On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators
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arXiv
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| Natura: | Preprint |
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2024
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| author | Detherage, Isabel Srivastava, Nikhil Stier, Zachary |
| author_facet | Detherage, Isabel Srivastava, Nikhil Stier, Zachary |
| contents | We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schrödinger operator on $θ\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $θ\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schrödinger operator on $\mathbb{R}^d$ with parameter $θ$, 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 $θ$; the opposite inequality holds up to a factor of $1-o(1)$ as $θ\rightarrow 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $θ$ with the property that discrete and continuous averages of $V$ on fundamental domains of $θ\mathbb{Z}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $θ\mathbb{Z}^d$ to low-energy functions for the continuous operator on $\mathbb{R}^d$, and vice versa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators Detherage, Isabel Srivastava, Nikhil Stier, Zachary Spectral Theory Mathematical Physics Functional Analysis We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schrödinger operator on $θ\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $θ\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schrödinger operator on $\mathbb{R}^d$ with parameter $θ$, 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 $θ$; the opposite inequality holds up to a factor of $1-o(1)$ as $θ\rightarrow 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $θ$ with the property that discrete and continuous averages of $V$ on fundamental domains of $θ\mathbb{Z}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $θ\mathbb{Z}^d$ to low-energy functions for the continuous operator on $\mathbb{R}^d$, and vice versa. |
| title | On the Ground State Energies of Discrete and Semiclassical Schrödinger Operators |
| topic | Spectral Theory Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2405.05907 |