Crossover phenomena and resonances in quantum systems

The finite-size scaling method is used to calculate the critical parameters of a two-parameter model Hamiltonian that exhibit resonances. We show the existence of a crossover phenomenon for the energy spectrums the transition from a bound state to a continuum is a "second-order phase transition" in one region and a "first-order phase transition" in another region. As the parameters varied. the numerical value of the critical exponent of the energy levels changed from two to one. In the zone adhere the critical exponent equals one, the system has narrow resonances and they disappear when the exponent is two. The method has potential applicability in predicting stable and metastable atomic and molecular states. We also show that finite-size scaling methods are capable if detecting multicritical points.