The Schrödinger operator with Morse potential on the right half line

Details The Schrödinger operator with Morse potential on the right half line The Schrödinger operator with Morse potential on the right half line

2007-12-19

arxiv.org/abs/0712.3238v6

Communications in Number Theory and Physics 3 (2009), No.2, 323--361

Mathematics

Spectral Theory

33 pages; v2 and v3 introduction revised, Polya and other refs. added, v4,v5 intro revised, typos corrected, v6 corrections to

Scientific paper

This paper studies the Schr\"odinger operator with Morse potential on a right half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains information on zeros of the Whittaker function $W_{\kappa, \mu}(x)$ for fixed real parameters $\kappa, x$, with x positive, viewed as an entire function of the complex variable $\mu$. In this case all zeros lie on the imaginary axis, with the exception, if $k<0$, of a finite number of real zeros. We obtain an asymptotic formula for the number of zeros of modulus at most T of form $N(T) = (2/\pi) T \log T + f(u) T + O(1)$. Some parallels are noted with zeros of the Riemann zeta function.

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