Hamiltonians generated by Parseval frames

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a $\D$-quasi basis, \cite{bell,bit}, rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the {\em mathematical } Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the {\em physical} system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$-connection for observables. Several examples are discussed.

and its maximal domain of definition is D(H) = {g ∈ H : N n=1 E n e n , g e n ∈ H} = {g ∈ H : N n=1 E 2 n | e n , g | 2 < ∞}. In view of (1.1), the physical part of H has the form E n e n , P g ϕ n = N n=1 E n P e n , P g ϕ n = N n=1 E n ϕ n , f ϕ n , where f = P g and ϕ n = P e n . Therefore, the operator H ph = P HP acts in a subspace H ph of H and E n ϕ n , f ϕ n , f ∈ D(H ph ) = P D(H).
(1. 2) In general, the set of vectors F ϕ = {ϕ n = P e n , n = 1, 2, . . . , N} loses the property of being an ONB of H ph and, instead, it turns out to be a Parseval frame 1 of H ph . For this reason, the operator H ph in (1.2) can be considered as a Hamiltonian H ϕ (= H ph ) generated by a Parseval frame F ϕ , in analogy with what discussed in [2,3]. The operators defined by (1.2) are particular case of more general concept of multipliers, which was introduced and studied in [7,22]. Furthermore, for finite Parseval frames (N < ∞), the operators H ph can be regarded as quantum observables that are obtained by using a finite version of the Klauder-Berezin-Toeplitz-type coherent state quantization [14] of a real-valued function f defined on a set of data {a 1 , . . . , a N } related to a physical system. The eigenvalues of H ph form the 'quantum spectrum' of a classical observable f whereas its 'classical spectrum' coincides with the set of values {E n = f (a n )} N n=1 [12,13]. These results can be generalized to the case of infinite Parseval frames with the use of POVM quantization developed in [15].
The main objective of this paper is the investigation of Hamiltonians generated by Parseval frames. As we will see, an interesting aspect is that, when we project from H to H ph , the eigenvalues of the observables are not preserved in general. This could have relevant consequences in concrete situations, where H is just a formal simple operator, while H ph is its really useful physical counterpart. In other words, while each E n is an eigenvalue for H, see (1.1), E n is not an eigenvalue for H ph if ϕ n = 1, despite of the fact that H and H ph share similar expansions (1.1) and (1.2), but in terms of families of vectors which have different properties. In particular, F ϕ is not an ONB, and therefore ϕ j , ϕ k = δ j,k , in general. We construct five explicit examples in Section II. 2 showing, among other features, how eigenvalues change.
In the present paper, we concentrate on the case of bounded operators. Unbounded Hamiltonians generated by Parseval frames have a lot of delicate properties and they will be considered in a forthcoming paper.
The paper is structured as follows: after short preliminaries about frames in Section II.1, we begin our analysis of the bounded Hamiltonians H ϕ . General methods of the calculation of eigenvalues and eigenvectors of H ϕ are presented in Theorem 6 and Corollary 7. In Section III, we introduce and study a physically motivated relation (E-connection) between Parseval frames and ONBs (Definition 8). It should be mentioned that this relation, when it exists, is not related to the possibility of getting an ONB in a larger space (see the Naimark dilation theorem 2, [18]) since it works in the same Hilbert space. In fact, it is more connected with what is stated in [19,Corollary 8.33] or [11,Theorem 5.5.5]. Conclusions are given in Section IV.

II.1 Frames and Parseval frames
Here all necessary information about frame theory are presented in a form convenient for our exposition. More details on frames can be found in monographs [11,19]. The papers [6,8,10,18] are recommended as complementary reading on the subject.
Let K be a complex Hilbert space with scalar product ·, · linear in the second argument. Let J denote a generic countable (or finite) index set such as Z, N, N ∪ {0}, etc. By |J| we denote the cardinality of J.

Definition 1 A set of vectors
(2.1) The optimal constants in (2.1) (maximal for A and minimal for B) are called the frame bounds. A frame F ϕ is called a tight frame if A = B, and is called a Parseval frame (PF in the following) if A = B = 1. The potential of a frame F ϕ is defined by FP[F ϕ ] = j,i∈J |(ϕ i , ϕ j )| 2 . Following [6], we recall that the excess e[F ϕ ] of F ϕ is the greatest integer n such that n elements can be deleted from the frame F ϕ and still leave a complete set, or ∞ if there is no upper bound to the number of elements that can be removed. Formula (2.1) with A = B = 1 is quite similar to the Parseval equality for ONBs and in fact it is well known that a PF is indeed an ONB if each vector in F ϕ is normalized: ϕ j = 1 for all j ∈ J, and vice-versa. On the other hand, some of the vectors in a PF may be the zero vector, while this is not allowed for ONBs. Similarly to ONBs, PFs have the maximality property: they cannot be enlarged to a PF by adding non-zero vectors (a unique way to enlarge is to add zero vectors).
It is well known that, given an ONB F e = {e j , j ∈ J} in a Hilbert space H containing K as a subspace, and an orthogonal projection P : H → K, the set F ϕ = {ϕ j = P e j , j ∈ J} is a PF for K. The inverse of this statement is the so-called Naimark theorem (see, for instance, [18, Propositions 1.1, 1.4]), which is crucial for our investigations. More precisely: is an ONB for 2 H = K ⊕ M. Each Parseval frame F ϕ determines an analysis operator θ ϕ : K → ℓ 2 (J): which is an isometry θ ϕ : K → ℓ 2 (J). The image set R(θ ϕ ) is a closed subspace of ℓ 2 (J) and, because of the Theorem 2, where the isometry θ ψ : M → ℓ 2 (J) is defined by the PF F ψ introduced in Theorem 2. The next statement is well known in the frame theory. For convenience of the reader we give its simple proof.
Proof: If c j ϕ j = 0, then f = c j e j = c j ϕ j + c j ψ j = c j ψ j due to (2.2). This means that f ∈ M and c j = e j , f = ψ j , f . Hence, θ ψ (f ) = {c j }.
Conversely, if {c j } ∈ R(θ ψ ), then there exists f ∈ M such that c j = ψ j , f = e j , f . Hence, f = c j e j = c j ϕ j + c j ψ j that gives c j ϕ j = 0, since f ∈ M. ✷ Lemma 5 The Gram matrix G ϕ = [ ϕ i , ϕ j ] i,j∈J of a PF F ϕ determines a bounded selfadjoint operator in ℓ 2 (J) which is the orthogonal projector in ℓ 2 (J) onto R(θ ϕ ).
Proof: In view of [19,Theorem 7.5], the Gram matrix G ϕ determines a bounded self-adjoint operator in ℓ 2 (J), which acts as follows G ϕ : {c j } j∈J → { j∈J ϕ i , ϕ j c j } i∈J . This operator coincides with the orthogonal projection in ℓ 2 (J) onto R(θ ϕ ) constructed in [11,Proposition 5.3.6] (here, we should take into account that the frame operator S in the formula (5.12) of [11] is the identity operator, since F ϕ is a PF and the scalar product in [11] is linear in the first argument). ✷

II.2 Working with bounded operators
Assume that {E j } j∈J is a bounded sequence of real numbers. Hence, there exist E min = inf j∈J E j and E max = sup j∈J E j such that −∞ < E min ≤E j ≤ E max < ∞. The Hamiltonian generated by a PF F ϕ and a collection of numbers {E j }: is a bounded everywhere defined self-adjoint operator in K. These properties of H ϕ can be established as follows: due to Theorem 2 there exists an ONB {e j } of H such that P e j = ϕ j , where P is the orthogonal projection in H on K. Therefore, H ϕ = P H e P , where is a bounded self-adjoint operator in H with eigenvalues {E j } and the corresponding eigenvectors {e j }. The spectrum of H e coincides with the closure of the set of its eigenvalues: In general, as already noticed, we cannot claim that the quantities {E j } in (2.5) are also eigenvalues of H ϕ . This will become evident in the examples below.
Let us assume that H ϕ f = µf for nonzero f ∈ K. Since F ϕ is a PF, f = j∈J ϕ j , f ϕ j and the eigenvalue equation takes the form: In view of Lemma 4, the relation (2.7) is equivalent to the condition . Summing up, this proves:

Theorem 6 The operator H ϕ has an eigenvalue µ if and only if there exists a sequence
where R(θ ϕ ) and R(θ ψ ) are the orthogonal subspaces of ℓ 2 (J), see (2.4). In this case, the corresponding eigenvector is f = j∈J c j ϕ j .
Lemma 5 allows one to carry out modifications of Theorem 6 which, sometimes, are more convenient for the calculation of eigenvalues. Simultaneously with the Gram matrices Corollary 7 Let H ϕ be a Hamiltonian generated by a PF F ϕ , see (2.5). The following are equivalent: Proof: Let F e = {e j = ϕ j ⊕ ψ j : j ∈ J} be an ONB of H determined in Theorem 2 and let be the corresponding analysis operator that isometrically maps H onto ℓ 2 (J). In view of (2.6), θ e (H e g) = Eθ e (g), where E is a bounded multiplication operator in ℓ 2 (J): where P is the orthogonal projection in H on K and θ e is defined by (2.9), we arrive at the conclusion that θ e maps K onto R(θ ϕ ) (see the decomposition (2.4)). Taking into account that H ϕ = P H e P , we obtain that H ϕ is unitary equivalent to the operator E ϕ acting in the subspace R(θ ϕ ) of ℓ 2 (J): where P = θ e P θ −1 e is the orthogonal projection operator in ℓ 2 (J) on R(θ ϕ ). (i) ⇐⇒ (ii). In view of (2.11), µ ∈ σ p (H ϕ ) if and only if there exists {c j } ∈ ℓ 2 (J) such that P{c j } = 0 and P(E − µI)P{c j } = 0. By Lemma 5, the operator P coincides with the Gram operator G ϕ . Therefore, the first condition takes the form G ϕ {c j } = 0, whereas the second one: To prove the equivalence of (i) and (iii) it suffices to repeat the previous arguments and rewrite the condition P(E − µI)P{c j } = 0 as (E − µI)P{c j } ∈ R(θ ψ ). Taking into account that G ψ is an orthogonal projection on R(θ ψ ) and (E − µI)P{c j } = D ϕ (µ){c j } we complete the proof. ✷

II.3 Examples
In this section we propose some examples of H and their related H ph , starting with purely mathematical examples, and then considering applications arising in concrete applications discussed in the literature. In particular, in Examples 1, 2, and 3, we will show how a physical Hamiltonian H ϕ = N j=1 E j ϕ j , · ϕ j , given in terms of some particular PF, can be rewritten in terms of an ONB of its eigenvectors, and how its related eigenvalues are different from the E j in the expansion above. Of course, in view of what we have discussed in the Introduction, H ϕ should be understood as the physical part of another, larger, Hamiltonian, H, which produces H ph after a suitable projection. This means that we have a first ONB, given by the eigenvectors of H, which, when projected to H ph , defines a PF. Hence, to find the physical containt of H ph , we need to diagonalize H ph , getting a second ONB, different from the first one, which can really be considered as the physical eigenvectors of the system. In Examples 4 and 5 these steps will be particularly emphasized.
Example 1. Mercedes frame.-In the Hilbert space H = C 3 , we consider the ONB F e = {e j , j ∈ J}, where J = {1, 2, 3} and The orthogonal projection of F e = {e 1 , e 2 , e 3 } onto the subspace The image sets of the analysis operators θ ϕ and θ ψ are subspaces of ℓ 2 (J) = C 3 : Let E 1 , E 2 , and E 3 be real quantities. Then the Hamiltonian 3 H ϕ = 3 j=1 E j ϕ j , · ϕ j generated by the Mercedes frame acts in C 2 (we identify K with C 2 ) and, due to Theorem 6, µ ∈ R is an eigenvalue of H ϕ if and only if the linear system Another way to obtain (2.12) is, of course, to present H ϕ (acting in C 2 ) in the matrix form Here N ± are normalization constants fixed to have ẽ ± = 1. This means that, using a bra-ket notation, the operator H ϕ can be rewritten in term of an ONB of its eigenstates: 3 We are assuming here that H ϕ arises from a different operator H, acting on a larger Hilbert space C 3 , after taking its projection H ϕ = P HP on the physical Hilbert space H ph = C 2 . Notice that we are not giving here the explicit form of H, since it is not really relevant. We will do this in Example 4 and Example 5, since in those cases H has a physical meaning.
Representing each vector f ∈ K as f = K i=1 x i e i and using (2.9), we define the subspaces R(θ ϕ ), R(θ ψ ) of ℓ 2 (J) = C K+1 : Let E 1 < E 2 < . . . < E K+1 be real quantities. Then the Hamiltonian H ϕ = K+1 j=1 E j |ϕ j ϕ j | generated by the frame F ϕ acts in K and, due to Theorem 6, its eigenvaluesẼ j coincide with µ ∈ R for which the linear system x. An elementary analysis shows that the largest eigen-valueẼ K of H ϕ coincides with E K+1 , while the other eigenvaluesẼ j , j = 1, . . . K − 1 are the roots of the equation The corresponding eigenfunctions are where µ j (j = 1, . . . K − 1) is a solution of (2.15). After the normalization of f j we obtain H ϕ = K j=1Ẽ j |f j f j |, which is a different way to write H ϕ = K+1 j=1 E j |ϕ j ϕ j |. Example 3.-The finite PF considered above is a key counterpart of an infinite PF which contains no subset that is a Riesz basis [9]. We investigate spectral properties of Hamiltonians generated by this curious PF.
Let K be a separable Hilbert space. Index an ONB for K as {e K j } K∈N,j=1,...,K . Set K K = span{e K 1 , e K 2 , . . . e K K }. The vectors ϕ K j ≡ ϕ j defined by (2.14) form a PF F ϕ K = {ϕ K j , j = 1, 2 . . . K + 1} of the space K K . Since K = ∞ K=1 ⊕K K , the collection of vectors F ϕ = ∞ K=1 ⊕F ϕ K is a PF for K (and it was constructed in [9]). Assume that {E K j } K∈N,j=1,...,K+1 is a bounded strictly increasing sequence, i.e. E 1 It is easy to verify (using the previous example) that the point spectrum of H ϕ involves the subset {E K K+1 } ∞ K=1 of the original quantities and the solutions of the equations, cf. (2.15): Let us now discuss a fourth, physically motivated, example, based on the anti-commutation relations (CAR) for two fermionic modes.
Example 4.-Let a 1 and a 2 be the operators satisfying the CAR {a j , a † k } = δ j,k 1 1, with {a j , a k } = 0, j, k = 1, 2. Here 1 1 is the identity operator in the Hilbert space of the system, which is H = C 4 . Calling η 0,0 the vacuum of a j , a j η 0,0 = 0, j = 1, 2, we can construct three more vectors acting on it with a † j : η 1,0 = a † 1 η 0,0 , η 0,1 = a † 2 η 0,0 and η 1,1 = a † 1 a † 2 η 0,0 . An explicit realization of these vectors and operators is the following: η 0,0 = δ 1 , η 1,0 = δ 2 , η 0,1 = δ 3 , η 1,1 = δ 4 , where {δ j } is the canonical ONB in C 4 , and In [4] these operators have been used to construct a dynamical system describing two populations moving in a two-dimensional lattice, and interacting adopting a sort of predator-prey mechanism. The dynamics has been produced by the sum of several copies of a single-cell term, H α (α labels the lattice cells), plus a global contribution responsible for the migration of the species. Here we only consider the single-cell term, which we rewrite as follows: where ω 1 , ω 2 and λ are parameters whose values can be fixed in different way, according to which aspect of the system we want to put in evidence, see [4]. We also refer to [4] for the meaning of this Hamiltonian, for the rationale for its introduction, and for the dynamics connected to it. The matrix expression for H is , whose eigenvalues are For concreteness, if we take λ = 2, ω 1 = 1 2 and ω 2 = 7 2 , then , while e 1 and e 2 are those previously introduced. Also, E 1 = 0, E 2 = 4, E 3 = − 1 2 , E 4 = 9 2 , and we can rewrite H = 4 j=1 E j |e j e j |. Due to the physical interpretation of the model, [4], if we consider the orthogonal projector P = 1 1 − P 1,0 , P 1,0 f = η 1,0 , f η 1,0 , P project the system on a space in which it is impossible to find the system in a state with high density of the first species, and low density of the other species. In other words, this state is forbidden for us. The (biological) reason for requiring this is that, for instance, we want to avoid a dominance of the first species on the second one. The computation of H ph is easy: as for the effect of P on the eigenvectors {e j } of H, we get four vectors ϕ j = P e j in H = C 4 . Removing the zero second component from each one of these vectors 4 we recover the following PF F ϕ = {ϕ j , 1 ≤ j ≤ 4} in K = C 3 , The operator H ph acting in K = C 3 coincides with the operator H ϕ = 4 j=1 E j |ϕ j ϕ j | generated by F ϕ . In view of (2.8) and (2.16), the Gram matrix G ϕ and B ϕ (µ) are: These matrices and Corollary 7 allow one to describe eigenvaluesẼ 1 ,Ẽ 2 , andẼ 3 of H ph = H ϕ . Precisely,Ẽ 1 = E 1 = 0,Ẽ 2 = E 2 = ω 1 + ω 2 , whileẼ 3 coincides with the solution µ of the equation (E 3 − µ) cos 2 β + (E 4 − µ) sin 2 β = 0. So we see how the eigenvalues of H ph are different from those of the original Hamiltonian H. Of course, these differences have consequences on the dynamics of the system, but this aspect will not be discussed in this paper. However, we want to stress that differences in the eigenvalues imply, among other differences, different stationary states. This might have serious, and very interesting consequences: a system which, in principle, should not evolve in time (since it is in a supposed stationary state), does indeed evolve. Or vice-versa. In both cases this can be related to the fact that the real Hamiltonian for the system is not H, but H ph . This is because of some constraint on the system. The last example we want to discuss here is again built in terms of anti-commutation relations (CAR) and fermionic modes, and is based on an ecological system considered first in [5] and then in [1].
Example 5.-The system we want to describe is made by two levels of organisms (L 1 and L 2 ), one compartment for the nutrients and another compartment for the garbage, see Figure 1 for a schematic view. The nutrients feed the organisms of L 1 , which feed those of L 2 . Moreover, when dying, the organisms of both levels contribute to increase the density of the garbage which, after some time, turns into nutrients. This is a simple example of closed ecosystem. More complicated systems (with more levels and with different kind of garbages) are considered in [5].
The dynamics of our system is described by the following Hamiltonian: where {a j , a † k } = δ j,k 1 1, and a 2 j = 0, for all j, k = 0, 1, 2, 3, and where ω j , ν j and λ j are real constants, whose meaning is explained in [1,5]. The zero-th fermionic mode (j = k = 0) is related to the nutrients, the 3-th mode to the garbage, while the two remaining modes describe the organisms of the various trophic levels. We again refer to [5] for the biological meaning of the various terms in H. Here, we just comment that, for instance, the term λ j a 3 a † 0 , describes the fact that the garbage is recycled by decomposers and transformed into nutrients: this is due to the fact that the density of the garbage decrease (because of the presence of the lowering operator a 3 ) , and simultaneously the density of the nutrients increases (because of the raising operator a † 0 ). The equations of motion can be deduced using the Heisenberg ruleẊ = i[H, X], and the solution (which is not essential here) can be found in [5]. Here we want to see how the idea introduced in this paper works for this particular example. For that, let e 0,0,0,0 be the vacuum of the system: a j e 0,0,0,0 = 0 for j = 0, 1, 2, 3. This corresponds to an essentially empty system: very low densities in all the levels (including garbage and nutrients). Analogously, e 0,0,1,0 is a vector describing a situation where only L 2 is filled, while all the other levels are essentially empty, and so on. We now fix, for concreteness, ω 0 = ω 3 = 2, ω 1 = 3, ω 2 = 4, λ 0 = 1, λ 1 = 2, λ 2 = 3, ν 0 = 1 and ν 1 = 3. The criterium for choosing some particular values of the parameters of the Hamiltonian is widely discussed in [1]. Here this choice is relevant just to allow a simple computation of the eigenvalues and the eigenvectors of H, which acts in H = C 16 and turns out to be the Hermitian matrix: Using any mathematical software one easily deduces the following list of eigenvalues {E j } 16 j=1 , ordered in decreasing absolute value: 13.6645, 11.4925, 11, 9.17202, 8.82798, 7, 6. What we are interested in here is to see the mathematical consequences if we assume that the second level, L 2 , cannot be filled because of, say, some particular reason: for instance, the organisms in L 2 receive nutrients from L 1 , see Figure 1, but the external conditions can be not sufficiently good to allow these organisms to survive. From a mathematical point of view, this situation can be described by introducing the following orthogonal projector where P i,j,0,k f = e i,j,0,k , f e i,j,0,k , for all vector f in H. The physical Hamiltonian H ph = P L 2 HP L 2 is an operator acting in the subspace K = span{δ 1 , δ 2 , δ 3 , δ 4 , δ 9 , δ 10 , δ 11 , δ 12 } of H, where {δ j } is the canonical ONB of C 16 : , whose eigenvalues are the following: whereẼ j are the eigenvalues above andφ j are the related eigenvectors, whose set is an ONB in H ph = C 8 . As in the previous example, the difference between E j andẼ j may have consequences on the dynamics of the system, but this aspect will not be considered in this paper.

E-connection between PFs and ONBs
The examples above suggests to introduce the following, physically motivated, relation between PFs and ONBs in a given Hilbert space K: Definition 8 Given a PF F ϕ = {ϕ j ∈ K, j ∈ J} and a bounded set of real numbers E = {E j , j ∈ J}, we say that the pair This is exactly what we have seen in the examples considered in Section II.3. In other words, (3.1) can be used to give two different representations of the same operator H ϕ (the physical Hamiltonian, in our examples), one in terms of a PF, F ϕ , and one, possibly more relevant for its physical interpretation, in terms of Fẽ.
If a pair (F ϕ , E) is E-connected to an ONB Fẽ, then the real numbersẼ k in (3.1) are eigenvalues of the self-adjoint operator H ϕ , whereasẽ k are the corresponding normalized eigenvectors. The eigenvaluesẼ k can be defined with the use of Theorem 6 and obviously, they are defined uniquely by the pair (F ϕ , E). Moreover, due to Remark 3, the cardinality |J ′ | coincides with the frame potential FP[F ϕ ] and it cannot exceed the cardinality |J| = If F ϕ is an ONB in K, then the pair (F ϕ , E) is E-connected to the same ONB F ϕ = Fẽ and E =Ẽ. A slightly more interesting example is the following: if ϕ 2k−1 = ϕ 2k = 1 √ 2 e k , where F e = {e k , k ∈ N} is an ONB of K, then the set F ϕ = {ϕ j , j ∈ N} is a PF in K and the pair (F ϕ , E) is E-connected to F e = Fẽ. In this case,Ẽ = {Ẽ k = 1 2 (E 2k−1 + E 2k ), k ∈ N}. Proof: The self-adjoint operator H ϕ = N j=1 E j |ϕ j ϕ j | acts in a finite dimensional space K with dim K = M. Hence its spectrum is discrete and the corresponding normalized eigenfunctions {ẽ k ∈ K, k = 1, . . . M} form an ONB of K.
If all E j are positive, then the set Fφ = {φ j = E j ϕ j , j = 1, . . . N} is a frame in K and the operator H ϕ turns out to be a frame operator S = N j=1 |φ j φ j | of Fφ. In this case, the first three relations in Proposition 9 follow from [10, Section 5] and [8, Corollary 2.3]. The inequality (3.2) follows from [10,Theorem 6.3].
✷ Another simple result is given by the following Lemma, where the validity of the resolution of the identity is used both for the ONB and for the PF.
Lemma 10 Let F ϕ be a PF and let the real numbers E j be the same for all j ∈ J (i.e., E j = λ for all j ∈ J). Then the pair (F ϕ , E) is E-connected to every ONB Fẽ of K and E =Ẽ = {λ}.
Proposition 11 Let F ϕ be a PF and let the set of real numbers E = {E j , j ∈ J} have only one accumulation point λ. Then there exist an ONB Fẽ and a set of numbersẼ k such that the pair (F ϕ , E) is E-connected to Fẽ.