# Resonant Excitation of Oscillator with Randomly Shifted Levels

## TL;DR

This paper investigates how a harmonic oscillator with randomly shifted energy levels responds to resonant excitation, revealing a threshold for excitation and applying the method to electron localization in variable cross-section wires.

## Contribution

It introduces a threshold-based analysis of resonant excitation in oscillators with random level shifts and extends the approach to electron localization in non-uniform wires.

## Key findings

- Existence of a threshold resonance field magnitude for excitation.
- Below threshold, excitation remains localized on lower levels.
- Above threshold, indefinite excitation occurs, requiring dissipation considerations.

## Abstract

The problem of resonant excitation of a harmonic oscillator the energy levels of which are slightly shifted under the action of a random potential is solved. It is shown that, in this case, there exists a threshold magnitude of the exciting resonance field, below which the excitation is localized on lower levels, and above which the oscillator is indefinitely excited so that it is necessary to take into account dissipative processes. A method similar to that developed for the oscillator is applied to examine the localization of electrons in a wire with cross-section varying along its length. It is shown, in particular, that there is no localization if this variation is superlinear.