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

Asymptotic behavior of the solutions and nodal points of Sturm-Liouville differential expressions

“…Using the integral representation (20) and estimate (19) with ρ λ ≡ π q C[0,π] ≤ π as ρ λ , as in [7, Chapter 1, § 2, Lemma 2.2], we conclude the existence of a constant B = B(q) depending only on the norm of the potential q in C[0, π] such that y(x, λ n j k ) − cos √ λ n j k x − β(x, λ n j k ) sin √ λ n j k x √ λ n j k…”

“…To the study of Sturm-Liouville operators with potential depending on the spectral parameter, we may also relate the cycle of works devoted to the sheaves of differential operators (for example, see [19,20]). …”

Inverse nodal problem for polynomial pencil of Sturm‐Liouville operator

“…Consider the following differential equation on the interval 0 ≤ x ≤ π where , λ is a spectral parameter and q d ( x ), d = 0,1,2,…, n − 1 are continuous complex‐valued functions on (see previous studies). Denote by y ( x , λ ) solution of with the boundary conditions …”

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

“…is investigated in [10] (also see [11] In this paper using the properties of transformation operators, the considering boundary value problem is investigated and asymptotic formula for the eigenvalues is obtained.…”