## 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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### References:

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