Transformation operators and asymptotic formulas for the eigenvalues of a polynomial pencil of Sturm--Liouville operators (Q1594591)

The authors consider the equation \[ -y'' +\bigl(q_0(x)+\lambda q_1(x)+\dots + \lambda^{n-1}q_{n-1}(x)\bigr)y =\lambda^{2n} y,\;x\in (0,a), \tag{1} \] with \(q_0\in C([0,a])\), \(q_i\in C^1([0,a])\) for \(i=1,\dots,n-1\), and \(\lambda\) a spectral parameter. By an eigenfunction the authors mean a solution to (1) satisfying the Dirichlet boundary conditions, i.e. the conditions \(y(0)=y(a)=0\). The first part of the article is devoted to constructing the transformation operators, i.e. operators taking the solution \(\exp\bigl((-1)^{j+1}i\lambda^n x\bigr)\) to the equation \(y''+\lambda^{2n}y=0\) into the solution to equation (1) satisfying the initial conditions \(y_j(0)=1\) and \(y_j'(0)=(-1)^{j+1}\lambda^n\). This operator allows the author to write out asymptotic representations for linear independent solutions to equation (1). In the last part of the article, the authors present asymptotic formulas for the eigenvalues.