We deal with the difficulties claimed by the author of [Ann. Phys. 206, 90 (1991)] while solving the Schrödinger equation for the ground states of two-dimensional anharmonic potentials. It is shown that the ground state energy eigenvalues and eigen-functions for the coupled quadratic and quartic potentials can be obtained by making some simple assumptions. Expressions for the energy eigenvalues and the eigenfunctions for the first and second excited states of these systems are also obtained.

We investigate the quasi-exact solutions of an analogous Schrödinger wave equation for two-dimensional non-Hermitian complex Hamiltonian systems within the framework of an extended complex phase space characterized by $x = x_{1} + ip_{3}$, $y = x_{2} + ip_{4}$, $p_{x} = p_{1} + ix_{3}$, $p_{y} = p_{2} + ix_{4}$. Explicit expressions for the energy eigenvalues and eigenfunctions for ground and first excited states of a two-dimensional $\mathcal{PT}$-symmetric sextic potential and some of its variants are obtained. The eigenvalue spectra are found to be real within some parametric domains.

Making use of an ansatz for the eigenfunction, we investigate closed-form solutions of the Schrödinger equation for an even power complex deictic potential and its variant in one dimension. For this purpose, extended complex phase-space approach is utilized and nature of the eigenvalue and the corresponding eigenfunction is determined by the analyticity property of the eigenfunction. The imaginary part of the energy eigenvalue exists only if the potential parameters are complex, whereas it reduces to zero for real coupling parameters and the result coincides with those derived from the invariance of Hamiltonian under $\mathcal{PT}$ operations. Thus, a non-Hermitian Hamiltonian possesses real eigenvalue, if it is $\mathcal{PT}$-symmetric.

An energy eigenvalue expression for a sextic anharmonic interaction potential is derived by solving the Schrödinger equation analytically within the framework of the Nikiforov–Uvarov (NU) method. The resultant expression is then used to calculate the mass spectra of some heavy and light mesons, viz., c$\bar{c}$, b$\bar{b}$ , b$\bar{c}$, c$\bar{s}$, b$\bar{s}$ and b$\bar{q}$, and the results are found to be in good agreement with the other theoretical and experimental studies. The energy eigenvalue expression is further used to compute some important thermodynamic quantities like partition function, specific heat capacity, free energy, mean energy, entropy and magnetisation.