The inverse spectral problem for differential pencils by mixed spectral data

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

“…Considering arbitrary 𝜎 ∈ L 2 (0, 2𝜋) such that q = 𝜎 ′ , we rewrite (20) in the form (4). From (7), we have that the eigenvalues {𝜇 k } k∈Z 0 of the boundary value problems (20) and (21) satisfy the asymptotics…”

“…Note that since the quasi-derivative is absent in boundary conditions (21), the eigenvalues {𝜇 k } k∈Z 0 do not depend on the choice of 𝜎 ∈ L 2 (0, 2𝜋) up to a constant function.…”

“…Remark Analogously one can study other types of boundary conditions and other partial inverse problems, including the ones from earlier studies 9,13,20,21,23,25 …”

“…In the recent two decades, there essentially increased an interest in the inverse spectral theory for second‐order quadratic differential pencils of the form $$$$ -{y}^{\prime \prime }+\left[q(x)+2\lambda p(x)\right]y={\lambda}^2y,\kern0.60em x\in \left(0,\pi \right), $$$$ under various types of boundary conditions including those polynomially dependent on the spectral parameter $$$ \lambda $$$ (see earlier studies 1–29 and references therein). Here, $$$ p(x) $$$ and $$$ q(x) $$$ are some, generally speaking, complex‐valued functions in appropriate classes.…”