A quantum-mechanical anharmonic oscillator with a most interesting spectrum

摘要

AbstractWe revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameterλ. The ground state can be obtained exactly and its energyE0=1is independent ofλ. This solution is valid only forλ0because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Padé and Borel–Padé summable forλ0. Whenλ/mml:mo>0the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyse by means of the expectation values?x2?.Highlights?A quasi-exactly solvable anharmonic oscillator that depends on a parameter.?The perturbation series for the lowest eigenvalue converges for positive values of the parameter.?The perturbation series for the eigenfunction and expectation values are divergent.?The perturbation series are Padé and Borel–Padé summable for positive values of the parameter.?For negative values of the parameter there is an infinite number of avoided crossings.]]>