Solution of the quantum harmonic oscillator plus a delta-function   potential at the origin: The oddness of its even-parity solutions
  We derive the energy levels associated with the even-parity wave functions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of solution of the harmonic oscillator problem. 
J. Viana-Gomes and N. M. R. Peres
quant-ph physics.ed-ph