Perfect Quantum State Revivals: Designing Arbitrary Potentials with Specified Energy Levels

It is known that there exist a limited number of analytic potentials with the unusual property that any bound quantum state therein will be periodic in time. This is known as a perfect quantum state revival. Examples of such potentials are the infinite well, quantum harmonic oscillator and the P\"oschl-Teller potentials; here, we present a general method of designing such potentials. A key requirement is that their energy eigenvalues have integer spacings (up to a prefactor). We first analyze the required conditions which permit quantum state revivals for potentials in general, and then we use techniques of iterated Hamiltonian intertwining to construct potentials exhibiting perfect quantum revivals. Our method can readily be extended to multiple dimensions.