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The microlocal Landau-Zener formula. (English) Zbl 0986.81027

Consider the behavior as \(h\to 0+\) of the solutions of \(-ihd\psi/dt=A(t)\psi,\) where \(t\in{\mathbb R},\) \(\psi(t)\in H,\) some Hilbert space, and \(A(t)\) is a linear operator defined on \(H.\) When \(A(t)\) does not have crossing eigenvalues, i.e. when the distance of one eigenvalue from the neighboring ones is bounded away from 0 by some fixed constant \(\delta\) (independent of \(h\)) for any time \(t,\) then one has the adiabatic approximation: when \(h\to 0+,\) if for some \(t_0\) the function \(\psi(t)\) belongs to the eigenspace of such an isolated eigenvalue, it belongs to it for all times \(t.\) It is then of interest studying the breakdown of the adiabatic approximation, in other words, what happens when two eigenvalues cross or nearly cross (avoided crossing) for some \(t_0.\) In this case, if \(\psi(t)\) belongs to the eigenspace of only one of the two eigenvalues for \(t<t_0,\) then it belongs to the direct sum of the eigenspaces of both eigenvalues for later times \(t>t_0,\) and one then defines a probability transition at the avoided crossing given by the Landau-Zener formula.
In this paper, the authors extend the aforementioned formula to more general cases, and in particular to the case of \(2\times 2\) \(h\)-pseudodifferential systems on the real line of the kind \[ \left(\begin{matrix} P_1(\varepsilon,h)&\varepsilon W(\varepsilon,h)\\ \varepsilon W(\varepsilon,h)^*& P_2(\varepsilon,h)\end{matrix} \right) \left(\begin{matrix} u\\ v\end{matrix}\right)=0, \] where \(P_1,P_2,W\) are \(0\)th-order \(h\)-pseudodifferential operators smoothly depending on \(\varepsilon,\) \(P_1\) and \(P_2\) are self-adjoint. Their analysis focusses on a point \(z_0\in T^*{\mathbb R}\) such that, upon denoting by \(p_j\) the principal symbol of \(P_j,\) \(p_j(z_0)=0,\) \(j=1,2,\) the differentials \(dp_1(z_0)\) and \(dp_2(z_0)\) are linearly independent at \(z_0,\) and the principal symbol \(w\) of \(W\) does not vanish at \(z_0.\) Then \(\varepsilon W(\varepsilon)\) induces an avoided crossing of the eigenvalues of the principal symbol. The authors consider hence \(z_0\in Z_1\cap Z_2,\) where \(Z_j:=\{p_j=0\},\) oriented according to the Hamiltonian vector field \(H_{p_j},\) and the transfer matrix \[ T=\left(\begin{matrix} t_{1,2}&t_{1,4}\\ t_{3,2}&t_{3,4}\end{matrix}\right), \] which relates microlocal incoming solutions to outgoing solutions by the condition that they admit an extension as a microlocal solution near \(z_0.\) In the end, they get the following Landau-Zener type formula \[ |t_{1,2}|=|t_{3,4}|+O(h^\infty)=\exp\left(-{\pi\over h}\left( {|w(z_0)|^2\over |\{p_1,p_2\}(z_0)|}\varepsilon^2+O(\varepsilon^3)+ O(h\varepsilon^2)\right)\right)+O(h^\infty) \] (here \(\{p_1,p_2\}\) denotes the Poisson bracket of \(p_1\) and \(p_2).\) They also give applications to the case of two coupled Schrödinger operators and give some perspectives about the global case and the case of higher dimension.

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q15 Perturbation theories for operators and differential equations in quantum theory
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
35Q40 PDEs in connection with quantum mechanics
35S30 Fourier integral operators applied to PDEs
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