### Gevrey-Type Resolvent Estimates at the Threshold for a Class of Non-Selfadjoint Schrödinger Operators

#### Abstract

*V*(

*x*), the derivatives of the resolvent of the non-selfadjoint Schröinger operator

*H*= −∆ +

*V*(

*x*) satisfy some Gevrey estimates at the threshold zero. As applications, we establish subexponential time-decay estimates of local energies for the semigroup

*e*

^{−tH},

*t*> 0. We also show that for a class of Witten Laplacians for which zero is an eigenvalue embedded in the continuous spectrum, the solutions to the heat equation converges subexponentially to the steady solution.

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J. Aguilar, J.M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians. Comm. Math. Phys. 22 (1971), 269-279.

L. Desvillettes, C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Commun. Pure Appl. Math., LIV(2001), 1-42.

B. Helffer, A. Martinez, Comparaison entre les diverses notions de résonances. (French) [Comparison among the various notions of resonance] Helv. Phys. Acta 60 (1987), no. 8, 992-1003.

B. Helffer, F. Nier, Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians. Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005. x+209 pp. ISBN: 3-540-24200-7

F. Hérau, F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151-218.

I. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Commun. in Math. Phys., 53(1977)(3), 285-294.

S. Nakamura, Low energy asymptotics for Schrödinger operators with slowly decreasing potentials, Commun. in Math. Phys., 161(1994), 63-76.

C. Villani, Hypocoercivity. Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141 pp. ISBN: 978-0- 8218-4498-4

X.P. Wang, Asymptotic behavior of resolvent of N-body Schrödinger operator near a threshold, Ann H. Poincaré, 4(2003), 553-600.

X.P. Wang, Asymptotic expansion in time of the Schrödinger group on conical manifolds, Ann. Inst. Fourier, Grenoble 56(2006) 1903-1945.

X.P. Wang, Time-decay of semigroups generated by dissipative Schrödinger operators. J. Differential Equations 253 (2012), no. 12, 3523-3542.

X. P. Wang, Large-time asymptotics of solutions to the Kramers-Fokker-Planck equation with a short-range potential. Comm. Math. Phys. 336 (2015), no. 3, 1435-1471.

X.P. Wang, Gevrey estimates of the resolvent for a class of non-seladjoint Schrödinger operators, preprint 2015, to appear.

D. Yafaev, The low-energy scattering for slowly decreasing potentials, Commun. Math. Phys., 85(1982), 177-196.

D. Yafaev, Spectral properties of the Schrödinger operator with a potential having a slow falloff, Funct. Anal. Appli., 16(1983), 280-286.

DOI: 10.6092/issn.2240-2829/5891

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