Uncertainty principles and singular potentials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908973132677120 |
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| author | Iosevich, A. Park, C. |
| author_facet | Iosevich, A. Park, C. |
| contents | We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-ε-ε')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15442 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uncertainty principles and singular potentials Iosevich, A. Park, C. Classical Analysis and ODEs Analysis of PDEs Spectral Theory 42B10, 42B37, 58J40 We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-ε-ε')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds. |
| title | Uncertainty principles and singular potentials |
| topic | Classical Analysis and ODEs Analysis of PDEs Spectral Theory 42B10, 42B37, 58J40 |
| url | https://arxiv.org/abs/2604.15442 |