Multiple well anharmonic oscillators and perturbation theory

Highly Accurate Calculation of the Real and Complex Eigenvalues of One-Dimensional Anharmonic Oscillators

“…One of them is an ordinary resonance embedded in the continuum and other one is some kind of strange resonance appearing in the point spectrum close to the ground state. Present results for the lowest resonance in the discrete spectrum of the three-well potential (11) give support to the conjecture that the analytical WKB formula for the resonance width derived several years ago [8] may not be correct. Present results improve considerably upon those reported earlier by Killingbeck [9] and Fernández [2].…”

“…Killingbeck [9] and Fernández [6] calculated this resonance more accurately and for smaller values of g and showed that |ℑE| g 2 e 1/(2g 2 ) does not exhibit a uniform behaviour as suggested by the earlier calculation of Benassi et al [8]. In particular, Killingbeck [9] suggested that |ℑE| g 2 e 1/(2g 2 ) exhibits a maximum.…”

“…Since V (x) behaves asymptotically as g 4k x 4k+2 when |x| → ∞ then one expects only bound states with positive eigenvalues when g is real. However, Benassi et al [8] proved that this family of potentials supports complex eigenvalues with all the properties of actual resonances. They calculated the lowest resonance for k = 1 and several values of g and compared the imaginary part with the WKB transmission coefficient through the barrier |ℑE W KB | ∼ Ag −2 e −1/(2g 2 ) .…”

“…Present calculation of |ℑE| g 2 exp 1/ 2g 2 for the oscillator (11) (solid line) and the results of Beanassi et al[8] (filled circles), Killingbeck[9] (crosses) andFernández[6] (empty circles). log |ℑE res | (circles) and log |ℜE res − E bs | (solid line) vs. g for the oscillator (11) with k = 1.…”

A Gauss elimination method for resonances

“…By varying the imbedded parameter β in the factor e −βx 2 /2 appearing in the Hill-series formalism it was found possible to produce both real and complex eigenvalues for H(g) at small g values. From a physical point of view the resonances can be regarded as referring to the outwards tunnelling of a wavepacket initially sited in the inner well; from a mathematical point of view they are associated with a complex scaled Hamiltonian [8]. The real eigenvalues are just the energies of the traditional bound states which would be expected for a potential which rises towards infinity at large x values.…”