Spectral problems for operators with crossed magnetic and electric fields

TL;DR

This paper derives a formula for the spectral shift function's derivative for operators with crossed magnetic and electric fields, proving finiteness of embedded eigenvalues and providing semiclassical asymptotics.

Contribution

It introduces a new representation formula for the spectral shift function derivative and advances understanding of embedded eigenvalues in magnetic-electric operators.

Findings

Finite number of embedded eigenvalues on  for the studied operators.

Derived semiclassical asymptotics for the spectral shift function.

Established a step towards proving absence of embedded eigenvalues.

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 $\R$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\R.$ Applying the formula for $\xi'(\lambda, B, \epsilon)$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h) = (hD_x - By)^2 + h^2D_y^2 + \epsilon x$ and $H(h) = H_0(h) + V(x,y).$