Systematic expansion for infrared oscillator basis extrapolations

Recent work has demonstrated that the infrared effects of harmonic oscillator basis truncations are well approximated by a partial-wave Dirichlet boundary condition at a properly identified radius $L$. This led to formulas for extrapolating the corresponding energy $E_L$ and other observables to infinite $L$ and thus infinite basis size. Here we reconsider the energy for a two-body system with a Dirichlet boundary condition at $L$ to identify and test a consistent and systematic expansion for $E_L$ that depends only on observables. We also generalize the energy extrapolation formula to nonzero angular momentum, and apply it to the deuteron. Formulas given previously for extrapolating the radius are derived in detail.

Figure 1

Ground-state wave functions for a square well potential of depth $V_0=4$ [see Eq. (19); lengths are in units of $R$ and energies in units of $1/R^2$ with ${\hslash }^2/\mu =1$] from solving the Schrödinger equation with a truncated harmonic oscillator basis with $\hslash \Omega =18$ and $N=8$ (dashed) and with a Dirichlet boundary condition at $r=L_2$ given from Eq. (1) (solid). The coordinate-space radial wave functions in (a) exhibit a difference at $r$ near 1.5, but the Fourier-transformed wave functions in (b) are in close agreement at low $k$, showing that the differences are high-momentum modes.

Figure 2

Bound-state energy for a square well of depth $V_0=4$ (lengths are in units of $R$ and energies in units of $1/R^2$ with ${\hslash }^2/\mu =1$) from solving the Schrödinger equation with a Dirichlet boundary condition at $r=L$. The diamonds are exact results for each $L$ while the horizontal dotted line is the energy for $L\to \infty$, $E_{\infty }=-1.5088$. The dashed, dot-dashed, and solid lines are predictions for the energy using the systematic correction formula Eq. (17) at LO (first term only), L-NLO (first two terms), and full NLO (all terms), respectively. The dotted curve on top of the solid line and the dot-double-dashed lines are respectively the NLO and N2LO predictions for the square well from the Taylor expansion method described in Sec. 2b.

Figure 3

Error plots of the energy correction at each $L$ for the square well of Fig. 2 ($V_0=4$) predicted at different orders by Eq. (17) and by the Taylor expansion method in Sec. 2b, each compared to the exact energy. Lines proportional to $L{e}^{-4k_{\infty }L}$ (dashes) and $L^2{e}^{-6k_{\infty }L}$ (with arbitrary normalization) are plotted for comparison to anticipated error slopes.

Figure 4

Bound-state energy for a square well of depth $V_0=1.83$ (units with $R=1$), which simulates a deuteron, from solving the Schrödinger equation with a Dirichlet boundary condition at $r=L$. The horizontal dotted line is the exact energy, $E_{\infty }=-0.1321$, and the other curves are as the same as in Fig. 2.

Figure 5

Comparison of the actual energy correction due to truncation to the energy correction predicted to different orders by Eq. (17) for a square well [Eq. (19)] with $V_0=1.83$ and $R=1$.

Figure 6

Deuteron energy versus $L_2$ [see Eq. (1)] for the chiral N${}^3$LO (500 MeV) potential of Ref. [29]. To eliminate the UV contamination we only plot results for $\hslash \Omega >49$ MeV. The dashed, dot-dashed, and solid lines are respectively the LO [first term in Eq. (17)], L-NLO [first two terms in Eq. (17)], and full NLO [all the terms in Eq. (17)] predictions for the energy correction. The horizontal dotted line is the deuteron energy.

Figure 7

Comparison of the actual energy correction due to HO basis truncation ($\hslash \Omega$ restricted to be greater than 49 MeV to eliminate UV contamination) for the deuteron to the energy correction predicted to different orders from Eq. (17). For the parameter $w_2$ in Eq. (17) we use the value reported in Ref. [28].

Figure 8

Error plots of the energy correction at each $L$ for (a) $l=1$ and (b) $l=2$ square-well states predicted at leading order by Eqs. (46) and (47) compared to the exact energy. Lines proportional to the expected L-NLO residual errors are plotted for comparison.

Figure 9

Residual error for the deuteron energy due to HO basis truncation as a function of $L=L_2$ (with $\hslash \Omega >49$ MeV to eliminate UV contamination) after subtracting $l=0$ energy corrections at different orders from Eq. (17) and the $l=2$ correction from Eq. (47). For the parameter $w_2$ in Eq. (17) we use the value reported in Ref. [28].

Figure 10

Deuteron radius squared versus $L_2$ for the chiral N${}^3$LO (500 MeV) potential of Ref. [29]. To eliminate the UV contamination we only plot results for $\hslash \Omega >49$ MeV. The solid, dot-dashed, and dashed lines are results from fitting Eq. (70) in the shaded region to find ${\langle r^2\rangle }_{\infty }$ and one, two, or three of the $c_i$ constants, respectively. The horizontal dotted line is the deuteron radius squared.