Starting from Bohr model, Eigenvalue Theorem by A. Weinstein is a historical consequence guaranteeing the existence of quantum states based on the classical data, that is the existence of Lagrangian submanifold in the phase space satisfying Maslov quantization condition. We need to wait a rigorous proof until the development of the Fourier integral operator theory by Hörmander.

In this talk, I will remark that the result can be generalized to “sub-Laplacian” case and discuss the behaviour of Lagrangian submanifolds under Riemannian submersion which is not necessarily harmonic. Finally I will give an example of a Lagrangian submanifold satisfying Maslov quantization condition other than torus.

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