Fabio Bagarello
We show how the notion of {\em pseudo-bosons}, originally introduced as operators acting on some Hilbert space, can be extended to a distributional settings. In doing so, we are able to construct a rather general framework to deal with generalized eigenvectors of the multiplication and of the derivation operators. Connections with the quantum damped harmonic oscillator are also briefly considered.
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We propose an alternative definition for pseudo-bosons. This simplifies the mathematical structure, minimizing the required assumptions. Some physical examples are discussed, as well as some mathematical results related to the biorthogonal sets arising out of our framework.
In this paper we extend some previous results on weak pseudo-bosons and on their related bi-coherent states. The role of {\em compatible} functions is discussed in details, and some examples are considered. The pseudo-bosonic ladder operators analysed in this paper generalize significantly those considered so far, and a class of new diagonalizable manifestly non self-adjoint Hamiltonians are deduced.
The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plan…
We construct a two-parameters example of {\em pseudo-bosons}, and we show that they are not regular, in the sense previously introduced by the author. In particular, we show that two biorthogonal bases of $\Lc^2(\Bbb R)$ can be constructed, which are not Riesz bases, in general.
After a brief review on $\D$-pseudo-bosons we introduce what we call {\em Riesz bi-coherent states}, which are pairs of states sharing with ordinary coherent states most of their features. In particular, they produce a resolution of the identity and they are eigenstates of two different annihilation operators which obey pseudo-bosonic commutation rules.
It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schrödinger equation expressed in terms of a non self-adjoint hamiltonian. We show how {\em pseudo-bosons}, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel.
It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass $N>0$ satisfies $N<N^*$, where the finite constant $N^*$ is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each $N<N^*$. In this paper, we prove that the above uniqueness conjecture holds for the particular case where $N>0$ is small enough.
In a series of recent papers the author has introduced the notion of (regular) pseudo-bosons showing, in particular, that two number-like operators, whose spectra are ${\Bbb N}_0:={\Bbb N}\cup\{0\}$, can be naturally introduced. Here we extend this construction to operators with rather more general spectra. Of course, this generalization can be applied to many more physical systems. We discuss several examples of our framework.
We discuss in which sense the so-called {\em regular pseudo-bosons}, recently introduced by Trifonov and analyzed in some details by the author, are related to ordinary bosons. We repeat the same analysis also for {\em pseudo-bosons}, and we analyze the role played by certain intertwining operators, which may be bounded or not.
It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator. We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable. This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.
We construct examples of pseudo-bosons in two dimensions arising from the Hamiltonian for the Landau levels. We also prove a no-go result showing that non-linear combinations of bosonic creation and annihilation operators cannot give rise to pseudo-bosons.
In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called {\em pseudo-bosons}. They arise from a special deformation of the canonical commutation relation $[a,a^\dagger]=\1$, which is replaced by $[a,b]=\1$, with $b$ not necessarily equal to $a^\dagger$. We start discussing some of their mathematical properties and then we discuss several examples.
We review some basic definitions and few facts recently established for $\D$-pseudo bosons and for pseudo-fermions. We also discuss an extended version of these latter, based on biorthogonal bases, which lives in a finite dimensional Hilbert space. Some examples are described in details.
We consider some modifications of the two dimensional canonical commutation relations, leading to {\em non commutative bosons} and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (FB), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.
We discuss two physical examples of the so-called {\em pseudo-bosons}, recently introduced in connection with pseudo-hermitian quantum mechanics. In particular, we show that the so-called {\em extended harmonic oscillator} and the {\em Swanson model} satisfy all the assumptions of the pseudo-bosonic framework introduced by the author. We also prove that the biorthogonal bases they produce are not Riesz bases.
We show how some recent models of PT-quantum mechanics perfectly fit into the settings of $\cal D$ pseudo-bosons, as introduced by one of us. Among the others, we also consider a model of non-commutative quantum mechanics, and we show that this model too can be described in terms of $\cal D$ pseudo-bosons.
We consider a fully pseudo-bosonic Swanson model and we show how its Hamiltonian $H$ can be diagonalized. We also deduce the eigensystem of $H^\dagger$, 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.
We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.
We show how the Zak $kq$-representation can be adapted to deal with pseudo-bosons, and under which conditions. Then we use this representation to prove completeness of a discrete set of bi-coherent states constructed by means of pseudo-bosonic operators. The case of Riesz bi-coherent states is analyzed in detail.
We consider the special type of pseudo-bosonic systems that can be mapped to standard bosons by means of generalized Bogoliubov transformation and demonstrate that a pseudo-Hermitian systems can be obtained from them by means of a second subsequent Bogoliubov transformation. We employ these operators in a simple model and study three different types of scenarios for the constraints on the model parameters giving rise to a Hermitian system, a pseudo-Hermitian system in which the second the Bogoliubov transformations is equivalent to the associated Dyson map and one in which we obtain D-quasi ba…
Extended coupled SUSY, pseudo-bosons and weak squeezed states
本文通过伪玻色子阶梯算符的平方构造满足变形su(1,1)李代数的本征态，分析其平方可积性与双正交性，并初步探讨双压缩态。
The first purpose of this paper is to show a method of constructing a regular biorthogonal pair based on the commutation rule: $ab-ba=I$ for a pair of operators $a$ and $b$ acting on a Hilbert space ${\cal H}$ with inner product $( \cdot | \cdot )$. Here, sequences $\{ ϕ_{n} \}$ and $\{ ψ_{n} \}$ in a Hilbert space ${\cal H}$ are biorthogonal if $( ϕ_{n} | ψ_{m})= δ_{nm}$, $n,m=0,1, \cdots$, and they are regular if both $D_ϕ \equiv Span \{ ϕ_{n} \}$ and $D_ψ \equiv Span \{ ψ_{n} \}$ are dense in ${\cal H}$. Indeed, the assumption to construct the regular biorthogonal pair coincide with the def…
In a series of recent papers one of us has analyzed in some details a class of elementary excitations called {\em pseudo-bosons}. They arise from a special deformation of the canonical commutation relation $[a,a^\dagger]=\1$, which is replaced by $[a,b]=\1$, with $b$ not necessarily equal to $a^\dagger$. Here, after a two-dimensional extension of the general framework, we apply the theory to a generalized version of the two-dimensional Hamiltonian describing Landau levels. Moreover, for this system, we discuss coherent states and we deduce a resolution of the identity. We also consider a diffe…
We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the beh…
The present paper is the third contribution of a series of works, where we investigate pseudo--bosonic operators and their connections with finite dimensional Lie algebras. We show that all finite dimensional nilpotent Lie algebras (over the complex field) can be realized by central extensions of Lie algebras of pseudo-bosonic operators. This result is interesting, because it provides new examples of dynamical systems for nilpotent Lie algebras of any dimension. One could ask whether these operators are intrinsic with the notion of nilpotence or not, but this is false. In fact we exibit both a…
In a recent paper we have considered an explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. We have introduced the so-called {\em pseudo-bosons}, and the role of Riesz bases in this context has been analyzed in detail. In this paper we consider a general construction of pseudo-bosons based on an explicit {coordinate-representation}, extending what is usually done in ordinary supersymmetric quantum mechanics. We also discuss an example arising from a linear modification of standard creation and annihilation operators, and we analyze its connect…
Extending our previous analysis on bi-coherent states, we introduce here a new class of quantum mechanical vectors, the \emph{bi-squeezed states}, and we deduce their main mathematical properties. We relate bi-squeezed states to the so-called regular and non regular pseudo-bosons. We show that these two cases are different, from a mathematical point of view. Some physical examples are considered.
In the first part of this paper we show how a simple system, a 2-dimensional quantum harmonic oscillator, can be described in terms of pseudo-bosonic variables. This apparently {\em strange} choice is useful when the {\em natural} Hilbert space of the system, $L^2({\bf R}^2)$ in this case, is, for some reason, not the most appropriate. This is exactly what happens for the $D_2$ type quantum Calogero model considered in the second part of the paper, where the Hilbert space $L^2({\bf R}^2)$ appears to be an unappropriate choice, since the eigenvectors of the relevant hamiltonian are not square-i…
Parallels between the notions of nonlinear pseudobosons and of an apparent non-Hermiticity of observables as shown in paper I (arXiv: 1109.0605) are demonstrated to survive the transition to the quantum models based on the use of unbounded metric in the Hilbert space of states.
In a recent paper, Trifonov suggested a possible explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. Although being rather intriguing, in his treatment many mathematical aspects of the model have just been neglected, making most of the results of that paper purely formal. For this reason we are re-considering the same model and we repeat and extend the same construction paying particular attention to all the subtle mathematical points. From our analysis the crucial role of Riesz bases clearly emerges. We also consider coherent states associate…
We have recently shown that pseudo-bosonic operators realize concrete examples of finite dimensional nilpotent Lie algebras over the complex field. It has been the first time that such operators were analyzed in terms of nilpotent Lie algebras (under prescribed conditions of physical character). On the other hand, the general classification of a finite dimensional nilpotent Lie algebra $\mathfrak{l}$ may be given via the size of its Schur multiplier involving the so-called corank $t(\mathfrak{l})$ of $\mathfrak{l}$. We represent $\mathfrak{l}$ by pseudo-bosonic ladder operators for $t(\mathfra…
The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.
In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space $\Lc^2(\mathbb{R})$, leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in \…
In a recent paper a pair of operators $a$ and $b$ satisfying the equations $a^\dagger a=bb^\dagger+γ\1$ and $aa^\dagger=b^\dagger b+δ\1$, has been considered, and their nature of ladder operators has been deduced and analysed. Here, motivated by the spreading interest in non self-adjoint operators in Quantum Mechanics, we extend this situation to a set of four operators, $c$, $d$, $r$ and $s$, satisfying $ dc=rs+γ\1$ and $cd=sr+δ\1$, and we show that they are also ladder operators. We show their connection with biorthogonal families of vectors and with the so-called $\D$-pseudo bosons. Some ex…
Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their immediate generalizations that we call pseudo-bosons and pseudo-fermions, and also matrix algebras, in the finitely generated case. We then recover q-bosons (and pseudo-q-bosons) by a completion operation.
We demonstrate that not all generalized Bogoliubov transformations lead to $\cal D$-pseudo-bosons and prove that a correspondence between the two can only be achieved with the imposition of specific constraints on the parameters defining the transformation. For certain values of the parameters we find that the norms of the vectors in sets of eigenvectors of two related apparently non self-adjoint number-like operators possess different types of asymptotic behavior. We use this result to deduce further that they constitute bases for a Hilbert space, albeit neither of them can form a Riesz base.…
We show that the metric operator for a pseudo-supersymmetric Hamiltonian that has at least one negative real eigenvalue is necessarily indefinite. We introduce pseudo-Hermitian fermion (phermion) and abnormal phermion algebras and provide a pair of basic realizations of the algebra of N=2 pseudo-supersymmetric quantum mechanics in which pseudo-supersymmetry is identified with either a boson-phermion or a boson-abnormal-phermion exchange symmetry. We further establish the physical equivalence (non-equivalence) of phermions (abnormal phermions) with ordinary fermions, describe the underlying Lie…
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