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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

#### 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

#### 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

#### 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.