Deficiency indices of the singular differential operator generated by the expression \(y^{IV}-2a(x^\alpha y')'+ bx^\beta y\) (Q1810289)

Suppose that \(L_0\) is the minimal symmetric operator in the space \(L_2([1,\infty))\) generated by the differential expression \[ \ell(y)= y^{\text{IV}}- 2a(x^\alpha y')'+ bx^\beta y, \] where \(\lambda\) is a complex parameter \(a\), \(b\), \(\alpha\), \(\beta\) are real constants. The author analyzes deficiency indices for \(L_0\) depending on the parameters \(a\), \(b\), \(\alpha\) and \(\beta\). To this end, the main auxiliary problem is the problem of determining an asymptotic formula for the solutions of the equation \(\ell(y)-\lambda y=0\) as \(x\to\infty\).