# On Mathematical Structure of Effective Observables

C. P. Viazminsky and James P. Vary
a) Department of Physics, University of Aleppo, Syria
and International Institute of Theoretical and Applied Physics,
Iowa State University, Ames, Iowa 50011, USA
b) International Institute of Theoretical and Applied Physics
and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA
###### Abstract

We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigenvalues of the full observable, while the second serves to determine its matrix elements, in any basis in one of the subspaces, in terms of quantities pertaining to the model space. We also show that if the Hamiltonian of a physical system possesses symmetries then these symmetries continue to hold for its effective representatives of the first type. Maximum information about the system can be obtained in terms of two sets of effective representatives. The first set of representatives is complete. Other observables that do not commute with all members of the complete set have only one type of representative.

## 1  Introduction

Effective operators are often used in nuclear, atomic and molecular physics. The general scheme aims to construct from the Hamiltonian of the system, acting on the Hilbert space of wave functions, an operator that acts on a low dimensional space, so that the eigenvalues of the latter operator are also eigenvalues of the full Hamiltonian of the given system [1-9]. The low dimensional space, we have mentioned, is called a model space and the operator acting on it to produce some of the eigenvalues of the full Hamiltonian is called an effective Hamiltonian, or an effective representative of the Hamiltonian. The latter requirement does not determine an effective representative uniquely. A general class of effective representatives was obtained by Suzuki  who also delineated forms according to the role of an arbitrary parameter, the starting energy, in the iterative method of solution., or according to their Hermiticity. Hermitian forms have been introduced or adopted by many researchers [10-17]. A standard non-Hermitian form [2, 8, 18] is relatively simple, and is commonly used for implementing the scheme of effective representatives.

Our present work, which is concerned with the effective representation of any observable in the standard non-Hermitian scheme, has the following objectives:

1. To establish the equivalence between the decoupling condition on the transformed observable and a corresponding condition on its transformed eigenfunctions.

2. To show that the decoupling equation always has solutions and to specify the maximum number of inequivalent solutions.

3. Starting from a complete set of observables associated with the physical system, we construct a complete set of effective representatives, and prove accordingly that the symmetries of the Hamiltonian are carried over to the effective representatives.

4. Two effective representatives can be constructed associated with every observable. The first representative corresponds to the standard non-Hermitian form and gives some of the eigenvalues of the original observable. The second representative is Hermitian and has the property that the matrix elements of the original observable, in any basis of the subspace that is mapped onto the model sace, can be calculated in terms of this representative and the projected basis in the model space.

## 2 The Model Space

The truncated Hilbert space of square integrable functions associated with the system, denoted by , consists of all columns with complex entries. is just the unitary space of complex numbers through the isomorphism where denotes the transpose. The standard basis in will be denoted by , so that

 e1=(1,0,....,0)t,e2=(0,1,....,0)t,.........,eN=(0,....,0,1)t (1)

Let be a distinct subset of elements of the set The subspace generated by the subset of basis elements will be denoted by and will be called a model space. The projection on will be denoted by , whereas will denote the projection on the orthogonal complement . It follows that  If it is desired, one may rearrange the order of the basis elements (1) so that the vectors are placed first. We shall assume that such reordering is done whenever it is necessary, and drop the index , if no ambiguity arises. The symbol accordingly, will denote a projection on some model space . The reordering operation is particularly useful when we have to represent vectors and operators in matrix form.

Let be an operator in such that

 S=QSP (2)

It follows that and hence Equation (2) implies also that

 S=(0d0s0) (3)

where is the nil matrix. Consider the transformation

 e−S:HN→HN,ψ→˜ψ=(1−S)ψ. (4)

Setting where , we write

 ˜ψ≡(˜α˜β)=(10−s1)(αβ)=(αβ−sα) (5)

It is apparent that the mapping is determined by given by (3), which in turn is determined by

Through an obvious isomorphism we may overlook as being a subspace of and consider it as a space on its own right. Hence, and whenever it is convenient, we may set and thus consider as d-vectors instead of being -vectors with vanishing components in . A similar statement is applicable to and to the vectors and hence we may set It is evident from (5) that if then and hence every point in the invariant subspace is a fixed point of the transformation

Let be a linearly independent set of vectors. Hence there exists at least one subspace in which the set of projections of these vectors is linearly independent. This last statement is equivalent to say that the rank of the matrix is . The symbol designates the inner product.

We shall choose the matrix S such that

(ii) The set of vectors where is the projection corresponding to is linearly independent.

Requirement (ii) can always be satisfied through reordering the basis if necessary. By (5), requirement (i) implies or

 sPψi−Qψi=0(i=1,.....,d) (6)

We write (6) collectively as a matrix equation  in which As its columns are linearly independent the matrix is invertible, and hence

 s=[QΨ][PΨ]−1 (7)

Therefore requirements (i) and (ii) yield equation (7). It is easy to see that equation (7), which embodies in it that the matrix is invertible, is fact equivalent to conditions (i) and (ii).

With so chosen, the matrix given by (3) has the property:  projects every vector of the -dimensional space , generated by the set of vectors , onto the model space  This follows immediately from requirement (i) and linearity of In other words, an arbitrary vector is mapped by to . The operator is not a projection operator as implied by the mathematical definition of a projection operator. The word ”project” however is used here in a geometrical sense to describe an operation in which every vector of a certain subspace (visualized as hyperplane) is mapped to a vector that has the same first components, whereas its remaining components are zeros (visualized as a vector in a coordinate hyperplane). Also if then its image is not in the model space. The proof of the last fact relies on the regularity of , which implies that the image of the independent set namely is linearly independent, and hence . The vector therefore has at least one non-vanishing component outside the space .

The operator , with given by (7), as projects the subspace orthogonally on , is thus determined solely by and , and is independent of the particular choice of a set of independent vectors in . Indeed if is another set of independent vectors in , then

 ψ′i=\stackunderj=1d∑cjiψj(i=1,...,d) (8)

where are constants. Denoting the matrix whose elements are by , and the matrices whose columns are and by and respectively, we write the last relation as . Equivalently we have and . Substituting from these equations for and in (7) we get which proves our assertion.

We finally note that as is invertible, the inverse image of every vector , which also is identifiable with ,  is retrievable as

## 3 Lee and Suzuki Transformation

Let be a Hermitian matrix, with an independent set of eigenvectors, and consider the eigenequation

 Oψi=Eiψi(i=1,.....,N) (9)

Applying the Lee and Suzuki similarity transformation  to the matrix and to the truncated space , we obtain

 ˜O˜ψi=Ei˜ψi(i=1,....,N), (10)

where we have used tilde to designate transformed quantities so that

 ˜O=e−SOeS,˜ψ=e−Sψ. (11)

Our work will be distinguished from that of Lee and Suzuki through our identification of additional freedom in the choice of S. Multiplying both sides of (10) byand injecting  conveniently in the right hand side we get

 P˜OP˜ψi+P˜OQ˜ψi=EiP˜ψi(i=1,....,N). (12)

In a similar way we get

 Q˜OP˜ψi+Q˜OQ˜ψi=EiQ˜ψi(i=1,....,N). (13)

We shall choose the transformation (11) such that there exists a subset with for which (i) the set of vectors is linearly independent, and (ii) Such a choice, as we have seen in the previous section, is certainly possible.

Proposition 1: Let be such that the set  is linearly independent. The following assertions concerning the Lee and Suzuki transformation are equivalent:

A1.

A2.

A3. (i) the decoupling equation holds, and

(ii) are eigenvectors of

Proof. We have seen in section 2 that the assertions A1 and A2 are equivalent (this expression of was first given by Navratil and Barrett  ) . To prove that assertion A1 implies A3, we set  in (12) and (13) to find that are eigenvectors of and   Due to the linear independence of   the later d equations imply that  . Conversely, if   are linearly independent eigenvectors of  then the N-vectors are eigenvectors of  It follows that the inverse image of these vectors coincides with a subset  of eigenvectors of The subset clearly fulfills assertion 1. Hence A1 is equivalent to A3.

## 4 The Effective Form

When the transformed operator is such that for some subset with we refer to the operator as an effective representative of the operator  corresponding to the model space and to the form taken by as an effective form. When is in an effective form corresponding to the model space the matrix elements are all zero except those for which , and consequently we make the identification in which is considered a matrix. In a similar way we treat as an matrix.

We elaborate here on the effective form and develop a more explicit framework. We write the eigenequation (9) as

 (abb+f)(αiβi)=Ei(αiβi)(i=1,.....,N) (14)

where the matrix has been partitioned to submatrices corresponding to and with is a matrix. By (11) the last equation is transformed to

 (a+bsb−s(a+bs)+b++fsf−sb)(αiβi−sαi)=Ei(αiβi−sαi) (15)

The later equation is equivalent to (12) and (13) together. It is easy to check that every , given as in proposition 1, puts into an effective form corresponding to some model space . In other words, every  is a solution to the decoupling equation

 Q˜OP≡−s(a+bs)+b++fs=0. (16)

To demonstrate the converse we assume that the later equation is satisfied by some , and hence the action of on any vector in is zero. In particular this action is zero for all vectors such that are eigenvectors of , and hence

 −s(a+bs)αi+b+αi+fsαi=0(i=1,....,N) (17)

Making use of (14) we reduce the last equation to the eigenequation

 (f−sb)(sαi−βi)=Ei(sαi−βi)(i=1,....,N) (18)

which is the same as embodied in equation (15) but now extended to all . However, not all vectors can be eigenvectors of because the later operator has only eigenvectors. It follows that there exists a subset consisting of elements of such that which implies that as given in proposition 1.

We list here the following comments on the effective form assuming from now on that is in such a form. i.e. the transformation (10) is such that .

1. If is the effective form corresponding to the model space then the right hand-side of the secular (characteristic) equation can be factorized to a product of two polynomials; one of which is of degree in

 det(Oeff−EId).det(Q˜OQ−EIN−d)=0. (19)

The eigenvalues of is the set of zeros of these two polynomials. In practical problems the secular equation of can be solved numerically as it is of  low degree in , whereas that of is of  high degree in and it is often hopeless to approach it for direct solution. One may apply the method of effective form described in the previous section afresh to the operator . Or alternatively one may pick up a new set of eigenvectors, say determine , and consequently a new effective form. Alternatively the matrix could be determined by iterative methods [4, 11, 7].

2. If is an eigenvector of corresponding to then by (12) which implies that the -vector is complex orthogonal to the rows of matrix and the vector is not necessarily zero. Therefore, if is an eigenvector of  then, though is an eigenvector of belonging to the eigenvalue there may exist another eigenvector  of that belongs to the same eigenvalue In the latter case and is an eigenvector of belonging to the eigenvalue . It is clear that is an eigenvector of that belongs to the eigenvalue . We summarize the latter observations by the following proposition

Proposition 2. Let be an eigenvector of belonging to the eigenvalue .

(i) If then is an eigenvalue of to which the eigenvector belongs.

(ii) If then is an eigenvector of belonging to the eigenvalue If in addition, then is not an eigenvector of unless In the latter case is a common eigenvalue of and to which the independent eigenvectors and belong. In the latter case the spectra of and intersect.

(iii) is an eigenvector of does not necessitate that However if the spectra of and do not intersect in then is an eigenvector of belonging to the eigenvalue

3. If the matrix  is singular for some choice of model space, say then we have to replace it by another such that the matrix is invertible. There certainly exists such a new choice of model space, otherwise the set would be linearly dependent.

We demonstrate here that for a given set of eigenvectors  , two legitimate choices of model spaces lead to two effective representatives which are related by a similarity transformation. Let and be two legitimate choices and denote the projections on the corresponding model spaces by and respectively. This leads to two distinct , say and , and hence to two distinct effective representatives and If is the set of eigenvalues to which belong, then

 Oeff[PΨJ]=[PΨJ]ΛJ,O′eff[P′ΨJ]=[P′Ψj]ΛJ (20)

where is a diagonal matrix with diagonal elements From (20) we deduce that

 Oeff=[PΨJ][P′ΨJ]−1O′eff[P′ΨJ][PΨJ]−1 (21)

which proves our claim.

Each independent set of eigenvectors provide at least one model space The number of possible choices of is not less than one and not greater than  which is of course the number of independent sets of projections . All such choices lead of course to the same set of eigenvalues

If the eigenvalues of are non-degenerate then different choices of out of the set of independent eigenvectors result in effective representatives with different spectra. The total choices of inequivalent effective representatives corresponding to is and within each of these there are a maximum number of equivalent representatives.

The above-identified freedoms are new and extend the work of Lee and Suzuki.

## 5 Spectral Representation of Oeff

Let be an effective representative of the operator in the model space , and let be the spectrum of , to which the vectors belong, so that Since each lies in the model space we have

 Pψi=\stackunderμ=1d∑ciμeμ(i∈J) (22)
 =\stackunderμ=1d∑c∗iμcjμ≡γij(i,j∈J) (23)

The matrix is clearly Hermitian, and determines the overlap the eigenvector of one with respect to another. Let

 χ=\stackunderj∈J∑bjPψj (24)

be an arbitrary vector in the model space, then

 =\stackunderj∈J∑γijbj(i∈J) (25)

Hence

 \stackunderi∈J∑γ−1ki=bk(k∈J) (26)

where is the inverse of the matrix . It is clear that always exists since are linearly independent. Applying to where are given by (26 )  we get

 Oeff∣χ>=\stackunderi,k∈J∑γ−1kiEkPψk (27)

This yields

 Oeff=\stackunderi,k∈J∑Eiγ−1ik∣Pψi>

which expresses in terms of quantities pertaining to the model space.

## 6 A Complete Set of Effective Representatives

Let be the Hamiltonian of a physical system and be a set of observables pertaining to the system so that the set of observables  is complete. It follows from the latter assumption that

 [Oρ,Oσ]=0(ρ,σ=1,....,c) (28)

The energy eigenvectors of any suitably chosen simple Hamiltonian could be taken as a basis for the Hilbert space of wave functions of the physical system. For example, these could be the eigenstates of the simple harmonic oscillator, when considering the bound states of the nucleus. Obeservables pertaining to the system are represented by Hermitian matrices in terms of this basis. Unless the matrices representing observables are given by recurrence formulae, we have to be content with finite matrix approximations, which imply truncating the infinite basis at some sufficiently large term . The space generated by the truncated basis will hopefully contain good approximations of all states of interest to the problem we consider.

It must be noted that, whenever the eigenvalue problem is to be solved numerically, which is usually the case in physically interesting problems, truncation is an inevitable task. We note that truncating an infinite basis by a finite one with a sufficiently large number of basis elements is justified by the fact that the sequence tends weakly to zero as tends to infinity. This means that for every wave function   the sequence of numbers  tends to zero as tends to inity [19, 20]. Alternatively, an upper cutoff, can be safely applied without seriously changing the low-lying properties.

It is noted that all the algebra carried out in the previous section, or to be carried out in the forthcoming discussion, is valid for infinite matrices as much as it is valid for finite ones, and hence we may replace N by without affecting the validity of these results.

The Hermitian commuting set of matrices is complete, and there exists accordingly a complete set of simultaneous eigenfunctions of the observables such that

 Oσψi=Eσiψi(i=1,...,N;σ=1,....,c) (29)

where are the eigenvalues of the observable to which the eigenvector     belongs. The eigenvectors given by (29) are preserved when the similarity transformation (11 ) is applied to the Hilbert space of wave functions and to the operators acting on , and hence

 ˜Oσ˜ψi=Eσi˜ψi(i=1,...,N;σ=1,...,c) (30)

Assume that the eigenvectors   are such that the set is linearly independent, and take

 s=[Q˜ψ1.......Q˜ψd][P˜ψ1......P˜ψd]−1 (31)

The matrix is the same for observables forming the complete set , for it is constructed of the same subset of the simultaneous eigenvectors of . The resulting transformed observables , have the same effective form, and hence have as a common subset of eigenvectors . Define a set of effective representatives

 Oσeff=P˜OσP(σ=1,....,c) (32)

and hence

 OσeffP˜ψi=EσiP˜ψi(i=1,....,d;σ=1,....,c) (33)

It follows, and since the set  is complete in the model space , that

 [Oρeff,Oσeff]=0(σ,ρ=1,...,c) (34)

The effective Hamiltonian and the effective representatives we have constructed have the virtue that the symmetries exhibited by the original Hamiltonian are carried over towith the effective representatives playing the role of generators of symmetry for

The matrices (32 ) are obviously non-Hermitian and, consequently, the expectation value of an effective representative in a state in the model space is generally a complex number. An exception to this fact is that when is an eigenvector of . In this case

 Pψi=/∥Pψi∥2=Ei (35)

## 7 A Second Type of Effective Representative.

The role of an effective operator seems limited to producing some of the eigenvalues of the original observable. However, we may enhance the scheme of ”effectiveness” and make a further step as follows: The matrix which is determined by iterative methods [4, 7, 17] and utilized to construct the effective representative can also be utilized to construct an effective representative of a second type that satisfy the property

 <ψ|O|ϕ>= (36)

for all . Using (11) and the definition of the adjoint operator, we have

 ⟨ψ|O|ϕ⟩ = ⟨e−S˜ψ∣O∣e−S˜ϕ⟩=⟨˜ψ∣e−S+Oe−S∣˜ϕ⟩ = ⟨P˜ψ∣e−S+Oe−S∣P˜ϕ⟩=⟨Pψ∣Pe−S+Oe−SP∣Pϕ⟩

The requirement ( 36) is fulfilled on taking

 ¯¯¯¯Oeff=Pe−S+Oe−SP (38)

In particular

We therefore associate with every observable two effective representatives. The first, , serves to determine some of the eigenvalues of and the projection of the corresponding eigenvectors on the model space; the second has the important property : the matrix elements of the original operator with respect to any basis in the space is given in terms of and the projected basis in the model space. It is evident that the last matrix can be calculated easily since is known whenever is known, and since the basis elements of the model space have finite components. We mention that the matrix is not the matrix of since is not orthogonal. In particular, and if then

 ψ==∥Pψ∥2<¯¯¯¯Oeff>Pψ

Expressed in words, the expectation value of the observable in the state is equal to the expectation value of its representative of the second type in the projection of the given state on the model space times the square norm of this projection.

For observables that do not commute with all elements of the complete set we can define only one effective representative, that is the effective representative of the second type  This serves to give a portion of the transition matrix of , namely that which corresponds to a basis of

A systematic study of the system is achieved by decomposing the space into linear subspaces with if so that

 HN=LinΨJ1⊕....⊕LinΨJa (39)

Now in each subspace we assign to every observable in a complete set of observables an effective representative of the first type and an effective observable of the second type . These effective representatives, of first or second type, differ from one subspace to another as does If representatives of the first type are all obtained, all eigenvalues of the full observable become known. Also if , then we have where denotes the projection on the model space corresponding to the subspace . Although similar relations are valid for every two vectors in the same subspace, one can not express in terms of representatives of second type when and belong to different subspaces, and consequently when they are arbitrary vectors in .

## 8 Towards Practical Applications

Following the traditional lines of thinking for many-body problems, we suggest that is developed for small sub-systems and used as an approximation for the full For example two and three body problems may be solved with high precision using current numerical techniques [16, 17]. A set of solutions is selected, is evaluated and the resulting is then used in many body problems within the appropriately restricted model space. Detailed tests will be needed for specific Hamiltonians to determine the efficacy of this approach and the utility of the various freedoms we have identified within the present work.

## 9 Acknowledgments

The authors thank Kenji Suzuki for a valuable discussion. Cesar Viazminsky acknowledges the generous financial support provided by the University of Aleppo in Syria. James P Vary acknowledges partial support by the U.S. Department of Energy under grant No. DE-FG02-87ER-40371, Division of High Energy and Nuclear Physics.

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