Decay estimates for one-dimensional wave equations with inverse power potentials

Costin, O.; Huang, M.

doi:10.1090/S0002-9947-2014-06345-9

Decay estimates for one-dimensional wave equations with inverse power potentials
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by O. Costin and M. Huang PDF

Trans. Amer. Math. Soc. 367 (2015), 3705-3732 Request permission

Abstract:

We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$, where $m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $t^{-m}$ for large $t$, which is consistent with existing mathematical and physical literature under slightly different assumptions.

Our results can be generalized to potentials consisting of a finite sum of inverse powers, the largest of which being $const.x^{-\alpha }$, where $\alpha >2$ is a real number, as well as potentials of the form $const.x^{-m}+O( x^{-m-\delta _1})$ with $\delta _1>3$.

References

Lars Andersson, Pieter Blue, and Jean-Philippe Nicolas, A decay estimate for a wave equation with trapping and a complex potential, Int. Math. Res. Not. IMRN 3 (2013), 548–561. MR 3021792, DOI 10.1093/imrn/rnr237
Michael Beals, Optimal $L^\infty$ decay for solutions to the wave equation with a potential, Comm. Partial Differential Equations 19 (1994), no. 7-8, 1319–1369. MR 1284812, DOI 10.1080/03605309408821056
P. Bizon, T. Chmaj, and A. Rostworowski, Anomalously small wave tails in higher dimensions. Physical Review D (Particles, Fields, Gravitation, and Cosmology) 76 (2007), no. 12, 124035.
E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, Wave propagation in gravitational systems: Late time behavior, Physical Review D (Particles, Fields, Gravitation, and Cosmology) 52 (1995), no. 4, 2118–2132.
O. Costin and M. Huang, Gamow vectors and Borel summability in a class of quantum systems, J. Stat. Phys. 144 (2011), no. 4, 846–871. MR 2826622, DOI 10.1007/s10955-011-0276-x
O. Costin, J. L. Lebowitz, and S. Tanveer, Ionization of Coulomb systems in $\Bbb R^3$ by time periodic forcings of arbitrary size, Comm. Math. Phys. 296 (2010), no. 3, 681–738. MR 2628820, DOI 10.1007/s00220-010-1023-x
Roland Donninger and Wilhelm Schlag, Decay estimates for the one-dimensional wave equation with an inverse power potential, Int. Math. Res. Not. IMRN 22 (2010), 4276–4300. MR 2737771, DOI 10.1093/imrn/rnq038
Piero D’ancona and Vittoria Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal. 227 (2005), no. 1, 30–77. MR 2165087, DOI 10.1016/j.jfa.2005.05.013
Piero D’Ancona and Luca Fanelli, $L^p$-boundedness of the wave operator for the one dimensional Schrödinger operator, Comm. Math. Phys. 268 (2006), no. 2, 415–438. MR 2259201, DOI 10.1007/s00220-006-0098-x
Roland Donninger, Wilhelm Schlag, and Avy Soffer, A proof of Price’s law on Schwarzschild black hole manifolds for all angular momenta, Adv. Math. 226 (2011), no. 1, 484–540. MR 2735767, DOI 10.1016/j.aim.2010.06.026
Michael Goldberg, Transport in the one-dimensional Schrödinger equation, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3171–3179. MR 2322747, DOI 10.1090/S0002-9939-07-08897-1
W. Schlag, Dispersive estimates for Schrödinger operators: a survey, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 255–285. MR 2333215
Robert S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis 5 (1970), 218–235. MR 0257581, DOI 10.1016/0022-1236(70)90027-3
Robert S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. MR 256219, DOI 10.1090/S0002-9947-1970-0256219-1
W Wasow, Asymptotic expansions for ordinary differential equations, Interscience Publishers, 1968.