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 \(f\in L^2\).


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: DOI arXiv


[1] Agmon S., Ann. Sc. Norm. Super. Pisa. Cl. Sci. (4) 2 pp 151– (1975)
[2] Arveson W., A Short Course on Spectral Theory (2002)
[3] DOI: 10.1007/s00220-008-0427-3 · Zbl 1148.35082
[4] DOI: 10.1002/cpa.1018 · Zbl 1031.35129
[5] Erdogan M.B., Dyn. Partial Differ. Equ. 1 pp 359– (2004)
[6] DOI: 10.1007/BF02789446 · Zbl 1146.35324
[7] Goldberg M., Geom. Funct. Anal. 16 pp 517– (2006)
[8] DOI: 10.1016/j.jfa.2008.11.005 · Zbl 1161.35004
[9] DOI: 10.1007/s00220-004-1140-5 · Zbl 1086.81077
[10] DOI: 10.1007/s00039-003-0439-2 · Zbl 1055.35098
[11] DOI: 10.1215/S0012-7094-79-04631-3 · Zbl 0448.35080
[12] DOI: 10.1002/cpa.3160440504 · Zbl 0743.35008
[13] DOI: 10.1353/ajm.1998.0039 · Zbl 0922.35028
[14] DOI: 10.1215/S0012-7094-87-05518-9 · Zbl 0644.35012
[15] Reed M., Methods of Modern Mathematical Physics. II. Fourier Analysis, Self Adjointness (1975) · Zbl 0308.47002
[16] Reed M., Methods of Modern Mathematical Physics. I. Functional Analysis (1980) · Zbl 0459.46001
[17] DOI: 10.1007/s00222-003-0325-4 · Zbl 1063.35035
[18] DOI: 10.1002/cpa.20066 · Zbl 1130.81053
[19] DOI: 10.1142/S0129055X04002175 · Zbl 1111.81313
[20] DOI: 10.2969/jmsj/04730551 · Zbl 0837.35039
[21] 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.