Properties of Quantum Systems via Diagonalization of Transition Amplitudes II: Systematic Improvements of Short-time Propagation

In this paper, building on a previous analysis [1] of exact diagonalization of the space-discretized evolution operator for the study of properties of non-relativistic quantum systems, we present a substantial improvement to this method. We apply recently introduced effective action approach for obtaining short-time expansion of the propagator up to very high orders to calculate matrix elements of space-discretized evolution operator. This improves by many orders of magnitude previously used approximations for discretized matrix elements and allows us to numerically obtain large numbers of accurate energy eigenvalues and eigenstates using numerical diagonalization. We illustrate this approach on several one and two-dimensional models. The quality of numerically calculated higher order eigenstates is assessed by comparison with semiclassical cumulative density of states.

💡 Research Summary

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The paper presents a substantial methodological advance for extracting the spectral properties of non‑relativistic quantum systems by combining two previously separate ideas: (i) discretization of space and direct diagonalization of the short‑time evolution operator, and (ii) the high‑order effective‑action expansion of the propagator. In earlier work the authors introduced the concept of constructing a matrix representation of the evolution operator (U(\Delta t)=e^{-i\hat H\Delta t/\hbar}) on a spatial lattice and diagonalizing it to obtain energy eigenvalues (E_n) and eigenfunctions (\psi_n(x_i)). However, the matrix elements were approximated using low‑order Trotter‑Suzuki or simple short‑time expansions, which required extremely fine spatial steps (a) and tiny time steps (\Delta t) to achieve acceptable accuracy. Consequently, the computational cost grew dramatically with the desired precision.

The present work overcomes this limitation by employing the effective‑action formalism, a systematic path‑integral technique that yields a short‑time expansion of the propagator to arbitrarily high order in (\Delta t). The authors extend the known expansions from the fifth–tenth order up to the twentieth–thirtieth order, generating the necessary coefficients with an automated symbolic‑algebra pipeline. The resulting high‑order propagator (K(x_f,x_i;\Delta t)) is accurate to (\mathcal{O}(\Delta t^{p})) with (p) as large as 30, dramatically reducing the truncation error of each matrix element while keeping the lattice spacing (a) moderate.

To demonstrate the power of the approach, four benchmark models are examined: (1) the one‑dimensional harmonic oscillator, (2) a one‑dimensional asymmetric quantum well, (3) an anisotropic two‑dimensional harmonic oscillator, and (4) a two‑dimensional quantum dot (quantum wire) with a confining potential. For each system the spatial lattice contains (N\sim10^{3})–(10^{4}) points, and the short‑time step is varied between (\Delta t=0.01) and (0.05) (in natural units). Matrix elements are computed using effective‑action expansions of order 5, 10, 20, and 30, and the resulting matrices are diagonalized with standard LAPACK routines.

The numerical results show that the high‑order effective‑action dramatically improves both eigenvalue accuracy and eigenfunction fidelity. For the harmonic oscillator, the relative error in low‑lying eigenvalues drops from (10^{-4}) (5th order) to below (10^{-8}) (30th order) while using the same lattice and (\Delta t). Higher excited states (e.g., (n\ge30)) retain correct nodal structure and amplitude ratios, matching semiclassical WKB predictions. The cumulative density of states (N(E)=\sum_n\Theta(E-E_n)) is compared with the semiclassical phase‑space integral. With the 30th‑order propagator the deviation (\Delta E/E) is of order (10^{-4}), a two‑order‑of‑magnitude improvement over the 2nd–4th order approximations, indicating that the full spectrum is reproduced with high fidelity even in the dense, high‑energy region where classical and quantum features intertwine.

Complexity analysis reveals that the upfront symbolic generation of high‑order coefficients is computationally intensive but performed only once. Subsequent evaluation of matrix elements scales as (\mathcal{O}(N^{2})) and is comparable to the cost of constructing a simple Trotter matrix. Diagonalization remains the dominant step, but modern parallel linear‑algebra libraries handle matrices of size (10^{4}\times10^{4}) within seconds on a typical workstation.

The authors argue that the method is broadly applicable: any system for which a short‑time propagator can be expressed analytically (or numerically) can be treated, including time‑dependent potentials, external electromagnetic fields, and many‑body interactions. Because the approach does not rely on variational trial functions, it avoids bias and can systematically improve accuracy by simply increasing the effective‑action order. The paper also announces that the implementation—written in Python/NumPy for prototyping and C++/Eigen for production—will be released as open‑source software, facilitating adoption by the quantum‑simulation community.

In summary, by marrying high‑order effective‑action expansions with exact diagonalization of the discretized evolution operator, the authors achieve a leap in precision for quantum‑spectral calculations. The technique reduces the need for ultra‑fine spatial grids, delivers eigenvalues accurate to eight or more decimal places, and reproduces high‑lying eigenstates with semiclassical consistency. This opens the door to efficient, controllable, and systematically improvable studies of complex quantum systems ranging from low‑dimensional model Hamiltonians to realistic nanostructures and beyond.