Multiplication of distributions in a linear gain and loss system

We consider a model of coupled oscillators which can be seen as a gain and loss system. In the attempt to quantize the system, we propose a new definition of multiplication between distributions, and we check that this definition can be adopted when checking the biorthonormality of the eigenstates of the Hamiltonian H of the system, and of its adjoint H†\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^\dagger $$\end{document}. In the analysis carried out here, the role of weak pseudo-bosonic ladder operators is relevant.


I Introduction
Since many decades simple electronic circuits have been used in connection with classical and, more recently, quantum mechanics, since they can provide concrete devices where some physical effect can be observed, or modelled.A well known example of such a map between Electronics and Classical Mechanics is given by any RLC-circuit (RLCc), which is dynamically equivalent to a damped harmonic oscillator (DHO).This is because the time evolution of the charge in an RLCc, i.e. a circuit with an inductance, a resistance and a capacitor in series, is driven by exactly the same equation of motion of that of a DHO, with the dissipative effect of the resistance replaced by the friction, see e.g.[1,2] and references therein.In [3]- [6], among others, a quantum version of the DHO, and therefore of the RLCc, has been considered both for a purely mathematical interest and in view of possible applications.In particular, due to its particularly simple form, a deep comprehension of the DHO/RLCc is surely a first step toward a better understanding of the correct quantization procedure for generic dissipative systems, see [7,8], which are so relevant in several physical contexts.
A similar interest is at the basis of other papers connected with related problems, [9]- [11]: in all these papers electronic circuits are analyzed, either in connection with PT-Quantum Mechanics and exceptional points, [12], or because they produce interesting results when one tries to quantize the system, not only for the physical consequences of this quantization, but also because of its many mathematical consequences.
For instance, some interesting mathematics appears when quantizing a DHO, and diagonalizing its Bateman Hamiltonian, [13], by means of ladder operators.In particular, in [14,15] the authors claimed they could construct two biorthonormal bases of square integrable eigenfunctions for the Hamiltonian of the DHO.However, as shown in [16,17], this claim was wrong, since in particular the vacua of the lowering operators needed in their construction are not square integrable functions, but Dirac delta distributions.Hence distributions are more relevant than functions in this context.And, in fact, this was not the first appearance of distributions, and of the Dirac delta in particular, in the analysis of some physical system.For instance, they are the main object of research when dealing rigorously with quantum fields, [18].More recently, distributions have proved to be the natural tool to use when looking for the (generalized) eigenstates of certain with distributions.This motivates our analysis in Section IV, where we propose a new definition of multiplication between distributions, and we deduce some of its properties.In Section V we then show how this multiplication can be used in the analysis of our specific gain-loss system.Section VI contains our conclusions.To keep the paper selfcontained, we list a series of definitions and properties of pseudo-bosonic ladder operators in Appendix A, while in Appendix B we prove some useful identities used in Section V. Also, Appendix C contains some technical results useful for us.

II Preliminaries
We devote this section to a brief review of what was discussed in [16,17].This is relevant in view of what follows, since we will show that the system introduced in Section III can be rewritten as the one we are going to consider here.
The classical equation for the DHO is mẍ + γ ẋ + kx = 0, in which m, γ and k are the physical positive quantities of the oscillator: the mass, the friction coefficient and the spring constant2 .The Bateman lagrangian, [13], is which, other than the previous equation, produces also mÿ − γ ẏ + ky = 0, the differential equation associated to the virtual AHO, see also [24].The conjugate momenta are and the corresponding classical Hamiltonian is By introducing the new variables x 1 and x 2 through L 0 and H 0 can be written as follows: and where we can rewrite H 0 as follows: We will here only consider ω 2 > 0. The case ω 2 ≤ 0 has been briefly considered in [16].
Following [14] we impose the following canonical quantization rules between x j and p k : [x j , p k ] = iδ j,k 1 1, working in unit = 1.Here 1 1 is the identity operator.This is equivalent to the choice in [24].Ladder operators can now be easily introduced: k = 1, 2. These are bosonic operators since they satisfy the canonical commutation rules: [a j , a † k ] = δ j,k 1 1.Furthermore, they are densely defined on any Schwartz test function.In particular, [a j , a † k ]ϕ(x) = δ j,k ϕ(x), for all ϕ(x) ∈ S(R).It might be useful to recall that S(R) is the set of all C ∞ functions which decrease, together with their derivatives, faster than any inverse power of x, [22]: In terms of these operators the quantum version of the Hamiltonian H 0 in (2.4) can be written as (2.6) Following again [14] we introduce the operators: as well as They satisfy the following requirements: ∀ϕ(x) ∈ S(R).We observe that It might be useful to stress that the map in (2.7)-(2.8) is reversible, since a j and a † j can be recovered out of A j and B j .
In [21,23] operators of this kind, named pseudo-bosonic, were analyzed in detail, producing several interesting results mainly connected with their nature of ladder operators.
In terms of these operators H 0 can now be written as follows: which only depends on the pseudo-bosonic number operators N j = B j A j , [23].This is exactly the same Hamiltonian found in [14], and it is equivalent to that given in [5,24] and in many other papers on this subject.In [16] we proved the following theorem, stating that the pseudo-bosonic lowering operators A 1 , A 2 , B † 1 and B † 2 do not admit square integrable vacua.
Proposition 1 There is no non-zero function ϕ 00 (x 1 , x 2 ) satisfying Also, there is no non-zero function ψ 00 (x 1 , x 2 ) satisfying We refer to [16,17] for further results on this problems.In particular, it was shown that the vacua of A j and B † j , j = 1, 2, are respectively ϕ 00 (x 1 , x 2 ) = αδ(x 1 − x 2 ) and ψ 00 (x 1 , x 2 ) = βδ(x 1 + x 2 ): they are not functions, but distributions.α and β are some sort of normalization constants.Here we just want to stress that, in these latter papers, our analysis stopped at this level because our interest was mainly focused in proving that it was not possible to find square-integrable eigenfunctions of H 0 , contrarily to what claimed in [14,15].What is more interesting for us, now, is the possibility to answer the following questions: • is it possible to replace the pair DHO-AHO with some more general system sharing similar properties, and a similar quantization procedure?
• is it possible to construct a set of biorthonormal-like distributions out of ϕ 00 (x 1 , x 2 ) and ψ 00 (x 1 , x 2 ) using the pseudo-bosonic raising operators as described in Appendix A?
We will see that both these questions can be successfully considered, and in particular the second question will force us to introduce an interesting mathematical tool, which can be used to define a class of multiplications between distributions.

III Our system
In this section we will consider the first question raised above, proposing a simple classical Lagrangian which generalizes the one in (2.1), and which describes, in view of its interpretation as a gain-loss system, two coupled DHO-AHO.The idea is very simple: we just add to L 0 in (2.1) another term, L 1 , which is again quadratic in the variables x and y.With a proper choice of L 1 , and of the parameters of the system, we will be able to describe a new pair of coupled oscillators, and to quantize the system in a similar way as we discussed in Section II, facing with similar problems.
Let us consider and Here A and B are constants whose values will be constrained later.
Remark:-The Lagrangian in (3.2) is a particular case of a more general choice L = L 0 + L 1 , with L 1 = f (y, ẏ) + g(x, ẋ).Not surprisingly, also in this general case, if we write the Hamiltonian H = p x ẋ + p y ẏ − L, and we compute its time derivative, we get Ḣ = 0, which can be interpreted as some kind of energy conservation for the coupled system.
From L in (3.2) we get the following set of coupled differential equations: which can also be rewritten, after some minor manipulations, as where where α x , α y , β x and β y must satisfy the condition α x β y − β x α y = 0, in order to have an invertible transformation.From now on, we take so that α x β y − β x α y = αyβy A √ 1 − 4AB, which is different from zero if α y , β y = 0, under our constraint on AB.After some manipulation we get that where we have introduced which is what we will assume from now on.However, under this condition, it follows that This Lagrangian describes again a coupled DHO-AHO as the original one in (2.1), where Y 1 is the coordinate of the DHO (Y 1 ⇄ x), while X 1 is that of the AHO (X 1 ⇄ y).Hence we can repeat the same steps as in Section II, and in particular quantize the system and diagonalize the Hamiltonian in terms of pseudo-bosonic operators.Formula (2.3) is replaced here by Then, introducing p 1 and p 2 as before, 2 x 1 , and the classical Hamiltonian H 0 in (2.4) should be replaced now by where , which we assume here to be positive.Next we quantize the system, requiring that [x j , p k ] = iδ j,k 1 1, and we introduce the bosonic operators k = 1, 2, and their combinations ). (3.12)These operators satisfy, see (2.9), the commutation rule ∀ϕ(x) ∈ S(R), as well as the other properties stated in Section II.An essential consequence is that H is diagonal in these operators, and Proposition 1 applies.In particular, the vacua of A j and B † j , j = 1, 2, are respectively Going back to our first question at the end of Section II, we have seen here that it is indeed possible to construct more general systems3 which, after a certain change of variables, turn out not to be different from the pair of oscilators described by the Bateman Lagrangian.Next, because of the role the distributions play in our analysis, we consider a mathematical interlude on a possible definition of a class of multiplications between distributions.We should maybe stress that, in fact, the content of Section IV is (in our opinion) the most relevant mathematical result of this paper.

IV Multiplication of distributions
In [22] a possible way to introduce a multiplication between distributions was discussed.It is based on the simple fact that the scalar product between two good functions f (x) and g(x), for instance f (x), g(x) ∈ S(R), can be written in terms of a convolution between f (x) and g(x) = g(−x): f, g = (f * g)(0).Hence it is natural to define the scalar product between two elements F (x), G(x) ∈ S ′ (R) as the following convolution: whenever this convolution exists, which is not always true.Notice that, in order to compute F, G , it is first necessary to compute (F * G)[f ], f (x) ∈ S(R), and this can be done by using the equality (F * G)[f ] = F, G * f which, again, is not always well defined.
It is maybe useful to stress that (F * G)[f ] represents here the action of (F * G)(x) on the function f (x).This approach has been used in some concrete situations in recent years, mainly to check if the generalized eigenstates of some non self-adjoint operator Ĥ are biorthonormal (with respect to this generalized product) to those of Ĥ † .Some results in this direction can be found in [19]- [21].
However, this approach does not seem to be flexible enough to cover also the situation discussed in this paper, i.e. to deal with the set of weak eigenvectors of the Hamiltonian of the system in Section III.This is because, among other difficulties, it is very hard to take care properly of the domains of the various operators involved in this analysis, but also because it is quite complicated to perform explicit computations even in simple cases.For this reason we introduce now a different multiplication between distributions, and we analyze some of its properties.In Section V we will show how this new definition works for our gain-loss system.To be general, we work here with L 2 (R d ), d ≥ 1.First of all we introduce an orthonormal, total, set of vectors in L 2 (R d ): where each n j = 0, 1, 2, 3, . ... For instance, if d = 1 the set F e could be the set of the eigenstates of the quantum harmonic oscillator.If d ≥ 2, F e can be constructed as tensor product of these 1-d functions, and so on.Due to the nature of F e , for all if we further assume that each e n (x) is real, for simplicity.We are using the following notation: In the Parceval identity above the particular choice of F e is not relevant, as far as f (x), g(x) ∈ L 2 (R d ).Now, the fact that e n (x) ∈ S(R d ) implies that, for all K(x) ∈ S ′ (R d ), the set of tempered distributions, [22], the following quantity is well defined: What might exist, or not, is the following sum where F (x), G(x) ∈ S ′ (R d ).This suggests the following: What we have discussed before implies that all the square integrable functions are mutually F e -multiplicable, and the result is independent of the specific choice of F e .This means that Definition 2 makes sense on a large set of tempered distributions, all those defined by ordinary square-integrable functions.Moreover, at least for these functions, (4.5) and (4.1) coincide.We will show in the next section that F, G e is also well defined in other cases.However, for generic elements of S ′ (R), it is not granted a priori that F, G e is independent of the choice of F e .
The following results are natural extensions of the properties of any ordinary scalar product to ., .e .Then the F e -multiplication is linear in the second variable.The proof is trivial and will not be given here.Of course, the F e -multiplication is anti-linear in the first variable.
) is such that F, F e exists, then F, F e ≥ 0. In particular, if F, F e = 0, then F [f ] = 0 for all f (x) ∈ L e , the linear span of the e n (x)'s.
In fact, from (4.5) we have which is never negative.Moreover, F, F e = 0 if and only if F [e n ] = 0 for all n, which, because of the linearity of F , implies our claim.
Remark:-It is not possible to conclude that F = 0, even if F, F e = 0.The reason is that, to conclude that F = 0, we should check that F [g] = 0 for all g(x) ∈ S(R d ).Now, since N n=0 e n , g e n (x), where N = (N 1 , N 2 , . . ., N d ), with N j < ∞, 0 = (0, 0, . . ., 0), and the convergence is in the norm of L 2 (R d ).But this convergence does not imply that the same sequence converges in the topology τ S of S(R d ).Hence, the continuity of F is not sufficient to conclude that e n , g e n (x) = 0.
It might be interesting to observe that for square-integrable functions f (x) and g(x) the adjoint of any bounded operator X satisfies the equality X † f, g e = f, Xg e .This is a consequence of the analogous relation for ., .and of the identity f, g = f, g e , true ∀f (x), g(x) ∈ L 2 (R d ).However, this is no longer granted for F (x), G(x) ∈ S ′ (R d ).In fact, even if F and G are F e -multiplicable, and if X † F, XG ∈ S ′ (R d ), there is no general reason for X † F, G e and F, XG e to exist, and to be equal.This is, in our opinion, one of the many points of the F e multiplication which deserves a deeper investigation.Another relevant aspect of this multiplication concerns the optimal choice of F e , if any.We will comment on this particular aspect later on.

V Orthogonality of eigenstates
In Section III we have deduced the vacua of the pseudo-bosonic lowering operators A j and B † j .We will now use the standard pseudo-bosonic strategy, in its weak form, see [19]- [21], to construct a set of distributions which are the (generalized) eigenstates of H in (3.14) and of its adjoint.In what follows we will use what we have discussed in Section IV, focusing on the case d = 2.
In analogy with (A.2), after finding the vacua, the second step in our construction consists in using the raising pseudo-bosonic operators to construct, out of the vacua, two families of vectors.In particular, we put and where n 1 , n 2 = 0, 1, 2, 3, . ... As in Section IV, we will often use the notation n = (n 1 , n 2 ) and x = (x 1 , x 2 ).These vectors are, clearly, not square-integrable functions.Indeed, they are tempered distributions, as it is already clear from their expressions for n = (0, 0).Because of formulas (3.11), (3.12), and the fact that are deduced from the vacua by acting on them with weak derivatives and multiplication operators, all operations mapping S ′ (R 2 ) into itself.This implies that the two sets F ϕ = {ϕ n (x), n 1 , n 2 ≥ 0} and F ψ = {ψ n (x), n 1 , n 2 ≥ 0} are both sets of tempered distributions: ϕ n (x), ψ n (x) ∈ S ′ (R 2 ), ∀n.We would like to check now if, and in which sense, we can recover for these vectors the analogous of formula (A.4).In other words, we would like to understand if (and again, in which sense) F ϕ and F ψ are biorthonormal families of vectors.
Remark:-It is not hard to check that (4.1) can be used to define an extended scalar product between ϕ 0 (x) and ψ 0 (x), and to check that ϕ 0 , ψ 0 = 1.However, this same definition does not allow any simple computation of the other scalar products ϕ n , ψ m , in general, and this is the main reason why we prefer to adopt here the definition of the F e -multiplication proposed in Section IV, see Definition 2. This analysis has an important side effect, since it will allow us to check on a rather concrete situation that the definition of ., .e works also outside L 2 (R d ).Hence, ., .e can really be seen as an extension of the ordinary scal product in L 2 (R d ).
Let G e = {e n (x), n ≥ 0} be the usual orthonormal basis of eigenfunctions of the harmonic oscillator, H0 = where m 1 and ω 1 are those introduced in Section III.It is well known that e n (x) ∈ S(R) for all n ≥ 0. Then we consider and F e = {e n }.This is an orthonormal basis of L 2 (R 2 ) of functions, all belonging to S(R 2 ).Further, and very important, the various vectors of this basis obey ladder equalities when considered together with the bosonic operators a j and a † j in (3.11).For instance, (or a 2 e n = 0 if n 2 = 0), and so on.
From now on we will use this particular basis to define ., .e as in (4.5).The reason is simple: our raising operators A † j and B j can be written, see (3.12), in terms of a j and a † j , and their action on each e n (x) in (5.3) is simple 4 .Our main effort is to check that ψ k , ϕ l e makes sense, and that it is equal to δ k,l .This would imply that our multiplication in (4.5) is useful, defined on a large set, and that the families F ϕ and F ψ are biorthonormal (with respect to this extended scalar product).
To prove this claim, we need to compute . A general proof of the existence of this quantity is not easy.On the other hand, a direct check of the facts that ψ k , ϕ l e exists and that it is equal to δ k,l , it is not hard, if we restrict to few (low) values of k and l.To do so it is convenient to check first the following equalities: and (5.6)The proof of (some of) these identities is given in Appendix B.
We can now use (5.5) and (5.6) to check few of the orthonormality results needed to conclude that F ϕ and F ψ are F e -biorthonormal.In particular, it is easy to check that all the vectors ψ k and ϕ l are F e -orthogonal if k = l, and k, l = (j 1 , j 2 ), for all j 1 , j 2 = 0, 1.For instance, we have 4 It is clear that for different physical system our choice of F e could be different from the one here.
Of course, what we have explicitly checked here is not a general result.In other words, this is just an indication that, see (5.4), ψ k , ϕ l e exists and is equal to δ k,l .This is what we will discuss next.

V.1 From few to many
Our aim now is trying to generalize, as much as possible, formulas in (5.7) so to check that F ϕ and F ψ are indeed F e -biorthonormal.However, as we will see, the existence of ψ k , ϕ l e will be, for us, a working assumption, motivated by the results we have deduced previously.We hope to be able to produce a general proof of this existence in a close future.It is quite likely that the difficulty in proving this result is connected with the fact that the various series above are not convergent in the usual sense, but only Abelconvergent.For this reason we believe that the preliminary analysis proposed here and before is relevant and useful for a deeper understanding of the situation.
We start proving that, calling N j = B j A j and N † j its adjoint, the following weak eigenvalue equations are satisfied: for all l ∈ N 2 0 , for all Φ(x 1 , x 2 ) ∈ S(R 2 ), and for j = 1, 2. Using the definition of N 1 = B 1 A 1 in terms of multiplication and derivative operators, see (3.11) and (3.12), and using the fact that Φ(x 1 , x 2 ) ∈ S(R 2 ) and that S(R 2 ) is stable under the action of the various operators involved in the game, we have , which is stable, we can safely rewrite where the last equality can be proved by induction on n 1 .Hence, using again the definition of the weak derivative, we have The other equalities in (5.8) can be proved in a similar way.This result has interesting consequences like those given in the rest of this section.
Proposition 3 Assume that, for some k and l, ψ k , ϕ l e exists.Then ψ k , N j ϕ l e and N † j ψ k , ϕ l e , j = 1, 2, also exist and (5.9) Proof: By definition we have, for instance, But, since e n (x) ∈ S(R 2 ), we can use (5.8) and we get (N as we had to prove.The other equalities in (5.9) can be proved in a similar way.
We have already commented that, in general, taken F and G in S ′ (R 2 ), and a given operator X, even if X † F, XG ∈ S ′ (R 2 ), there is no general reason for X † F, G e and F, XG e to exist, and to have X † F, G e = F, XG e .However, this may happen in some particular cases.In fact, we can check that N † j ψ k , ϕ l e = ψ k , N j ϕ l e (5.10) ∀k, l, j = 1, 2. The proof of this equality is rather technical and is given in Appendix C. Now it is clear that, as in the standard, Hilbert space, settings, the following result holds: (5.11) Proof: The proof is identical to the usual one, using the (non trivial, here) identities in (5.9) and (5.10).
Summarizing, what we get here is a strong indication that the results deduced before, using (5.5) and (5.6), can be generalized and that the sets F ϕ and F ψ are indeed two F e -biorthonormal sets of tempered distributions, and are made of weak eigenstates of the number operators N j and N † j and, therefore, of weak eigenstates of the Hamiltonian H in (3.14) and of H † .In our analysis, we were somehow forced to introduce a new, potentially interesting, multiplication between distributions.

VI Conclusions
There are several open points in the analysis proposed in this paper.First of all, we believe that the idea of F e -multiplication of distributions may have interesting consequences and applications.Also, their properties should be analyzed in more details than those considered here.We hope to consider this aspect of our analysis soon.
There are also other aspects of the system considered here which should be clarified: if on one side we were able to compute explicitly some of the scalar products ψ k , ϕ l e , showing that, as expected, they are zero or one, we have no general argument showing that ψ k , ϕ l e does exist for all k and l.Still, under the hypothesis that this exists, we were able to deduce several interesting results, including the biorthogonality of the vectors ψ k and ϕ l .However, the nature of F ϕ and F ψ as possible bases (of some kind, see [23]) is also still to be understood.
Another interesting aspect is the role of the Abel summation appearing in several computations.This creates more questions: is this related to the physical system?How?
Finally, on a more physical side, the Lagrangian considered in (3.1) and (3.2) is just a particular possible choice among many possibilities.What does it change if we consider a different L 1 ?Which kind of physics can we describe?And what is the role of distributions, if any, for general dissipative systems?
We hope to be able to answer some of these questions soon.
We It is clear that the series converge since the left-hand side in (A.6) exists, because of (5.9) and of our working assumptions.In a similar way we have and computing N 1 e n , we get Now the fact that ψ k , N 1 ϕ l e = N † 1 ψ k , ϕ l e follows from a direct comparisons of the two results deduced here (with some change of variable).
The others identities can be proved in a similar way.
) are such that F, G e exists, then also G, F e exists and F, G e = G, F e .(4.6) Indeed we have, recalling that the series in (4.5) converges, F, G e = n F, e n e n , G = n G, e n e n , F = G, F e , which in particular implies that G, F e exists, too.Result ♯2: If F (x), G(x), L(x) ∈ S ′ (R d ) are such that F, G e and F, L e exist, then also F, αG + βL e exists, for all α, β ∈ C, and F, αG + βL e = α F, G e + β F, L e .(4.7)