Airy Quantization for Half-Line Schrödinger Operators with Mixed Boundary Conditions and Perturbed Linear Potentials
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This paper develops a self-adjoint spectral framework for half-line Schrödinger operators with a linear potential and mixed boundary conditions in the Airy setting. We construct the corresponding self-adjoint realization and prove that it is bounded from below and has compact resolvent, implying a purely discrete spectrum. Within this framework, we derive a unified Airy quantization law for Dirichlet, Neumann, and Robin boundary conditions, and establish the simplicity of the mixed spectrum, orthogonality of eigenfunctions, interlacing of the positive roots, and closed normalization formulas. We also obtain the Weyl asymptotics of the positive mixed eigenvalues and identify the critical Robin parameter ρc=−Ai′(0)/Ai(0), which separates a strictly positive spectrum from the appearance of exactly one negative eigenvalue. We further analyze the dependence of the spectrum and the normalized eigenfunctions on the boundary condition, providing a spectral interpretation of the transition between Dirichlet, Neumann, and Robin realizations. For perturbed operators of the form Fx+εW(x), with W∈L∞(0,∞), we obtain explicit first-order corrections to eigenvalues and eigenfunctions in terms of Airy-function integrals, illustrating the theory with exponentially and rationally decaying bounded perturbations. Numerical values for ρ=0 and ρ=0.5 show that the rational perturbation produces larger first-order shifts and that the response to the Robin parameter depends on the state. The analysis shows that the boundary interaction alone controls the energy levels and the spatial localization of the bound states, while the dominant confining behavior of the linear potential preserves the discreteness of the spectrum under bounded perturbations. The results provide a unified analytical framework connecting self-adjoint spectral theory, Airy special functions, and one-dimensional quantum models with boundary interactions.
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