Spectral Theorem approach to the Characteristic Function of Quantum Observables II

TL;DR

This paper presents a spectral theorem-based method for computing the characteristic functions of quantum observables, offering an analytical alternative to traditional algebraic approaches, with applications to both infinite and finite-dimensional systems.

Contribution

It introduces a spectral theorem approach to derive the characteristic functions of quantum operators, expanding analytical tools beyond algebraic Lie algebra methods.

Findings

Computed resolvents for key quantum operators.

Derived vacuum characteristic functions using Stone's formula.

Demonstrated applicability to finite and infinite-dimensional systems.

Abstract

We compute the resolvent of the anti-commutator operator $XP+PX$ and of the quantum harmonic oscillator Hamiltonian operator $\frac{1}{2}(X^2+P^2)$. Using Stone's formula for finding the spectral resolution of an, either bounded or unbounded, self-adjoint operator on a Hilbert space, we also compute their Vacuum Characteristic Function (Quantum Fourier Transform). We also show how Stone's formula is applied to the computation of the Vacuum Characteristic Function of finite dimensional quantum observables. The method is proposed as an analytical alternative to the algebraic (or Heisenberg) approach relying on the associated Lie algebra commutation relations.