Rayleigh-Ritz Variational Method in The Complex Plane

We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition $|\alpha| < \tfrac{1}{2}$ for generalized Gaussian trial functions $\psi(z) = e^{\alpha z^2 + \beta z}$ through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator ($\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + \lambda x^4$), adaptive Gaussian ans\"atze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions ($\psi_n(z) = z^n$) in the Segal--Bargmann space -- while providing rigorous upper bounds $E_n = n + \tfrac{1}{2} + \tfrac{3\lambda}{4}(2n^2 + 2n + 1)$ for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., $x^3 + x^4$), showing that displacement parameters are essential to capture parity breaking and stabilization effects ($E_0 \approx \tfrac{1}{2} + \tfrac{3\mu}{4} - \tfrac{9\lambda^2}{4} + \cdots$).