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

Xue Ping Wang


In this article, we show that under some coercive assumption on the complex-valued potential 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.


Gevrey estimate of resolvent; threshold spectral analysis; non-selfadjoint Schrödinger operators; Witten Laplacian

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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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