Surpassing the Temple Lower Bound

2020

RSC Adv.

Section: Introductionmentioning

confidence: 99%

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

2020

RSC Adv.

“…The Weinstein lower bound 4 is based on the variance associated with the approximate eigenvalues of the Hamiltonian and Stevenson verified its validity 5 and further generalized the method. 6 These two approaches are the most important lower bound methods based on the variance of the Hamiltonian and several theoretical studies [7][8][9][10][11][12][13][14][15][16] and practical implementations [17][18][19] have improved their performance. A significant step forward was the development of Lehmann's optimal inclusion intervals, which provided an optimization of the basis set that maximizes Temple's lower bound.…”

Section: Introductionmentioning

confidence: 99%

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

Martinazzo

2021

Preprint

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.

“…The Weinstein lower bound 4 is based on the variance associated with the approximate eigenvalues of the Hamiltonian, and Stevenson verified its validity 5 and further generalized the method 6 . These two approaches are the most important lower bound methods based on the variance of the Hamiltonian and several theoretical studies 7 – 16 and practical implementations 17 – 19 have improved their performance. A significant step forward was the development of Lehmann’s optimal inclusion intervals, which provided an optimization of the basis set that maximizes Temple’s lower bound 20 – 23 .…”

Section: Introductionmentioning

confidence: 99%

Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues

Martinazzo

2021

Sci Rep

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 method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT 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 iSCLBT 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 iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds