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1. Eigenvalues of Operators with Analytic Coefficients

Let a(x,\xi) be an analytic symbol defined on an appropriate 2-dimensional phase. We are interested in the distribution of eigenvalues of a^w(x,hD), the Weyl-quantization of a(x,\xi), in the semi-classical limit as h \rightarrow 0^+.

    1. Eigenvalues and Curves


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      Problem 1.1.

      Do the eigenvalues of a^w(x,hD) lie on or cluster around curves in the complex plane \mathbb{C}?

        • Bohr-Sommerfeld


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          Problem 1.2.

          Is there a Bohr-Sommerfeld quantization rule for a^w(x,hD)? Is there a link to the usual Bohr-Sommerfeld rule for Hermitian operators?

            • A non-Hermitian Operator from Physics


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              Problem 1.3.

              What is the distribution of eigenvalues of the non-Hermitian Hamiltonian H = \omega a^\dagger a + \chi (a^\dagger a)^2 - i \gamma a a^\dagger + \beta (a^\dagger + a)?
              Here a and a^\dagger are the raising and lowering operators respectively and \omega, \chi, \gamma, and \beta are real parameters.

                  Cite this as: AimPL: Non-Hermitian quantum mechanics and symplectic geometry, available at http://aimpl.org/nhquantumsymp.