PT-Symmetric Matrix Quantum Mechanics

Abstract

Recently developed methods for PT-symmetric models are applied to quantum-mechanical matrix models. We consider in detail the case of potentials of the form $V=-(g/N^{p/2-1})Tr(iM)^{p}$ and show how the calculation of all singlet wave functions can be reduced to solving a one-dimensional PT-symmetric model. The large-N limit of this class of models exists, and properties of the lowest-lying singlet state can be computed using WKB. For $p=3,4$, the energy of this state for small values of $N$ appears to show rapid convergence to the large-N limit. For the special case of $p=4$, we extend recent work on the $-gx^{4}$ potential to the matrix model: we show that the PT-symmetric matrix model is equivalent to a hermitian matrix model with a potential proportional to $+(4g/N)Tr\Pi^{4}$. However, this hermitian equivalent model includes an anomaly term $\hbar\sqrt{2g/N}Tr\Pi$. In the large-N limit, the anomaly term does not contribute at leading order to the properties of singlet states.