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Distribution of matrix elements and level spacing for classically chaotic systems. (English) Zbl 0833.58018
This paper considers the semi-classical behaviour of matrix elements of observables. The system under consideration is the quantized version of a classical chaotic system.
Let \(p(x, \xi)\in C^\infty(T^*(X))\) be a classical Hamiltonian such that on some energy shell \(p(x, \xi)= E\), the corresponding classical dynamics is ergodic (or mixing). The quantized counterpart is described by the Hamiltonian operator, \[ P(\hbar)= -\hbar^2 \Delta+ V \] and consider a classical energy interval such that in this interval \(P(\hbar)\) has purely discrete spectrum; \(P(\hbar)\varphi_j= E_j(\hbar)\varphi_j\) for \(\{\varphi_j\}\) an orthonormal system of bound states.
For a classical observable \(a(x, \xi)\), the matrix elements of the corresponding quantum observable \(A(\hbar)\) are studied, i.e. \[ A_{j_k}(\hbar)= \langle A(\hbar) \varphi_j, \varphi_k\rangle. \] The paper discusses and makes precise under what conditions and in what sense the expected results \[ \lim_{\hbar\to 0} A_{j_k}(\hbar)= 0,\quad k\neq j \] holds true.

MSC:
53D50 Geometric quantization
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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