Differential and Integral Equations

Fredholm properties of Schrödinger operators in $L^P(\mathbbR^N)$

P. J. Rabier and C. A. Stuart

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We consider real potentials $V$ such that the Schrödinger operator $-\Delta+V$ maps the Sobolev space $W^{2,p}(\mathbb{R}^{N})$ continuously into $L^{p}(\mathbb{R}^{N})$ for a range of values of $p$ which includes 2. Let $\sigma_{e}$ denote the essential spectrum of $-\Delta+V$ as a self-adjoint operator in $L^{2}(\mathbb{R}^{N}).$ If $\lambda\notin$ $\sigma_{e},$ we show that for all $p$ in the range considered, $-\Delta+V-\lambda:W^{2,p}% (\mathbb{R}^{N})\rightarrow L^{p}(\mathbb{R}^{N})$ is a Fredholm operator of index zero, that ker {$-\Delta+V-\lambda$\} is independent of $p$ and that $L^{p}(\mathbb{R}^{N})=$ker {$-\Delta+V-\lambda$\}$\oplus$\{$-\Delta +V-\lambda$\}$W^{2,p}(\mathbb{R}^{N}).$

Article information

Differential Integral Equations, Volume 13, Number 10-12 (2000), 1429-1444.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)
Secondary: 35J10: Schrödinger operator [See also 35Pxx] 35Q40: PDEs in connection with quantum mechanics


Rabier, P. J.; Stuart, C. A. Fredholm properties of Schrödinger operators in $L^P(\mathbbR^N)$. Differential Integral Equations 13 (2000), no. 10-12, 1429--1444. https://projecteuclid.org/euclid.die/1356061133

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