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larger couplings or other more extreme situations, but new physical behavior will not occur. There are how- ever many other topics in polaron physics that remain unsettled. With a few singular exceptions none of them have been accessed with the same rigor as the present problem. Within the framework of the Holstein model, transport at finite temperature and collective effects at fi- nite density are probably the two most important issues. Proceeding beyond the Holstein model, in particular on trying to extend the discussion to polarons in novel ma- terials such as graphene $^{50}$, new questions emerge. These include polaron formation in the presence of impurities 29 and disorder $^{51,52}$, the influence of phonon dispersion, of different types of electron-phonon coupling, of lattice ge- ometry, anisotropy 48 or confinement, or the dynamics of polaron formation 53 out of equilibrium, realized e.g. in quantum dots or molecular aggregates. Some of these questions are directly connected with the present study. Does, e.g., longer-ranged electron-phonon interaction fa- vor the formation of large and heavy polarons? Although the polaron becomes larger then, its mass will decrease, and the existence of a large heavy polaron with corre- sponding optical signatures may crucially depend on a number of material parameters. The classification of these parameters and of their influence on polaron prop- erties is central to giving studies such as the present one relevance beyond the rigorous treatment of a model. To make such investigations possible is certainly the moti- vation to develop powerful high-precision numerics for the polaron problem, while their application provides us with the necessary information to understand real po- larons and the models to describe them. 

This work was supported by the US Department of Energy, Center for Integrated Nanotechnologies, at Los Alamos National Laboratory (Contract DE-AC52- 06NA25396) and Sandia National Laboratories (Contract DE-AC04-94AL85000). A.A. and H.F. are grateful for hospitality provided at the Los Alamos National Labo- ratory. 

The SOS | n 〉 $_{σ}$, with n ≥ 0, form a Hilbert space basis for a single harmonic oscillator. For σ ̸ = 0, they are characterized through two properties: (i) orthonormality 〈 n | m 〉 σ = δ$_{nm}$ , (ii) the n -th SOS is a linear combination 

of the n + 1 coherent states | mσ 〉 $_{c}$, 0 ≤ m ≤ n . In particular, | 0 〉 σ = | 0 〉 c = | vac 〉 . 

## Appendix A: Shifted Oscillator States 

## Acknowledgments 

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These conditions imply that the SOS can be con- structed through Gram-Schmidt-orthogonalization of the equally spaced coherent states | mσ 〉 $_{c}$, in the order of in- creasing m = 0 , 1 , 2 , . . . , ∞ . Moreover, the operator b is given by a upper triangular matrix in the SOS basis. It is possible to obtain explicit expressions for the coefficients 

where α = e −| σ | 2 and the q-binomials [ n $_{m}$]$_{q}$ (for a defini- tion, see e.g. Ref. 54) are polynomials in q . All relevant matrix elements can also be computed with a numerical Gram-Schmidt procedure. 

The above definition applies to all σ ̸ = 0. The limit σ → 0 exists, and the SOS reproduce the Fock states in this limit. In the opposite limit σ → ∞ , they become asymptotically equal to the coherent states | mσ 〉 $_{c}$, since the overlap between different coherent states is small. ( n ) 

From the explicit expressions for c m we find that 〈 n | b | n 〉 σ = nσ . For real σ , it follows that the oscilla- tor elongation in the n -th SOS is given by 2 nσ . Thus, SOS for σ > 0 ( σ < 0) should be used for states with positive (negative) elongation. 

To quantify the benefit of SOS over Fock states for groundstate calculations, we consider a shifted oscillator with Hamiltonian 

The groundstate of this Hamiltonian, with energy − ϵ$_{p}$ = − g $^{2}$ω$_{0}$ , is a coherent state | g 〉 $_{c}$, which contains g 2 bosons on average. In a numerical calculation an approximate groundstate of H$_{osc}$ is calculated in a truncated Hilbert space, e.g. consisting of the first N + 1 Fock states. The corresponding error in the groundstate energy is shown in Fig. 10. As expected, the error is large when N < g $^{2}$. 

Using SOS instead of Fock states, the error depends on σ . In the present example, the error is zero for σ = g/N , since the coherent state | g 〉 c is a linear combination of the first two SOS. For a meaningful error estimate we there- fore fix σ = g $^{max}$/N for the maximal g max considered. The virtue of the SOS is that the error is small also for all intermediate 0 < g < g max (see Fig. 10). Note that the error drops to zero at those values of g for which the true groundstate is contained in the truncated Hilbert space spanned by the N + 1 SOS. 

We also see how the SOS prefer positive elongations for σ > 0, and the error becomes large if σ and g have opposite sign. In a real calculation, the variational de- termination of the parameter σ would prevent this situ- ation. For the Holstein model (1), only positive elonga- tions occurs and always σ > 0. Note that in the worst case the optimal σ is zero, and the Fock states are re- covered. Therefore, the introduction of SOS can only improve groundstate results obtained with Fock states. In the given example, the error with 10 SOS is one order of magnitude smaller than that with 100 Fock states. 

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