SPECTRAL PROPERTIES OF THE BICONFLUENT HEUN DIFFERENTIAL-EQUATION

The spectrum of zeros of the polynomial solutions of the biconfluent Heun differential equation is investigated by two different methods. First, the spectral Newton sums (i.e., the sums of the rth powers of the zeros) are given in a rigorous and recurrent way. Second, the density of zeros (i.e., the number of zeros per unit interval) is calculated in an explicit way within the so-called semiclassical or BKW approximation; this is done by using a general theorem which applies to a linear second-order differential equation under certain conditions. Applications to the radial Schrodinger equations associated to some quantum mechanical systems (rotating harmonic oscillator, confinement potentials) as well as to the Bessel differential equation are also shown.