Spectral Distribution of the Weber Operator Perturbed by the Dirac Delta Function

Abstract

We consider the Sturm–Liouville operator in $L^2[0,+\infty )$ generated by the expression

$$ l_{a,b}:=-\frac {d^2}{dx^2}+\frac {x^2-2}{4}+a\delta (x-b),\quad a,b>0, $$

and the boundary condition $y(0)=0$. It is proved that the eigenvalues $\lambda _n$, $n=1,2,\ldots$, of this operator satisfy the inequalities $1<\lambda _1\le 2$, $2n-1\le \lambda _n\le 2n$, $n=2,3,\ldots$