Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on $\mathbb{R}^2$
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917402793476096 |
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| author | Liu, Xuefeng |
| author_facet | Liu, Xuefeng |
| contents | We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator $H=-Δ+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the restricted eigenvalue problem is solved with a rigorous eigenvalue bound, where Liu's eigenvalue bound along with the Composite Enriched Crouzeix--Raviart (CECR) finite element method proposed plays a central role. Two concrete potentials are studied: the radially symmetric ring potential $V_1(x)=(|x|^2-1)^2$ and the Cartesian double-well $V_2(x)=(x_1^2-1)^2+x_2^2$. To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27823 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on $\mathbb{R}^2$ Liu, Xuefeng Numerical Analysis 65N25 We propose a rigorous method for computing two-sided eigenvalue bounds of the Schrödinger operator $H=-Δ+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the restricted eigenvalue problem is solved with a rigorous eigenvalue bound, where Liu's eigenvalue bound along with the Composite Enriched Crouzeix--Raviart (CECR) finite element method proposed plays a central role. Two concrete potentials are studied: the radially symmetric ring potential $V_1(x)=(|x|^2-1)^2$ and the Cartesian double-well $V_2(x)=(x_1^2-1)^2+x_2^2$. To author's knowledge, this paper reports the first rigorous eigenvalue bounds for Schrödinger operators on an unbounded domain. |
| title | Rigorous Eigenvalue Bounds for Schrödinger Operators with Confining Potentials on $\mathbb{R}^2$ |
| topic | Numerical Analysis 65N25 |
| url | https://arxiv.org/abs/2603.27823 |