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arXiv:2511.20381

Matrix approximations of operators

BERTRAND G. GIRAUD, Karataglidis, S., Murulane, K., R. Peschanski

Published Nov 25, 2025• 0 citations
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Abstract

The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak approximation of the $δ$ function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of $-δ''$ is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.

DOI
10.48550/arxiv.2511.20381

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The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak approximation of the δδδ function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of −δ′′-δ''−δ′′ is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.
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In this thesis, we will show how certain classes of quantum many-body Hamiltonians with su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on kkk. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of kkk equations and diagonalizing ktimeskk\\times kktimesk matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have su2\\su{2}su2 spectrum generating algebras. Next, we extend the technique to the non-compact su1,1\\su{1,1}su1,1 algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a kkk-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of 10times1010\\times 1010times10 matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.
For both identical and distinguishable particles and for an arbitrary polynomial approximation to the realistic two-body forces, the A-body Hamiltonian may be converted into an infinite block-three-diagonal matrix in the properly arranged, translationally invariant, oscillator basis. As a consequence, the exact Green function and all the projections of eigenstates of the microscopic Schrodinger equation become expressible in terms of the matrix continued fractions. This generalises the recent reformulation of the one-dimensional A=1 anharmonic problem by Graffi and Greechi (1975). The quick convergence of this non-perturbative method of solving the many-body bound-state problem is demonstrated for the simplest three- and four-body examples. With the core-possessing type V(r)=-r 2 +r 4 of the spin- and isospin-independent two-body interaction, only the 10 -3 error in the three-bosonic ground-state energy arises from restricting the formulae to 5*5 dimensional matrices.
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Multi-mode entanglement is investigated in a system composed of N coupled identical harmonic oscillators interacting with a common environment. We treat the problem generally by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an N -mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form that can be expressed in terms of multiples of 3 × 3 and 4 × 4 matrices. This simple property allows us to treat the problem to some extent analytically. We illustrate the advantage of working in the transformed basis using a simple example of three harmonic oscillators and find that the entanglement can persist for long times due to the presence of constants of motion for the covariance matrix elements. We find that, in contrast to what one would expect, a strong damping of the oscillators leads to a better stationary entanglement than in the case of a weak damping.