Rayleigh-Ritz Variational Method in The Complex Plane

TL;DR

This paper systematically studies the Rayleigh-Ritz variational method in the Segal--Bargmann space for quantum oscillators, deriving normalizability conditions, and analyzing various trial functions for accuracy and applicability.

Contribution

It provides rigorous conditions for trial functions, compares different ansätze, and extends the method to asymmetric potentials with displacement parameters.

Findings

Normalizability condition |α| < 1/2 derived for Gaussian trial functions.

Exact ground state recovery for harmonic oscillator with appropriate trial family.

Gaussian ansätze yield perturbative energy expansions for anharmonic oscillators.

Abstract

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 ($E_0 \approx \tfrac{1}{2} + \tfrac{3\mu}{4} - \tfrac{9\lambda^2}{4} + \cdots$).