One-dimensional hydrogen atom: A singular potential in quantum mechanics

A generalized Laplace transform approach is developed to study the eigenvalue problem of the one-dimensional singular potential V = -e2/\x\. Matching of solutions at the origin that has been a matter of much controversy is, thereby, made redundant. A discrete and non-degenerate bound-state spectrum results. Existing arguments in the literature that advocate (a) a continuous spectrum, (b) a degeneracy of energy levels as a result of a hidden O(2) symmetry, (c) an infinite negative energy state and (d) an impenetrable barrier at the origin are found to be untenable. It is argued that a judicious use of generalized functions, coupled with some classical considerations, enables the conventional method of solving the problem to recover precisely the same results which are shown to be in accord with an accurate semiclassical analysis of the problem

ENERGY-SPECTRUM OF THE POTENTIAL V = AX2 + X4

Suitable sequences of quasi-exactly solvable Hamiltonians are shown to provide stringent upper bounds to the energy eigenvalues of the bound state potential V = ax2 + x4. Procedures to convert these bounds into even further improved energy estimates are developed. For the quartic anharmonic oscillator (a > 0) case a simple argument is provided to indicate that the conventional small-parameter energy expansion does not converge as a Taylor series. An accurate closed-form parametrization of the entire quartic (a = 0) spectrum is noted. The energy difference between the lowest-lying levels of a quartic double well (a < 0) is satisfactorily recovered and for deep wells a useful expression is deduced for it empirically

The Coulomb-diamagnetic problem in two dimensions

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