Department of Physics, Washington University,

St. Louis, Missouri 63130, USA

E-mail: qwang@hbar.wustl.edu

**Calculation of the operator C in PT-symmetric
quantum mechanics**

**Abstract:**

It has been shown that if a Hamiltonian H has an unbroken *PT* symmetry, then it also possesses a hidden symmetry represented by the linear operator *C* [1]. The operator *C* commutes with both H and *PT*. The inner product with respect to *CPT* is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary.

We will give a formal derivation of the operator *C*. Then we will construct the operator *C* for two different types of non-Hermitian *PT*-symmetric Hamiltonians. The first is H = ^{1}/_{2}p^{2}+^{1}/_{2}x^{2}+iex^{3}. We use perturbative techniques to calculate *C* to the third order in e for this theory. The second is H = ^{1}/_{2}p^{2}+^{1}/_{2}x^{2}-ex^{4}. For this theory nonperturbative methods must be used [2].

**References:**

- C. M. Bender, D. C. Brody and H. F. Jones, Phys. Rev. Lett. V.89, 270402 (2002).
- C. M. Bender, P. N. Meisinger and Q. Wang, J. Phys. A: Math. Gen. V.36, 1973-1983 (2003).