# Perturbation Theory and Generalizations of Aleksandrov-Clark Theory

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Delaware (DE) · $266,853

## Abstract

Many closed physical systems are modeled mathematically by self-adjoint differential operators.  Analysis of the spectrum is of central importance in understanding the long-term behavior of such systems. In some examples, the spectrum can be interpreted of as the color of light, or the energy range at which a semi-conductor allows electron travel. To obtain differential operators that are self-adjoint, boundary conditions must usually be imposed; think of a vibrating string that is clamped down or otherwise restricted at both ends, or a vibrating drum. In many examples, we know the complete spectrum of an operator with one set of boundary conditions, while the physical behavior of the same system with other boundary conditions is hard or impossible to compute directly. Perturbation theory will be developed to allow access to the spectral information to a wide class of operators under changing boundary conditions, even though this behavior is often quite unstable. The project will provide research opportunities for graduate students and postdoctoral researchers and engage the PI in conference organization; outcomes of the project will be disseminated widely through research articles and conference presentations.

More specifically, the two parts of this project’s intellectual merit are trace class perturbations of self-adjoint operators on Hilbert space, and Aleksandrov-Clark theory on the polydisk. The first part concerns the singular spectrum under infinite rank perturbati

## Key facts

- **NSF award ID:** 2452894
- **Awardee organization:** University of Delaware (DE)
- **SAM.gov UEI:** T72NHKM259N3
- **PI:** Constanze D Liaw
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** EXP PROG TO STIM COMP RES
- **Estimated total:** $266,853
- **Funds obligated:** $266,853
- **Transaction type:** Standard Grant
- **Period:** 07/01/2025 → 06/30/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2452894

## Citation

> US National Science Foundation, Award 2452894, Perturbation Theory and Generalizations of Aleksandrov-Clark Theory. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2452894. Licensed CC0.

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