A completely solvable new \(\mathcal{{PT}}\)-symmetric periodic potential with real energies

Non-Hermitian Hamiltonians respecting parity–time symmetry are well known to be associated with real spectrum, so long as \(\mathcal{{PT}}\) symmetry is exact or unbroken, with energies turning to complex conjugate pairs as this symmetry breaks down spontaneously. Such potentials are characterised by an even real part and an odd imaginary part. However, exactly solvable quantum mechanical models are very few. In this work, we conduct an exact analytical study of a new, periodic, \(\mathcal{{PT}}\)-symmetric potential. The energies are observed to be real always, as there is no scope for spontaneous breakdown of \(\mathcal{{PT}}\) symmetry. Using the principles of supersymmetric quantum mechanics, we find its partner Hamiltonian, sharing the same energy spectrum, with the possible exception of the ground state. Incidentally, the partner is also \(\mathcal{{PT}}\) symmetric. Using Mathematica, we plot the exact eigenfunctions of both the partner Hamiltonians. Additionally, supersymmetric quantum mechanics (SUSY QM) helps us to find a totally new exactly solvable, non-trivial Hamiltonian, with the same energy spectrum.