Existence of Density Functionals for Excited States and Resonances

We show how every bound state of a finite system of identical fermions, whether a ground state or an excited one, defines a density functional. Degeneracies created by a symmetry group can be trivially lifted by a pseudo-Zeeman effect. When complex scaling can be used to regularize a resonance into a square integrable state, a DF also exists.

DFs for resonant states have received much less attention. We want to study this problem here. First we will address two related issues, namely that of a unified theory for ground and excited states and that of a theory for non degenerate and degenerate ones. A generalized existence theorem can be constructed by modifying the Hamiltonian in such a way that the spectrum is shuffled but the eigenstates are left unchanged, and by making a systematic use of the Legendre transform (LT) for a detailed analysis of the density.
A reminder of the HK proof is useful here. Consider a finite number A of identical fermions, with a † r and a r their creation and annihilation operators at position r, and the physical Hamiltonian, u i are the kinetic, two-body interaction and one-body potential energies, respectively. For simplicity, we consider such fermions as spinless and isospinless and work at zero temperature. Both v and u may be either local or non local. Next, embed the system into an additional one-body, external field, W = A i=1 w i , to observe its (non linear!) response. The Hamiltonian becomes K = H + W. It is understood that w is local, r|w| r ′ = w(r) δ( r − r ′ ), although a DF theory with non-local potentials exists [20]. The usual Rayleigh-Ritz variational principle, where |ψ is just an A-particle, antisymmetric, square normalized, otherwise unrestricted wave function, applied to F M = min ψ F, with F = ψ|K|ψ , generates ψ min , the exact GS of K, with the exact eigenvalue F M . The minimum is assumed to be non degenerate, smooth, reached. Clearly, ψ min and F M are parametrized by w. An infinitesimal variation δw triggers an infinitesimal displacement δψ min , with δF M = ψ min |δW|ψ min . There is no first order contribution from δψ min . Define the one-body density matrix in coordinate representation, n( r, r ′ ) = ψ min |a † r a r ′ |ψ min . Its diagonal, ρ( r) = n( r, r), is the usual density deduced from |ψ min | 2 by integrating out all particles but one. Since δF M = d r ρ( r) δw( r), then δF M /δw( r) = ρ( r). Freeze t, v and u and consider F M as a functional of w alone. The HK process then consists in a Legendre transform of F M , based upon this essential result, δF M /δw = ρ. This LT involves two steps: i) subtract from F M the functional product of w and δF M /δw, i.e. the integral d r w( r) ρ( r), leaving F M = ψ min |H|ψ min ; then ii) set ρ, the "conjugate variable of w", as the primary variable rather than w; hence see F M as a functional of ρ.
Step ii) is made possible by the one-to-one (1 ↔ 1) map between w and ρ, under precautions such as the exclusion of trivial variations δw that modify w by a constant only, see for instance [16] and [19]. The 1 ↔ 1 map is proven by the usual argument ad absurdum [1]: if distinct potentials w and w ′ generated ψ min and ψ ′ min (distinct!) with the same ρ, then two contradictory, strict inequalities would occur, M . An inverse LT returns from F M to F M , because δF M /δρ = −w. Finally, the GS eigenvalue E 0 of H obtains as E 0 = min ρ F M [ρ]; the GS wave function ψ 0 of H is the wave function ψ min when w vanishes; that density providing the minimum of F M is the density of ψ 0 .
Consider now any excited bound eigenstate ψ n of H, with its eigenvalue E n . Then, trivially, ψ n is a GS of the semipositive definite operator (H − E n ) 2 . Since E n is not known a priori, consider rather an approximate value E n , obtained by any usual technique (configuration mixing, generator coordinates, etc.) and assume that E n is closer to E n than to any other eigenvalue E p . Then ψ n is a GS of (H − E n ) 2 . The possible degeneracy degree of this GS is the same whether one considers H or If there is no degeneracy of either ψ n or its continuation as a functional of w, then the HK argument holds as well for K as it does for K. Hence a trivial existence proof for a DF concerning ψ n . But most often, ψ n belongs to a degenerate multiplet. Degeneracies are almost always due to an explicitly known symmetry group of H. Notice however that the external potential w does not need to show the same symmetry; hence, in general for K, there is no degeneracy of its GS; a unique ψ min emerges to minimize the expectation value of K. However, for that subset of zero measure in the space of potentials where w shows the symmetry responsible for the degeneracy, and in particular for the limit w → 0, precautions are necessary. Consider therefore an (or several) additional label(s) g sorting out the members ψ ng of the multiplet corresponding to that eigenvalue (E n − E n ) 2 of (H − E n ) 2 . There is always an operator G related to the symmetry group, or a chain of operators G j in the reduction of the group by a chain of subgroups, which commute with H and can be chosen to define g. For simplicity, assume that one needs to consider one G only. Then define g as an eigenvalue of G and assume, obviously, that the spectrum of G is not degenerate, to avoid a reduction chain of subgroups. It is obvious that, given some positive constant C, and given any chosen γ among the values of g, there is no degeneracy for the GS of even if w has the symmetry. When several labels become necessary with a subgroup chain reduction, it is trivial to use a sum j C j (G j − γ j ) 2 of "pusher" terms. Finally a DF results, now from the HK argument with K. We stress here that pusher terms, because they commute with H, do not change the eigenstates of either H nor (H − E n ) 2 . Only their eigenvalues are sorted out and reorganized. Note that the pusher expectation value vanishes for ψ nγ . Naturally, when w is finite, eigenstates of K differ from those of K, but what counts is the information given by the DF when w vanishes. A simplification, avoiding cumbersome square operators H 2 , is worth noticing. Consider the operator, K = H + C (G − γ) 2 + W. At the limit where w vanishes, there is always a choice of a positive constant C which makes the lowest state with quantum number γ become the GS. This leads to a more restricted density functional that is of interest for the study of an yrast line.
That DF, F M [ρ], based upon K, provides the ex- where ψ min , square normalized to unity, is also constrained by the facts that ψ min |a † r a r |ψ min = ρ( r) and K|ψ min = ε|ψ min for the eigenvalue ε = F M . It may be interesting to find a DF that provides the expectation value of H itself. This can be done by taking the derivative of F M [ρ] with respect to E n , at constant ρ. We suppose that this derivative exists, which is the case for a discrete spectrum at least. With the notation |ψ = d|ψ /d E n , and using the fact that ψ min |W|ψ min + ψ min |W|ψ min = w( r) dρ( r)/d E n d r = 0, one can write: Therefore we can define a new DF, such that F D [ρ] = ψ min |H|ψ min and F D [ρ nγ ] = E n for the density ρ nγ of the eigenstate ψ nγ of H at energy E n . Furthermore one finds that δFD δρ [ρ nγ ] = 0, because δ ψmin| δρ H|ψ min + ψ min |H δ|ψmin δρ = E n δ ψmin|ψmin δρ = 0 for ψ min = ψ nγ . Hence the functional F D [ρ] is stationary at the exact density ρ = ρ nγ . It is not expected to be minimal at ρ nγ , however, unless the resulting eigenstate corresponds to the absolute GS when w vanishes.
Resonances may be defined as special eigenstates of H if one uses an argumentà la Gamow, allowing some radial Jacobi coordinate r ≥ 0 to show a diverging, exponential increase of the resonance wave function at infinity of the form exp(ipr), where the channel momentum p is complex and ℑp < 0. It is well known that those eigenvalues E n describing resonances are complex numbers, with ℑE n < 0. There have been extensive discussions in the literature about the physical, or lack of, meaning of such non normalizable wave functions and about the wave packets which might be used to replace them, [21,22,23,24]. The point of view we adopt in this note is based upon the Complex Scaling Method (CSM) [25,26,27,28]: a modest modification of H transforms narrow resonances into square integrable states; then there is no difference between the diagonalization for bound states and that for resonances. The cost of the CSM, however, is a loss of hermiticity: the CSM Hamiltonian H ′ is non hermitian, somewhat similar to an optical Hamiltonian [25,26,27,28].
Given the ket eigenstate equation, (H ′ − E n )|ψ n = 0, where |ψ n is now a square integrable resonance wave function, we can consider the hermitian conjugate equation, ψ n |(H ′ † − E * n ) = 0. Clearly, ψ n is a GS, as both a ket and a bra, of the hermitian and semipositive definite operator, Q exact = (H ′ † − E * n ) (H ′ − E n ), with eigenvalue 0. Applying the same argument as before, but now to Q exact instead of (H − E n ) 2 , demonstrates the existence of a DF around the targeted resonant state.
In practice we do not know E n exactly. Given a sufficiently close estimate E n of E n , an approximate GS eigenvalue |E n − E n | 2 occurs for Q apprx = (H ′ † − E * n ) (H ′ − E n ), at first order with respect to ∆Q = Q apprx − Q exact . Since ψ n is not a ket eigenstate of H ′ † = H ′ − 2i ℑH ′ , it is also perturbed at first order in ∆Q. Still one can copy the construction for F D [ρ], see Section 3, if one interprets the operator d/d E * n as d/dℜ E n + id/dℑ E n . The resulting functional F D [ρ] is linear in H ′ . For E n = E n the functional will be stationary at the density of the exact resonant state. While provinding a proof of existence, the construction of the exact functional for H ′ requires the knowledge of the exact eigenvalue E n . This might be an inconvenient limitation but fortunately calculations of numbers such as E n are usually much easier and much more precise than calculations of wave functions ψ n and/or of their densities.
If the resonance has good quantum numbers (QN)s inducing degeneracies, the same pusher terms as those which have been discussed above can be added to create a unique GS, from the operator, Q exact + C (G − γ) 2 . The HK argument, implemented with the full operator, then proves that DFs exist for those resonances regularized by the CSM. Notice, however, that a simplified theory, with an "yrast suited" operator K ′ , linear with respect to H ′ , is not available here, since the restoration of hermiticity forces a product H ′ † H ′ upon our formalism.
We now consider a special case of rather wide interest in nuclear and atomic physics. i) Good parity of eigenstates of H 0 = T + V or H = H 0 + U when u is restricted to be even, is assumed in the following. Hence our eigendensities, quadratic with respect to the states, have positive parities. ii) We also assume that the number of fermions is even. iii) The QNs in which we are interested in this Section are the integer angular momentum L and magnetic label M of an eigenstate Ψ LM of H, where it is understood that the two-body v and one-body u interactions conserve angular momentum. When w is switched on and is not rotationally invariant, eigenstates of K, K, or K may still tolerate such labels LM by continuity. First, consider w = 0. The density ρ LM comes from the product Ψ * LM Ψ LM , but it does not transform under rotations as an {LM } tensor. Rather, it is convenient to define "auxiliary densities", σ λ0 ( r) = It can therefore be written as the product of a spherical harmonic and a radial form factor, σ λ0 ( r) = Y λ0 (r) τ λ (r) = (2λ + 1)/4π L λ (cos β) τ λ (r), where L λ is a Legendre polynomial and the angle β is the usual polar angle, counted from the z-axis. Conversely, This provides a "Fourier analysis" of ρ LM in angular space. The density is parametrized by scalar form factors, τ λ . Since L is here an integer and furthermore ρ L−M and ρ LM are equal, and since Clebsch-Gordan coefficients have the symmetry property There are thus (L + 1) scalar functions, τ 0 , τ 2 , ..., τ 2L , to parametrize (L + 1) distinct densities ρ L0 , ρ L1 , ..., ρ LL . Because of the quadratic nature of the density observable, the even label λ for angular "modulation" of ρ runs from zero to twice L, with a "2L cut-off"; a signature, necessary if not sufficient, for an "L-density". Reinstate now w as the LT conjugate of ρ LM . It makes sense to study situations where w is restricted to expansions with (L + 1) arbitrary scalar form factors, w( r) = 2L even λ=0 Y λ0 (r) w λ (r). With inessential factors such as (−) L−M L − M L M | λ 0 omitted for simplicity in the following, every pair {r τ λ , r w λ } is conjugate. An eigendensity of K, K, K may have an infinite number of multipole form factors, but, with such restricted potentials w, only τ 0 , τ 2 , ..., τ 2L are chosen by the LT relating F M and F M .
It can make even more sense to restrict w to one multipole only, w( r) = Y λ0 (r) w λ (r), with λ = 0, or 2, ... or 2L, to study each multipole of ρ separately. For simplicity we now use the easier version of the theory, with that operator K which is suited to the yrast line. Add therefore to H a pusher term Z LM leaving intact the eigenstates, Here the subscript λ specifies that w is reduced to one multipole only. Then L is the total angular momentum operator and L z is its third component. This note offers theorems for the existence of exact DFs for every excited bound state, and even narrow resonances, and every set of good QNs used in nuclear, atomic and molecular physics. Furthermore, the densities used as arguments of our DFs do not need to be fully three-dimensional ones; they can be radial form factors of multipole components of the states under study. Our existence theorems, though, suffer from the usual plague of the field: constructive algorithms are missing and empirical approaches will have to be designed. What is the corresponding (KS) theory [3]? In its usual form, the task of calculating the kinetic energy part of the DF is actually left to the solution of a Schrödinger equation, and this can be trivially generalized to any one-body part. Published studies of the KS formalism are actually dedicated to calculations of the functional derivative, δV xc /δρ( r), of the exchange and correlation part of the DF, coming from the two-body part V of the DF. Our present use of modified Hamiltonians, or even squares of H, introduces two-body operators, but also three-and four-body operators. For the versions where no squares of H occur, see the yrast suited operator K and Section 3, the nature of the three-and four-body terms, typically coming from ( L. L) 2 , is not forbidding, because of obvious factorization properties. Hence a KS theory might be realizable for such simplified versions. With squared Hamiltonians, however, a KS theory seems out of reach at present. A systematic analysis of solvable models on a basis of "modes" [19], however, may help to extrapolate such models into practical rules. For the discussion of differentiability, representability and fine topological properties of the w-and ρ-spaces, we refer again to [16]. Up to our understanding of the topology of the variational spaces, flat or curved [29], of general use in nuclear, atomic and molecular theory, the validity domain of our existence theorems is quite large. We have not used the time dependent formalism, although much progress has been made in deriving excitation energies from it [30]. A generalization of our arguments to finite temperatures seems plausible, however, and insofar as inverse temperature may be viewed as an imaginary time, a generalization to a time dependent theory is not excluded.
B.G.G. thanks T. Duguet and J. Meyer for stimulating discussions, and also the hospitality of the Hokkaido University and Kyoto Yukawa Institute for part of this work. S.M.A.R. thanks D. Van Neck for interesting discussions.