is a unit vector in the spectral subspace corresponding to \(E_j (t)\). The expansion orders, \(\nu_k (\varepsilon)\), have the form \(\varepsilon^{n (k)/ (p+1)} (\log (\varepsilon) )^{m(k)}\), where \(p\) is the order of the zero of \(E_A (t)- E_B (t)\), and \(n(k)\) and \(m(k)\) are certain non-negative integers.

MSC:

47A55 Perturbation theory of linear operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
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