## A dispersive bound for three-dimensional Schrödinger operators with zero energy eigenvalues.(English)Zbl 1223.35265

Consider $$V\in L^p(\mathbb{R}^3)\cap L^q(\mathbb{R}^3)$$ for exponents $$p<\frac32<q$$. Here all function spaces are complex and over $$\mathbb{R}^3$$. Denote by $$H:=-\Delta+V$$ the corresponding Schrödinger operator, which may not be symmetric. A resonance $$\Psi$$ of $$H$$ is a distributional solution of $$H\Psi=\lambda^2\Psi$$, for some $$\lambda\in\mathbb{R}$$, such that $$\Psi\in L^3_{\text{weak}}\smallsetminus L^2$$. Denote by $$X_1$$ the set of $$\Psi\in L^2$$ that are solutions of $$H\Psi=0$$, that is, the zero energy eigenfunctions. Define inductively, if $$X_k\subseteq L^1$$, the space $X_{k+1}:=\{\Psi\in L^3_{\text{weak}}\mid H\Psi\in X_k\}.$ The main result is a bound on the $$L^\infty$$-norm of the time evolution of initial values $$f\in L^1$$ away from the generalized eigenspaces of $$H$$, in terms of the $$L^1$$-norm of $$f$$. Suppose that $$H$$ has no resonances, that $$X_k\subseteq L^1$$ for each $$k\in\mathbb{N}$$, and that $$\bigcup_{k\in\mathbb{N}}X_k$$ is finite dimensional. Setting $$P$$ to the sum of all spectral projections to generalized eigenspaces of eigenvalues of $$H$$, there is $$C>0$$ such that $\|e^{-itH}(I-P)f\|_\infty\leq C|t|^{-3/2}\|f\|_1$ for all $$f\in L^1$$ and $\|e^{-itH}(I-P)f\|_2\leq C\|f\|_2$ for all $$f\in L^2$$.

### MSC:

 35Q41 Time-dependent Schrödinger equations and Dirac equations 81U30 Dispersion theory, dispersion relations arising in quantum theory 35J10 Schrödinger operator, Schrödinger equation 47D08 Schrödinger and Feynman-Kac semigroups
Full Text:

### References:

  Agmon S., Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) 2 pp 151– (1975)  Arveson W., A Short Course on Spectral Theory (2002)  DOI: 10.1007/s00220-008-0427-3 · Zbl 1148.35082  DOI: 10.1002/cpa.1018 · Zbl 1031.35129  Erdogan M.B., Dyn. Partial Differ. Equ. 1 pp 359– (2004)  DOI: 10.1007/BF02789446 · Zbl 1146.35324  Goldberg M., Geom. Funct. Anal. 16 pp 517– (2006)  DOI: 10.1016/j.jfa.2008.11.005 · Zbl 1161.35004  DOI: 10.1007/s00220-004-1140-5 · Zbl 1086.81077  DOI: 10.1007/s00039-003-0439-2 · Zbl 1055.35098  DOI: 10.1215/S0012-7094-79-04631-3 · Zbl 0448.35080  DOI: 10.1002/cpa.3160440504 · Zbl 0743.35008  DOI: 10.1353/ajm.1998.0039 · Zbl 0922.35028  DOI: 10.1215/S0012-7094-87-05518-9 · Zbl 0644.35012  Reed M., Methods of Modern Mathematical Physics. II. Fourier Analysis, Self Adjointness (1975) · Zbl 0308.47002  Reed M., Methods of Modern Mathematical Physics. I. Functional Analysis (1980) · Zbl 0459.46001  DOI: 10.1007/s00222-003-0325-4 · Zbl 1063.35035  DOI: 10.1002/cpa.20066 · Zbl 1130.81053  DOI: 10.1142/S0129055X04002175 · Zbl 1111.81313  DOI: 10.2969/jmsj/04730551 · Zbl 0837.35039  DOI: 10.1007/s00220-005-1375-9 · Zbl 1079.81021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.