Spectral Theory for Schrodinger Operators with Interface Transmission Conditions
Abstract
Interfaces influence transport across biological membranes, habitat boundaries, and composite media. Motivated by these applications, we study two classes of Schrodinger eigenvalue problems with transmission conditions allowing jumps in the solution or its normal derivative. We develop a weighted variational framework that accommodates asymmetric interface coupling and yields self-adjoint realizations in suitable weighted Hilbert spaces. For both problems and their adjoints, we establish discrete spectra, variational eigenvalue characterizations, and eigenfunction completeness in arbitrary spatial dimensions. We prove spectral continuity and identify limiting spectra in singular parameter regimes. We also extend Courant's nodal domain theorem to eigenfunctions that may be discontinuous across the interface. The spectral results further extend to divergence-form elliptic operators under suitable assumptions on the coefficients. These results provide a foundation for analyzing elliptic and parabolic equations with interface transmission conditions.
Related
- Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators
- Smooth spectral statistics of random perturbations of Schrödinger operators near Anosov energy levels
- Generic Spectral Determination of Semiclassical Schrödinger Operators with ℤ2-Symmetry
- Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators
- ℛ-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators
BibTeX
@misc{su2026spectral,
title = {{Spectral Theory for Schrodinger Operators with Interface Transmission Conditions}},
author = {Linlin Su and Yantao Wang and Zhannan Zhuang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21543},
eprint = {2610.21543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21543},
}
RIS
TY - UNPB
TI - Spectral Theory for Schrodinger Operators with Interface Transmission Conditions
AU - Linlin Su
AU - Yantao Wang
AU - Zhannan Zhuang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21543
UR - https://arxiv.org/abs/2610.21543
AB - Interfaces influence transport across biological membranes, habitat boundaries, and composite media. Motivated by these applications, we study two classes of Schrodinger eigenvalue problems with transmission conditions allowing jumps in the solution or its normal derivative. We develop a weighted variational framework that accommodates asymmetric interface coupling and yields self-adjoint realizations in suitable weighted Hilbert spaces. For both problems and their adjoints, we establish discrete spectra, variational eigenvalue characterizations, and eigenfunction completeness in arbitrary spatial dimensions. We prove spectral continuity and identify limiting spectra in singular parameter regimes. We also extend Courant's nodal domain theorem to eigenfunctions that may be discontinuous across the interface. The spectral results further extend to divergence-form elliptic operators under suitable assumptions on the coefficients. These results provide a foundation for analyzing elliptic and parabolic equations with interface transmission conditions.
ER -