Half-Line non-self-adjoint Schrödinger operators with polynomial potentials: Asymptotics of eigenvalues

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For integers m≥ 3, we study the non-self-adjoint eigenvalue problems -u′′(x)+(xᵐ+P(x))u(x)=E u(x), 0≤ x<+∞, with the boundary conditions u(+∞)=0 and αu(0)+βu′(0)=0 for some α, β∈ with |α|+|β|=0, where P(x)=a₁ xᵐ⁻¹+a₂ xᵐ⁻²+...+aₘ₋₁ x is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues Eₙ. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.