Mouez Dimassi and Vesselin Petkov
Resonances for magnetic Stark Hamiltonians in two dimensional case
(344K, postscript)
ABSTRACT. We study the resonances of the two-dimensional Schr\"odinger operator $P_1(B, \beta) =(D_x - B y)^2 + D_y^2 + \beta x + V(x,y),\: B > 0, \beta > 0$, with constant magnetic and electric fields. We define the resonances of $P_1(B, \beta)$ and the spectral shift function $\xi(\lambda)$ related to $P_1(B, \beta)$ and $P_0(B, \beta) = P_1(B, \beta) - V(x, y)$ without any restriction on $B$ and $\beta$. For strong magnetic fields ($B \to \infty)$ we obtain a representation of the derivative of $\xi(\lambda)$, a trace formula for $\tr(f(P_1(B, \beta)) - f(P_0(B, \beta)))$ and an upper bound for the number of the resonances lying in $\{z \in \C:\: |\Re z - (2n-1) B|\leq \alpha B,\: \ii\: z \geq \mu \ii\: \theta\},\: 0 < \alpha < 1,\: 0 < \mu < 1, \: \ii\: \theta < 0.$
Moreover, for $B \to \infty$ we examine the free resonances domains and show that the resonances are included in the neighborhoods $ \{z \in \C: |\Re z - (2n - 1)B| \leq C_0\}$, where $(2n -1)B$ are the Landau levels and $C_0 > 0$ is a constant independent on $B$ and $n \in \N^*.$