Ultraviolet extrapolations in finite oscillator bases

The use of finite harmonic oscillator spaces in many-body calculations introduces both infrared (IR) and ultraviolet (UV) errors. The IR effects are well approximated by imposing a hard-wall boundary condition at a properly identified radius $L_{\mathrm{eff}}$. We show that duality of the oscillator implies that the UV effects are equally well described by imposing a sharp momentum cutoff at a momentum ${\Lambda }_{\mathrm{eff}}$ complementary to $L_{\mathrm{eff}}$. By considering two-body systems with separable potentials, we show that the UV energy corrections depend on details of the potential, in contrast to the IR energy corrections, which depend only on the $S$-matrix. An adaptation of the separable treatment to more general interactions is developed and applied to model potentials as well as to the deuteron with realistic potentials. The previous success with a simple phenomenological form for the UV error is also explained. Possibilities for controlled extrapolations for $A>2$ based on scaling arguments are discussed.

Figure 1

(a) Relative error in the deuteron energy, computed in harmonic-oscillator bases, for a wide range of oscillator parameters $N$ and $\Omega$ as a function of $L_2(N,\Omega )$. Red labels mark the minimum and maximum $N$ along sequences of constant $\Omega$ (indicated by blue labels along the sequence of crosses). These calculations use the ${\mathrm{N}}^3\mathrm{LO}NN$ potential with a 500-MeV regulator cutoff from Ref. [12], which was evolved by the similarity renormalization group [13] to $\lambda =2{\text{fm}}^{-1}$. (b) Subset of calculations from (a) for which the UV correction can be neglected compared to the IR correction (“raw”), with LO and NLO corrections subtracted as described in the text. Inset: Curves for the lowest values of $L_2$.

Figure 2

(a) Oscillator calculations of the relative error in the deuteron energy for a wide range of oscillator parameters $N$ and $\Omega$ as a function of ${\Lambda }_2(N,\Omega )$. These are the same calculations as in Fig. 1. (b) Subset of calculations from (a) for which the IR correction can be neglected compared to the UV correction. Inset: Linear plot.

Figure 3

Calculations of the relative error in the deuteron energy as a function of ${\Lambda }_2(N,\Omega )$. Circles represent a wide range of oscillator parameters $N$ and $\Omega$ that are IR converged. The series of lines shows energies for which the Hamiltonian has been smoothly cut off with exponent $n$. The solid line corresponds to a sharp cutoff.

Figure 4

Relative error of deuteron binding energy plotted vs lengths ${\Lambda }_2,{\Lambda }_0$, and ${\Lambda }_{\kappa ,\max }$ (multiplied by factors 2, 1, and $1/2$, respectively, to separate the curves). Inset: The same values on a linear scale and without the separation factors.

Figure 5

Test of the extrapolation law, (16), for a contact $a=-0.1$ fm and the quartic regulator, (23), with $\lambda =2$ fm. Points: solution of the quantization condition, (11). Line: fit of ${\kappa }_{\infty }$ and $A$ from Eq. (25).

Figure 6

Oscillator calculations (with $b=2.5$ fm and $n=6,...,16)$ and extrapolations for a contact $a=-0.1$ fm and quartic regulator, (23), with $\lambda =2$ fm. Circles and short-dashed line: direct-quantization result and fit, as in Fig. 5. Squares: oscillator result with $\Lambda ={\Lambda }_0\left(n\right)$. Dotted line: fit of Eq. (25) to squares. Diamonds: oscillator result with $\Lambda ={\Lambda }_2\left(n\right)$. Long-dashed line: fit of Eq. (25) to diamonds.

Figure 7

Oscillator calculations (with $b=2.5$ fm and $n=6,...,34)$ and extrapolations for a contact $a=-0.1$ fm and Gaussian regulator, (27), with $\lambda =2$ fm. Symbols and curves are as in Fig. 6.

Figure 8

Logarithmic plot of $\Delta {\kappa }_{\Lambda }$ as defined in Eq. (28) for a contact $a=-0.2$ fm and quartic regulator, (23), with $\lambda =1.8$ fm. Dotted line: result for $\Lambda ={\Lambda }_0$. Thick-dashed line: result for $\Lambda ={\Lambda }_2$. Thin-dashed line: result for $\Lambda ={\Lambda }_2^{\left(1\right)}$ (including the first subleading correction). Inset: The small improvement from ${\Lambda }_2$ to ${\Lambda }_2^{\left(1\right)}$.

Figure 9

Logarithmic plot of $\Delta {\kappa }_{\Lambda }$ as defined in Eq. (28) for a contact $a=-0.33$ fm and Gaussian regulator, (27), with $\lambda =1.66$ fm. Curves and inset are as in Fig. 8.

Figure 10

Oscillator calculations (with $b=2.0$ fm and $n=4,...,64)$ for a spherical step potential with $V_0=-4{\mathrm{fm}}^{-1}$ and $R=1\mathrm{fm}$ and its separable approximation. Dashed line: direct-quantization result according to Eqs. (31) and (36). Squares: oscillator result with the separable approximation, (36). Circles: oscillator result with the original (full) potential, (35). The horizontal dotted line indicates the exact result for the binding momentum.

Figure 11

Oscillator calculations (with $b=2.5$ fm and $n=4,...,32)$ for a Pöschl-Teller potential with $\beta =3$ and $\alpha =2/3{\mathrm{fm}}^{-1}$ and its separable approximation. Dashed line: direct-quantization result according to Eqs. (31) and (B3). Squares: oscillator result with the separable approximation, (B3). Circles: oscillator result with the original (full) potential, (40). The horizontal dotted line indicates the exact result for the binding momentum.

Figure 12

Binding momentum as a function of $b$ obtained from oscillators with different basis sizes. Circles: $n=6$. Boxes: $n=12$. Diamonds: $n=24$. The horizontal dotted line indicates the exact result for the binding momentum.

Figure 13

Comparison of UV extrapolations for an oscillator calculation (fixed $b=4.0\mathrm{fm}$, running $n=4,...,12)$ with a $\beta =3,\alpha =2/3{\mathrm{fm}}^{-1}$ Pöschl-Teller potential. Circles: oscillator results. Dotted line: exponential extrapolation (fit $E)$. Dashed line: Gaussian extrapolation (fit $G)$. Solid line: simplest separable extrapolation (fit $\eta )$. The horizontal dotted line indicates the exact result for the binding momentum.

Figure 14

Same as Fig. 13, but now with fixed basis size $n=12$ and running oscillator length $b=3.5,\cdots ,5.5$ fm.

Figure 15

Comparison of UV extrapolations for a deuteron state calculated with the Entem-Machleidt N3LO (500-MeV cutoff) potential: (a) SRG-evolved down to a resolution scale $\lambda =2.0{\mathrm{fm}}^{-1}$ and (b) with the “bare” (unevolved) interaction. Circles: oscillator results. Dotted line: exponential extrapolation (fit $E)$. Dashed line: Gaussian extrapolation (fit $G)$. Solid line: simplest separable extrapolation (fit $\eta )$. Dotted horizontal lines indicate the exact result for the binding momentum.

Figure 16

Comparison of UV extrapolations for a deuteron state calculated with the Epelbaum et al. N3LO (550/600-MeV cutoff) potential: (a) SRG-evolved down to a resolution scale $\lambda =2.0{\mathrm{fm}}^{-1}$ and (b) with the “bare” (unevolved) interaction. Symbols and curves are as in Fig. 16.

Figure 17

Comparison of UV extrapolations for a deuteron state calculated with the Epelbaum et al. N3LO (550/600-MeV cutoff) potential. Circles: oscillator results with fixed $\Omega =4\mathrm{MeV}$ and $n=4,...,12$. Squares: oscillator results with fixed $n=10$ and $\Omega =3,...,12$ MeV (in steps of 1.5 MeV. Curves are as in Fig. 16 and show fits to the squares only.

Figure 18

Comparison of UV extrapolations for a deuteron state calculated with the Epelbaum et al. N2LO (550/600-MeV cutoff) potential. Circles: oscillator results. Solid line: simplest separable extrapolation (fit $\eta )$. Long-dashed line: general separable extrapolation (fit $\eta ,\mathrm{gen}.)$. Short-dashed line: modified general separable extrapolation (fit $\eta ,\mathrm{gen}{.}^{\prime }$). The dotted horizontal line indicates the exact result for the binding momentum.

Figure 19

Relative error for the deuteron energy from HO basis truncation as a function of ${({\Lambda }_2/\lambda )}^4$ for $(N,\Omega )$ values for which the IR correction can be neglected. Several SRG-evolved potentials are used, all with the same initial potential as in Fig. 15. The dashed line shows the expected slope (up to prefactors) for ${\Lambda }_2/\lambda \gg 1$ according to the analysis in Sec. 3a3. Inset: The relative error is plotted against the unscaled ${\Lambda }_2$.

Figure 20

Relative error for the deuteron energy from HO basis truncation as a function of ${({\Lambda }_2/\lambda )}^2$ for $(N,\Omega )$ values for which the IR correction can be neglected. The potential is the same as in Fig. 15. The solid line is an approximate fit to a region near ${\Lambda }_2/\lambda =1$.

Figure 21

Relative error for the deuteron energy from HO basis truncation as a function of ${({\Lambda }_2/\lambda )}^2$ for $(N,\Omega )$ values for which IR corrections can be neglected for the SRG-evolved N3LO potential by Entem and Machleidt (left) and the bare potential (right). Also indicated are the regions where the fit to a Gaussian was attempted.

Figure 22

(a) Wave functions and (b) corresponding separable form factor for a Pöschl-Teller potential with $\alpha =2/3$ and $\beta =3$. Solid lines: results from oscillator calculation with $b=4.0$ fm and $n=4$. Dashed lines: exact (analytically known) results for comparison.

Figure 23

(a) Wave functions and (b) corresponding separable form factor for a Pöschl-Teller potential with $\alpha =2/3$ and $\beta =3$. Solid lines: results from oscillator calculation with $b=4.0$ fm and $n=8$. Dashed lines: exact (analytically known) results for comparison.