A fully pseudo-bosonic Swanson model Fabio Bagarello Dipartimento di Ingegneria

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A fully pseudo-bosonic Swanson model

Fabio Bagarello

Dipartimento di Ingegneria,

Universit`a di Palermo,

I-90128 Palermo, Italy, and

INFN, Catania

e-mail: fabio.bagarello@unipa.it

Abstract

We consider a fully pseudo-bosonic Swanson model and we show how its Hamilto-

nian Hcan be diagonalized. We also deduce the eigensystem of H†, using the general

framework and results deduced in the context of pseudo-bosons. We also construct, using

diﬀerent approaches, the bi-coherent states for the model, study some of their properties,

and compare the various constructions.

arXiv:2210.11326v1 [math-ph] 20 Oct 2022

I Introducion

In the standard literature on quantum mechanics one of the main axioms of any well established

approach to the analysis of the microscopic world is that the observables of a physical system,

S, are represented by self-adjoint operators. This is, in particular, what is required to the

Hamiltonian of S. Since few decades, however, it has become more and more evident that this

is only a suﬃcient condition to require, but it is not also necessary. This (apparently) simple

remark is at the basis of thousands of papers, and of many monographs. Here we only cite

some of these latter, [1]-[5], where many other references can be found.

The use of non self-adjoint Hamiltonians opens several lines of research, both for its possible

implications in Physics, and for the mathematical issues raised by this extension. In particular,

in a series of papers and in the books [6, 7], a particular class of non self-adjoint Hamiltonians

has been analyzed in detail, together with their connections with a special class of coherent

states. These Hamiltonians are constructed in terms of pseudo-bosonic operators which are,

essentially, suitable deformations of the bosonic creation and annihilation operators. These

deformations are again ladder operators, and this is the reason why we were, and still are,

interested in ﬁnding the eigenstates of these new annihilation operators. Several examples

have been constructed along the years, by us and by other authors, [8]-[13]. In particular,

one Hamiltonian which has become very famous in the literature on P T -quantum mechanics

is related to what is now called the Swanson model, [13, 14, 15]. The Hamiltonian for this

model is Hs=ωsc†c+αc2+βc†2, where α, β, ωs∈Rand where [c, c†] = 11. Of course, since

cand c†are unbounded, the above expressions for Hsand [c, c†] are simply formal. To make

them rigorous, we should add, in particular, details on their domains of deﬁnition. A more

mathematical approach to Hs, closer to what is relevant for us here, can be found in [15, 6, 7].

In particular, we have shown that Hscan be rewritten in a diagonal form in terms of pseudo-

bosonic operators, and this has been used to analyze in detail its spectrum and its eigenvectors.

In particular, we have shown that the set of these eigenvectors is complete in H, but it is not a

basis. There are many papers devoted to the Swanson model, in one of its various expressions.

Some other paper on this model are the following: [16]-[20], just to cite a few.

In this paper we will focus on a particular version of a fully pseudo-bosonic extension of

Hs, i.e. on a version in which the pair of bosonic operators (c, c†) are replaced, from the very

beginning, by operators (a, b) satisfying certain properties, see Section II. Moreover, to simplify

the general treatment, and without any particular loss of generality, we will also restrict to

choosing α=β. Notice that, while this choice trivializes the original model, in the sense that

Hs=H†

s, does not change at all the lack of self-adjointness of the Hamiltonian Hwe will

introduce later, see (3.1).

The paper is organized as follows: after a review on pseudo-bosons, in Section II, we propose

our fully pseudo-bosonic Swanson model, and we ﬁnd the eigenvalues and the eigenvectors of the

Hamiltonian of the system, and of its adjoint. We prove that the sets of these eigenvectors are

complete and biorthonormal in L2(R), while they are not bases. This will be done in Section

III. Section IV is focused on bi-coherent states and on their properties. Section V contains our

conclusions, and plans for the future.

II Preliminaries

This section is devoted to some preliminary deﬁnitions and results on pseudo-bosons (PBs).

This will be needed in the next sections, where the modiﬁed Swanson Hamiltonianl will be

introduced and analyzed.

Let aand bbe two operators on H, with domains D(a) and D(b) respectively, a†and b†

their adjoint, and let Dbe a dense subspace of H, stable under the action of a,band their

adjoints. It is clear that D ⊆ D(a]) and D ⊆ D(b]), where c]=c, c†, and that a]f, b]f∈ D for

all f∈ D. Then both abf and baf are well deﬁned, ∀f∈ D.

Deﬁnition 1 The operators (a, b)are D-pseudo-bosonic (D-pb) if, for all f∈ D, we have

a b f −b a f =f. (2.1)

Sometimes, to simplify the notation, rather than (2.1) one writes [a, b] = 11. It is not surprising

that neither anor bare bounded on H. This is the reason why the role of Dis so relevant, here

and in the rest of these notes.

Our working assumptions for dealing with these operators are the following:

Assumption D-pb 1.– there exists a non-zero ϕ0∈ D such that a ϕ0= 0.

Assumption D-pb 2.– there exists a non-zero Ψ0∈ D such that b†Ψ0= 0.

Notice that, if b=a†, then these two assumptions collapse into a single one and (2.1)

becomes the well known canonical commutation relations (CCR), for which the existence of a

vacuum which belongs to an invariant set (S(R), for instance) is guaranteed. Then, for CCR,

Assumptions D-pb 1 and D-pb 2 are automatically true.

In [7] it is widely discussed the possibility that [a, b] = 11 can be extended outside L2(R).

This gives rise, as we will brieﬂy comment later in Section II.1, to the so-called weak PBs

(WPBs), in which a central role is no longer played by L2(R), but by other functional spaces.

In the present situation, the stability of Dunder the action of band a†implies, in particular,

that ϕ0∈D∞(b) := ∩k≥0D(bk) and that Ψ0∈D∞(a†). Here D∞(X) is the domain of all the

powers of the operator X. Hence

ϕn:= 1

√n!bnϕ0,Ψn:= 1

√n!a†nΨ0,(2.2)

n≥0, are well deﬁned vectors in Dand, therefore, they belong to the domains of a],b]and

N], where N=ba and N†is the adjoint of N. We introduce next the sets FΨ={Ψn, n ≥0}

and Fϕ={ϕn, n ≥0}.

It is now simple to deduce the following lowering and raising relations:











b ϕn=√n+ 1 ϕn+1, n ≥0,

a ϕ0= 0, aϕn=√n ϕn−1, n ≥1,

a†Ψn=√n+ 1 Ψn+1, n ≥0,

b†Ψ0= 0, b†Ψn=√nΨn−1, n ≥1,

(2.3)

as well as the eigenvalue equations Nϕn=nϕnand N†Ψn=nΨn,n≥0, where, more explicitly,

N†=a†b†. Incidentally we observe that this last equality should be understood, here and in

the following, on D:N†f=a†b†f,∀f∈ D.

As a consequence of these equations, choosing the normalization of ϕ0and Ψ0in such a way

hϕ0,Ψ0i= 1, it is easy to show that

hϕn,Ψmi=δn,m,(2.4)

for all n, m ≥0. The conclusion is, therefore, that Fϕand FΨare biorthonormal sets of

eigenstates of Nand N†, respectively. Notice that these latter operators, which are manifestly

non self-adjoint if b6=a†, have both non negative integer eigenvalues and, because of that, they

are called number-like (or simply number) operators. The properties we have deduced for Fϕ

and FΨ, in principle, does not allow us to conclude that they are (Riesz) bases or not for H.

In fact, this is not always the case, [7], even if sometimes (for regular PBs, see below), this is

exactly what happens. With this in mind, let us introduce for the following assumption:

Assumption D-pb 3.– Fϕis a basis for H.

This is equivalent to assume that FΨis a basis as well, [21, 22]. While Assumption D-pb 3, is

not always satisﬁed, in most of the concrete situations considered so far in the literature, it is

true that Fϕand FΨare total in H=L2(R). For this reason, it is more reasonable to replace

Assumption D-pb 3 with this weaker version:

Assumption D-pbw 3.– Fϕand FΨare G-quasi bases, for some subspace Gdense1in H.

This means that, ∀f, g ∈ G, the following identities hold

hf, gi=X

n≥0hf, ϕnihΨn, gi=X

n≥0hf, Ψnihϕn, gi.(2.5)

It is obvious that, while Assumption D-pb 3 implies (2.5), the reverse is false. However,

if Fϕand FΨsatisfy (2.5), we still have some (weak) form of resolution of the identity, and,

from a physical and from a mathematical point of view, this is enough to deduce interesting

results. For instance, if f∈ G is orthogonal to all the Ψn’s (or to all the ϕn’s), then fis

necessarily zero: FΨand Fϕare total in G. Indeed, using (2.5) with g=f∈ G, we ﬁnd

kfk2=Pn≥0hf, ϕnihΨn, fi= 0 since hΨn, fi= 0 (or hf, ϕni= 0) for all n. But, since

kfk= 0, then f= 0.

For completeness we brieﬂy discuss the role of two intertwining operators which are intrinsi-

cally related to our D-PBs. We only consider the regular case here. More details can be found

in [6].

In the regular case, Assumption D-pb 3 holds in a strong form: Fϕand FΨare biorthonormal

Riesz bases, so that we have

f=∞

n=0 hϕn, fiΨn=∞

n=0 hΨn, fiϕn,(2.6)

∀f∈ H. Looking at these expansions, it is natural to ask if sums like Sϕf=P∞

n=0 hϕn, fiϕn

or SΨf=P∞

n=0 hΨn, fiΨnalso make some sense, or for which vectors they do converge, if any.

In our case, since Fϕand FΨare Riesz bases, we know that an orthonormal basis Fe={en}

exists, together with a bounded operator Rwith bounded inverse, such that ϕn=Renand

Ψn= (R−1)†en,∀n. It is clear that, if R= 11, all these sums collapse and converge to f. But,

what if R6= 11?

The ﬁrst result follows from the biorthonormality of Fϕand FΨ, which easily implies that

SϕΨn=ϕn, Sψϕn= Ψn,(2.7)

1Notice that Gdoes not need to coincide with D, even if sometimes this happens.

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Afullypseudo-bosonicSwansonmodelFabioBagarelloDipartimentodiIngegneria,UniversitadiPalermo,I-90128Palermo,Italy,andINFN,Cataniae-mail:fabio.bagarello@unipa.itAbstractWeconsiderafullypseudo-bosonicSwansonmodelandweshowhowitsHamilto-nianHcanbediagonalized.WealsodeducetheeigensystemofHy,usingthegenera...

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A fully pseudo-bosonic Swanson model Fabio Bagarello Dipartimento di Ingegneria

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