## Differential and Integral Equations

- Differential Integral Equations
- Volume 30, Number 5/6 (2017), 329-386.

### Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

William R. Green and Ebru Toprak

#### Abstract

We investigate dispersive estimates for the Schrödinger operator $H=-\Delta +V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion \begin{align*} e^{itH}\chi(H) P_{ac}(H) & =O(\frac 1{\log t}) A_0+ O(\frac 1 t )A_1+O((t\log t)^{-1})A_2 \\ & + O(t^{-1}(\log t)^{-2})A_3. \end{align*} Here, $A_0,A_1:L^1(\mathbb R^4)\to L^\infty (\mathbb R^4)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

#### Article information

**Source**

Differential Integral Equations, Volume 30, Number 5/6 (2017), 329-386.

**Dates**

First available in Project Euclid: 18 March 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1489802418

**Mathematical Reviews number (MathSciNet)**

MR3626580

**Zentralblatt MATH identifier**

06738553

**Subjects**

Primary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)

#### Citation

Green, William R.; Toprak, Ebru. Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy. Differential Integral Equations 30 (2017), no. 5/6, 329--386. https://projecteuclid.org/euclid.die/1489802418