## Differential and Integral Equations

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356061133

**Mathematical Reviews number (MathSciNet)**

MR1787075

**Zentralblatt MATH identifier**

0989.47036

**Subjects**

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

#### Citation

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