Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

Details Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

2004-11-07

arxiv.org/abs/math/0411143v1

Mathematics

Spectral Theory

31 pages, 1 figure

Scientific paper

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problem $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $\lambda_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues. Also, we show for arbitrary $\mathcal{PT}$-symmetric polynomial potentials of degree $m\geq 3$ and all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.

Affiliated with

Also associated with

No associations

Rating

Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-25267