Abstract :
Given two self-adjoint operators A and V = V
+ − V
−
, we study the motion of
the eigenvalues of the operator A t = A − tV as t increases. Let α > 0 and let λ
be a regular point for A. We consider the quantities N
+ V λ α , N
− V λ α , and
N0 V λ α defined as the number of eigenvalues of the operator A t that pass
point λ from the right to the left, from the left to the right, or change the direction
of their motion exactly at point λ, respectively, as t increases from 0 to α > 0.
We study asymptotic characteristics of these quantities as α→∞. In the present
paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys. 193
(1998), 233–243] are extended and given new applications to differential operators.
