# Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type theorems

## TL;DR

This paper develops a unified framework for understanding how the growth of generalized eigenfunctions relates to spectral properties of Schrödinger operators on non-compact Riemannian manifolds, extending previous results with diverse examples.

## Contribution

It unifies and generalizes existing theorems connecting eigenfunction growth conditions to spectral inclusion for generalized Schrödinger operators.

## Key findings

- Provides a comprehensive set of growth conditions linking eigenfunctions to spectrum
- Includes diverse examples illustrating different growth behaviors
- Extends classical results to more general geometric settings

## Abstract

Let $H$ be a generalized Schr\"odinger operator on a domain of a non-compact connected Riemannian manifold, and a generalized eigenfunction $u$ for $H$: that is, $u$ satisfies the equation $Hu=\lambda u$ in the weak sense but is not necessarily in $L^2$. The problem is to find conditions on the growth of $u$, so that $\lambda$ belongs to the spectrum of $H$. We unify and generalize known results on this problem. In addition, a variety of examples is provided, illustrating the different nature of the growth conditions.