Resonance computations and nonlinear eigenproblems

In this section we describe the ideas behind the codes for squarepot.m and splinepot.m.

If the support of $V$ is contained in a compact interval $[-L, L]$, we can compute both resonance solutions and ordinary eigenvalues of the Schrödinger problem (8) by writing appropriate boundary conditions at $\pm L$:

$$\begin{array}{ll} (H_V - \lambda^2) u = 0 & \mbox{ for } x \i... ...\partial_x - i \lambda) u = 0 & \mbox{ at } x = -L. \end{array}$$ (30)

In terms of $\lambda$, this is a quadratic eigenvalue problem. We can introduce a new variable $\psi = \lambda u$ to convert this problem to a linear eigenvalue problem in two fields:

$$\begin{array}{ll} H_V u - \lambda \psi = 0 & \mbox{ for } x \... ...\partial_x + i \lambda) u = 0 & \mbox{ at } x = -L. \end{array}$$ (31)

We now discretize the boundary and domain operators to get a finite-dimensional generalized eigenvalue problem. For small discretizations with up to a few hundred unknowns, we can solve this generalized eigenvalue problem using MATLAB's eig command, which uses the dense eigensolvers in LAPACK [2]. For larger discretizations, we use MATLAB's eigs to call ARPACK, a standard Arnoldi-based iterative eigensolver [19].

For the calculations shown in this note, we used a high-order pseudospectral collocation method to discretize the operators [33], [5]. We partition the support interval $[-L, L]$ into subintervals, and approximate $u$ by a high-order polynomial on each subinterval. At the Chebyshev points on the interior of each subinterval, we insist that the domain differential equations be satisfied exactly, while at the junctions between neighboring intervals, we insist that the solution $u$ and the first derivative $\partial_x u$ must both be continuous. Assuming that the potential is smooth except possibly at the endpoints of the subintervals, the collocation scheme we use is spectrally accurate; that is, the error asymptotically decreases faster than any algebraic function of the order of the collocation scheme. As a simple check on the accuracy of the computed eigenvalues of (31), we increase the order of the method by 50%, recompute the eigenvalues, and compare the results obtained from the coarser and the finer discretization.

We can write the analogue of (30) in higher dimensions, with a Dirichlet-to-Neumann (DtN) map - or some approximation to a DtN map - in place of the boundary conditions at $\pm L$. In more than one space dimension, this boundary map ceases to be a linear function of $\lambda$, and so we cannot easily convert the problem into a linear eigenvalue problem. Researchers are studying these more complicated nonlinear eigenvalue problems for a variety of engineering problems [3]. Many of these problems involve resonances in models of elastic, acoustic, or electromagnetic resonators with radiation losses.