Quantum harmonic oscillators with nonlinear effective masses in the weak density approximation

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions in the weak density approximation, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schrödinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.