Event Title
Approximations of spectra of Schr\"odinger operators with complex potentials on $\mathbb{R}^d$
Location
Cupples I Room 218
Start Date
7-19-2016 5:00 PM
End Date
19-7-2016 5:20 PM
Description
We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega \subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $\Omega_n$, and of boundary conditions on $\partial \Omega_n$ such as mixed Dirichlet/Robin type. In particular, $\Re Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Our results are illustrated by numerical computations for several examples, such as complex harmonic and cubic oscillators for $d=1,2,3$. The talk is based on a joint work with S. B\"ogli and C. Tretter.
Approximations of spectra of Schr\"odinger operators with complex potentials on $\mathbb{R}^d$
Cupples I Room 218
We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$ with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains $\Omega \subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains $\Omega_n$, and of boundary conditions on $\partial \Omega_n$ such as mixed Dirichlet/Robin type. In particular, $\Re Q$ need not be bounded from below and $Q$ may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Our results are illustrated by numerical computations for several examples, such as complex harmonic and cubic oscillators for $d=1,2,3$. The talk is based on a joint work with S. B\"ogli and C. Tretter.