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A ground state alternative for singular Schr\"odinger operators
Analysis of PDEs · Submitted 30 Nov 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(\Omega)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(\Omega;\mathbb{R})$, or there is a sequence $\phi_k\in C_0^{\infty}(\Omega)$ and a function $\phi>0$ satisfying $L\phi=0$ such that $\mathbf{a}[\phi_k]\to 0$, $\phi_k\to\phi$ locally uniformly in $\Omega\setminus\{x_0\}$.
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The abstract, sentence by sentence
Let $\mathbf{a}$ be a quadratic form associated with a Schr\"odinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $\Omega\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(\Omega)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(\Omega;\mathbb{R})$, or there is a sequence $\phi_k\in C_0^{\infty}(\Omega)$ and a function $\phi>0$ satisfying $L\phi=0$ such that $\mathbf{a}[\phi_k]\to 0$, $\phi_k\to\phi$ locally uniformly in $\Omega\setminus\{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $\Omega$. In the latter case, one has an inequality of Poincar\'e type: there exists $W>0$ such that for every $\psi\in C_0^\infty(\Omega;\mathbb{R})$ satisfying $\int \psi \phi dx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u \psi dx|^2$ for all $u\in C_0^\infty(\Omega;\mathbb{R})$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathbf{a}$ be a quadratic form associated with a Schr\"odinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $\Omega\subset \mathbb{R}^d$. | background 0.98 | 0.17 | 0.02 | asserted 0.76 |
| s1 | If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(\Omega)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(\Omega;\mathbb{R})$, or there is a sequence $\phi_k\in C_0^{\infty}(\Omega)$ and a function $\phi>0$ satisfying $L\phi=0$ such that $\mathbf{a}[\phi_k]\to 0$, $\phi_k\to\phi$ locally uniformly in $\Omega\setminus\{x_0\}$. | result 0.97 | 0.55 | 0.76 | asserted 1.00 |
| s2 | This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $\Omega$. | interpretation 0.82 | 0.43 | 0.30 | asserted 1.00 |
| s3 | In the latter case, one has an inequality of Poincar\'e type: there exists $W>0$ such that for every $\psi\in C_0^\infty(\Omega;\mathbb{R})$ satisfying $\int \psi \phi dx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u \psi dx|^2$ for all $u\in C_0^\infty(\Omega;\mathbb{R})$. | result 0.98 | 0.66 | 0.17 | asserted 1.00 |
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Original abstract
Let $\mathbf{a}$ be a quadratic form associated with a Schr\"odinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $\Omega\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(\Omega)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(\Omega;\mathbb{R})$, or there is a sequence $\phi_k\in C_0^{\infty}(\Omega)$ and a function $\phi>0$ satisfying $L\phi=0$ such that $\mathbf{a}[\phi_k]\to 0$, $\phi_k\to\phi$ locally uniformly in $\Omega\setminus\{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $\Omega$. In the latter case, one has an inequality of Poincar\'e type: there exists $W>0$ such that for every $\psi\in C_0^\infty(\Omega;\mathbb{R})$ satisfying $\int \psi \phi dx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u \psi dx|^2$ for all $u\in C_0^\infty(\Omega;\mathbb{R})$.
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