An Improved Lower Bound to the Ground-State Energy

2020

RSC Adv.

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“…The rst expression on the right-hand side is the generalization of Temple's lower bound for the ground state derived in ref 40. and 41.…”

mentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Rontó

2020

RSC Adv.

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“…5-12. In the past year a different approach has been suggested and shown to improve upon Temple's method (13,14). It improved upon Temple's original bound; however, to ensure rapid convergence it was necessary to use an approximate estimate for overlap matrix elements (14).…”

mentioning

confidence: 99%

Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

Martinazzo

Proc. Natl. Acad. Sci. U.S.A.

2020

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The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple,Proc. R. Soc. A Math. Phys. Eng. Sci.119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

“…Recently, novel lower bound methods 45,46 have been developed based on the Lánczos construct, [47][48][49] which significantly improved Temple's lower bound for energy levels, as tested on quartic oscillators. The Lánczos algorithm provides an orthogonalization of states on the Krylov space, and from the resulting tridiagonal matrix, the approximate eigenvalues and the corresponding variances can be readily determined for the original Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

Rontó

Martinazzo

2021

Preprint

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Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.