2014/02/17 by Sylwia Kondej, Kondej, Sylwia, Vladimir Lotoreichik +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1402.3995
openalex publication_date 2014/02/17 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We consider a self-adjoint two-dimensional Schr "odinger operator\nH\α\μ, which corresponds to the formal differential expression n-
Delta -
alpha
mu, where \μ is a finite compactly supported positive\nRadon measure on mathbb R2 from the generalized Kato class and \α\n>0 is the coupling constant. It was proven earlier that \σ rm\ness(H\α\μ) = [0,+\∞). We show that for sufficiently small\n\α the condition sharp\σ rm d(H\α\μ) = 1 holds and that\nthe corresponding unique eigenvalue has the asymptotic expansion \n
lambda(
alpha) = -(C_
mu + o(1))
exp
Big(-
tfrac4
pi
alpha
mu(
mathbb\nR2)
Big),
qquad
alpha
rightarrow 0+, with a certain constant C_\μ >\n0. We obtain also the formula for the computation of C_\μ. The asymptotic\nexpansion of the corresponding eigenfunction is provided. The statements of\nthis paper extend Simon's results, see citeSi76, to the case of\npotentials-measures. Also for regular potentials our results are partially new.\n