Spectral Gap Stability for Dirichlet Schrödinger Operators

This work studies the stability of the spectrum of a Dirichlet Schrödinger operator under bounded potential perturbations. Let Ω be a bounded domain in Euclidean space and consider the operator A = −ΔD + μI + V (x) acting on the Hilbert space L² (Ω), where −ΔD denotes the Dirichlet Laplacian, μ > 0 is a constant shift, and V (x) is a bounded real-valued potential. The main result establishes an explicit lower spectral bound for the operator when the perturbation remains below the spectral floor of the reference operator. In particular, if the L∞ norm of the potential satisfies ‖V‖∞ < λ₁ (−ΔD) + μ then the spectrum of A remains bounded away from zero and satisfies inf Spec (A) ≥ λ₁ (−ΔD) + μ − ‖V‖∞. This quantitative bound yields several stability consequences: • exponential decay of the semigroup generated by −A• resolvent estimates in the left half-plane• persistence of the spectral gap under additional bounded perturbations• discrete spectrum due to compact resolvent• an equivalent spectral characterization via the Birman–Schwinger operator• control of the pseudospectrum through resolvent bounds The analysis relies on classical tools from operator theory, including perturbation theory for self-adjoint operators, the spectral theorem, and semigroup methods. The results provide a concise stability framework for Dirichlet Schrödinger operators subject to bounded perturbations.