Let $d \in \{3,4,5,\ldots\}$. Consider $L = -\frac{1}{w} \, \operatorname{div}(A \, \nabla u) + \mu$ over its maximal domain in $L^2_w(\mathbb{R}^d)$. Under certain conditions on the weight $w$, the coefficient matrix $A$ and the positive Radon measure $\mu$ we obtain upper and lower bounds on $N(\lambda,L)$–the number of eigenvalues of $L$ that are at most $\lambda \ge 1$. Furthermore we show that the eigenfunctions of $L$ corresponding to those eigenvalues are exponentially decaying. In the course of proofs, we develop generalized Poincaré and weighted Young convolution inequalities as the main tools for the analysis.
Tan Duc Do; Le Xuan Truong. Spectral asymptotics for generalized Schrödinger operators. Annales Fennici Mathematici, Tome 48 (2023) no. 2, pp. 703-727. doi: 10.54330/afm.140863
@article{AFM_2023_48_2_a11,
author = {Tan Duc Do and Le Xuan Truong},
title = {Spectral asymptotics for generalized {Schr\"odinger} operators},
journal = {Annales Fennici Mathematici},
pages = {703--727},
year = {2023},
volume = {48},
number = {2},
doi = {10.54330/afm.140863},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.54330/afm.140863/}
}
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AU - Tan Duc Do
AU - Le Xuan Truong
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JO - Annales Fennici Mathematici
PY - 2023
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EP - 727
VL - 48
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.54330/afm.140863/
DO - 10.54330/afm.140863
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