Computing Generalized Eigenfunctions in Rigged Hilbert Spaces

TL;DR

High-order convergence in key topologies is proved, including weak-star convergence for distributional eigenfunctions, uniform convergence on compact sets for locally smooth generalized eigenfunctions, and convergence in seminorms for separable Frechet spaces, covering the majority of physical scenarios.

Abstract

We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions, such as asymptotics or other analytic properties. Instead, we carefully sample the range of the resolvent operator to construct smooth and accurate wave packet approximations to generalized eigenfunctions. We prove high-order convergence in key topologies, including weak-star convergence for distributional eigenfunctions, uniform convergence on compact sets for locally smooth generalized eigenfunctions, and convergence in seminorms for separable Frechet spaces, covering the majority of physical scenarios. The method's performance is illustrated with applications to both differential and integral operators, culminating in the computation of spectral measures and generalized eigenfunctions for an operator associated with Poincare's internal waves problem. These computations corroborate experimental results and highlight the method's utility for a broad range of spectral problems in physics.

Figures (14)

• Figure 1: Branches of the inverse of $p(x)=x^3-x$, denoted $p_k^{-1}$ for $k=1,2$, are plotted in the left panel. The associated spectral measure of the multiplication operator $[Pu](x)=(x^3-x)u(x)$ with respect to $f(x)=(2+x)\cos(2\pi x)$ has singularities at the endpoints and the origin.
• Figure 2: Branches of the inverse of the Fourier multiplier $p(x)=x^2$ associated with the operator $[Au](x)=-u"(x)$. The associated spectral measure of $A$ with respect to $f(x)=\exp(-\pi x^2)$ has a single singularity at the endpoint $\lambda=0$ of the spectrum.
• Figure 3: Approximate generalized eigenfunctions ${u_\lambda^{(\epsilon)}}$ with $\epsilon=0.01$ at $\lambda=0.1$ (blue) and $\lambda=0.01$ (red) are plotted in the left panel. The peaks correspond to locations of intersections of the plot in the left panel of \ref{['fig:mult_op']} with vertical lines at $\lambda$. Notice the "ghost" of the generalized eigenfunction at $x=1$ appearing as $\lambda\rightarrow 0^+$, where the spectral measure $\rho_f$ has a jump discontinuity. The relative error in ${\langle u_\lambda^{(\epsilon)}|\phi\rangle}$ is plotted against $\epsilon>0$ (right panel) for $\lambda=0.1$ (blue), $\lambda = 0.01$ (red), and $\lambda = 0.001$ (yellow) with $\phi=(1+x)\cos(\pi x)$.
• Figure 4: An approximate generalized eigenfunction $u_\lambda^{(\epsilon)}$ with $\epsilon=0.01$ at $\lambda=1$ (blue) is compared point-wise with the generalized eigenfunction $u_\lambda(x)=\cos(x)$ in the left panel (gray). The maximum pointwise relative error in ${u_\lambda^{(\epsilon)}}$ over $x\in[-10,10]$ is plotted against $\epsilon$ (right panel) for $\lambda=10$ (blue), $\lambda = 1$ (red), and $\lambda = 0.1$ (yellow). The error increases as $\lambda$ approaches the endpoint of the spectrum $(\lambda=0)$ where $\rho_f$ is singular.
• Figure 5: Rational kernels of order $m=1,\ldots,6$ with equispaced poles in $[-1+i,1+i]$ are plotted in the left panel. A higher order approximation ($m=3$) of the generalized eigenfunction in \ref{['fig:diff_op_poisson']} improves the point-wise accuracy over a large interval with the same value of $\epsilon=0.01$.

Theorems & Definitions (18)

• Definition 1: $m$th order kernel
• Theorem 1
• proof
• Theorem 2: Weak-$^*$ convergence rates
• proof
• Theorem 3: Uniform convergence on compact sets
• proof
• Theorem 4: Convergence in Suslin spaces
• proof
• Lemma 1