## Dispersive estimates for Schrödinger operators in dimensions one and three.(English)Zbl 1086.81077

Summary: We consider $$L^{1}\rightarrow L^{\infty}$$ estimates for the time evolution of Hamiltonians $$H = - \Delta + V$$ in dimensions $$d=1$$ and $$d=3$$ with bound $$t^{-\frac d2}$$. We require decay of the potentials but no regularity. In $$d =1$$ the decay assumption is $$\int (1+| x|)| V(x)| dx <\infty$$, whereas in $$d=3$$ it is $$| V(x)\}| \leq C(1+| x|)^{-3-}$$.

### MSC:

 81U05 $$2$$-body potential quantum scattering theory 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35P15 Estimates of eigenvalues in context of PDEs 47E05 General theory of ordinary differential operators 47F05 General theory of partial differential operators 47N50 Applications of operator theory in the physical sciences
Full Text:

### References:

 [1] Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151-218 (1975) · Zbl 0315.47007 [2] Artbazar, G., Yajima, K.: The Lp-continuity of wave operators for one dimensional Schrödinger operators. J. Math. Sci. Univ. Tokyo 7(2), 221-240 (2000) · Zbl 0976.34071 [3] Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. XXXII, 121-251 (1979) · Zbl 0388.34005 [4] Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions results in L2(Rm), m? 5. Duke Math. J. 47(1), 57-80 (1980) · Zbl 0437.47009 [5] Jensen, A.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Results in L2(R4). J. Math. Anal. Appl. 101(2), 397-422 (1984) · Zbl 0564.35024 [6] Jensen, A., Kato, T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583-611 (1979) · Zbl 0448.35080 [7] Jensen, A., Nenciu, G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717-754 (2001) · Zbl 1029.81067 [8] Journé, J.-L., Soffer, A., Sogge, C.D.: Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44(5), 573-604 (1991) · Zbl 0743.35008 [9] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258-279 (1965/1966) · Zbl 0139.31203 [10] Katznelson, Y.: An introduction to harmonic analysis. New York: Dover, 1968 · Zbl 0169.17902 [11] Rauch, J.: Local decay of scattering solutions to Schrödinger?s equation. Commun. Math. Phys. 61(2), 149-168 (1978) · Zbl 0381.35023 [12] Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York-London: Academic Press [Harcourt Brace Jovanovich, Publishers], 1978 · Zbl 0401.47001 [13] Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155, 451-513 (2004) · Zbl 1063.35035 [14] Schlop, W.: Dispersive estimates for Schrödinger operators in dimension two, preprint 2004, to appear in Comm. Math. Phys. [15] Weder, R.: estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential. J. Funct. Anal. 170(1), 37-68 (2000) · Zbl 0943.34070 [16] Weder, R.: The Wk,p-continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208(2), 507-520 (1999) · Zbl 0945.34070 [17] Yajima, K.: The Wk,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47(3), 551-581 (1995) · Zbl 0837.35039 [18] Yajima, K.: Lp-boundedness of wave operators for two-dimensional Schrödinger operators. Commun. Math. Phys. 208(1), 125-152 (1999) · Zbl 0961.47004
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.