Spectral asymptotics for the Schrödinger operator with a non-decaying potential

Article type: Research Article

Authors: Dimassi, Mouez^{a; *} | Fujiié, Setsuro^b

Affiliations: [a] Mouez Dimassi, IMB, (UMR CNRS 7539), Université Bordeaux 1, 33400 Talence, France. E-mail: mdimassi@u-bordeaux.fr | [b] Ritsumeikan University, 1-1-1 Noji-Higashi, 525-8577 Kusatsu, Japan. E-mail: fujiie@fc.ritsumei.ac.jp

Correspondence: [*] Corresponding author. E-mail: mdimassi@u-bordeaux.fr.

Abstract: We study Schrödinger operators H(h)=−h2Δ+V(x) acting in L2(Rn) for non-decaying potentials V. We give a full asymptotic expansion of the spectral shift function for a pair of such operators in the high energy limit. In particular for asymptotically homogeneous potentials W at infinity of degree zero, we also study the semiclassical asymptotics to give a Weyl formula of the spectral shift function above the threshold maxW and Mourre estimates in the range of W except at its critical values.

Keywords: Stark Schrödinger operator, spectral shift function, asymptotic expansions

DOI: 10.3233/ASY-211754

Journal: Asymptotic Analysis, vol. 130, no. 3-4, pp. 335-365, 2022