تقرير
On the Ground State Energies of Discrete and Semiclassical Schr\'odinger Operators
| العنوان: | On the Ground State Energies of Discrete and Semiclassical Schr\'odinger Operators |
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| المؤلفون: | Detherage, Isabel, Srivastava, Nikhil, Stier, Zachary |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Spectral Theory, Mathematical Physics, Mathematics - Functional Analysis |
| الوصف: | We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $\theta\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schr\"odinger operator on $\mathbb{R}^d$ with parameter $\theta$, arising from the same choice of potential. We show that: the discrete g.s.e. is at most the continuous one for continuous periodic $V$ and irrational $\theta$; the opposite inequality holds up to a factor of $1-o(1)$ as $\theta\rightarrow 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $\theta$ with the property that discrete and continuous averages of $V$ on fundamental domains of $\theta \mathbb{Z}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $\theta \mathbb{Z}^d$ to low-energy functions for the continuous operator on $\mathbb{R}^d$, and vice versa. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.05907 |
| رقم الأكسشن: | edsarx.2405.05907 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |