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For quantum systems with competing potentials, the conventional perturbationtheory often yields an asymptotic series and the subsequent numerical outcomebecomes uncertain. To tackle such kind of problems, we develop a generalsolution scheme based on a new energy dissection idea. Instead of dividing thepotential energy into "unperturbed" and "perturbed" terms, a partition of thekinetic energy is performed. By distributing the kinetic energy term in partinto each individual potential, the Hamiltonian can be expressed as the sum ofthe subsystem Hamiltonians with respective competing potentials. The totalwavefunction is expanded by using a linear combination of the basis sets ofrespective subsystem Hamiltonians. We first illustrate the solution procedureusing a simple system consisting of a particle under the action of doubledelta-function potentials. Next, this method is applied to the prototypesystems of a charged harmonic oscillator in strong magnetic field and thehydrogen molecule ion. Compared with the usual perturbation approach, this newscheme converges much faster to the exact solutions for both eigenvalues andeigenfunctions. When properly extended, this new solution scheme can be veryuseful for dealing with strongly coupling quantum systems.

Split Kinetic Energy Method for Quantum Systems with Competing Potentials