Number of eigenvalues for dissipative Schrödinger operators under perturbation

Xue Ping Wang (1)

1 Equations aux dérivées partielles

In this article, we prove that $0$ is not an accumulating point of the eigenvalues for a class of dissipative Schrödinger operators $H = -\Delta + V(x)$ on $\bR^n$, $n \ge 2$, with a complex-valued potential $V(x)$ such that $\Im V(x) \le 0$ and $\Im V \neq 0$. If $\Im V$ is sufficiently small, we show that $N(V) = N( \Re V )+ k$, where $k$ is the multiplicity of the zero resonance of the selfadjoint Schrödinger operator $-\Delta + \Re V$ and $N(W)$ the number of eigenvalues of $-\Delta + W$, counted according to their algebraic multiplicity.

Mots clés: Complex eigenvalues; threshold resonance; non-selfadjoint operators; dissipative Schrödinger operators

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Xue Ping Wang. Number of eigenvalues for dissipative Schrödinger operators under perturbation. Journal de Mathématiques Pures et Appliquées, 2011, 96 (5), pp.409-422. ⟨hal-01005832⟩