Spectral problems for operators with crossed magnetic and electric fields

Abstract

We obtain a representation formula for the derivative of the spectral shift function $\xi (\lambda ;B,\epsilon )$ related to the operators $H_0(B,\epsilon )=(D_x-By{)}^2+D_y^2+\epsilon x$ and $H(B,\epsilon )=H_0(B,\epsilon )+V(x,y),B>0,\epsilon >0$. We prove that the operator $H(B,\epsilon )$ has at most a finite number of embedded eigenvalues on $\mathbb{R}$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\mathbb{R}.$ Applying the formula for ${\xi }^{\prime }(\lambda ,B,\epsilon )$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h)=(h{D}_x-By{)}^2+h^2{D}_y^2+\epsilon x$ and $H(h)=H_0(h)+V(x,y).$

References in corpus (1)

• Spectral shift function for operators with crossed magnetic and electric fields

Cited by in corpus (1)

• Absence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fields