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Narrowly avoided crossings. (English) Zbl 0627.58035
In order to create a degeneracy in a quantum mechanical system without symmetries we must vary two parameters in the Hamiltonian. When only one parameter, \(\lambda\) say, is varied, there is only a finite closest approach \(\Delta\) E of two eigenvalues, and never a crossing. Often the gaps \(\Delta\) E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as \(\hslash \to 0:\Delta E=A e^{-S/\hslash}.\)
This paper reports the results of numerical calculations on a system with two parameters, \(\epsilon\), \(\lambda\), which is completely integrable when \(\epsilon =0\). It is found that the gaps \(\Delta\) E obtained by varying \(\lambda\) decrease exponentially as \(\hslash \to 0\), consistent with the tunnelling conjecture. When \(\epsilon =0\), \(\Delta E=0\) because the system is completely integrable. As \(\epsilon\to 0\), the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping \(\hslash\) fixed, the gaps are of size \(\Delta E=O(\epsilon^{\nu})\), where \(\nu\) is usually very close to an integer. Theoretical arguments are presented which explain this result.

MSC:
58Z05 Applications of global analysis to the sciences
81Q99 General mathematical topics and methods in quantum theory
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