Search results for "Biorthogonal system"
showing 10 items of 23 documents
A concise review on pseudo-bosons, pseudo-fermions and their relatives
2017
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.
Biorthogonal Wavelet Transforms
2015
Wavelets in the polynomial and discrete spline spaces were introduced in Chaps. 8 and 10, respectively. In both cases, the wavelets’ design and implementation of the transforms were associated with perfect reconstruction (PR) filter banks. In this chapter, those associations are discussed in more detail. Biorthogonal wavelet bases generated by PR filter banks are investigated and a few examples of compactly supported biorthogonal wavelets are presented. Conditions for filters to restore and annihilate sampled polynomials are established (discrete vanishing moment property). In a sense, the material of this chapter is introductory to Chap. 12, where splines are used as a source for (non-spli…
Generalized Riesz systems and quasi bases in Hilbert space
2019
The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.
Biorthogonal Multiwavelets Originated from Hermite Splines
2015
This chapter presents multiwavelet transforms that manipulate discrete-time signals. The transforms are implemented in two phases: 1. Pre (post)-processing, which transforms a scalar signal into a vector signal (and back). 2. Wavelet transforms of the vector signal. Both phases are performed in a lifting way. The cubic interpolating Hermite splines are used as a predicting aggregate in the vector wavelet transform. Pre(post)-processing algorithms which do not degrade the approximation accuracy of the vector wavelet transforms are presented. A scheme of vector wavelet transforms and three pre(post)-processing algorithms are described. As a result, we get fast biorthogonal algorithms to trans…
Biorthogonal vectors, sesquilinear forms, and some physical operators
2018
Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular we discuss what happens when they forms two $\D$-quasi bases.
$$\mathscr {D}{-}$$ D - Deformed and SUSY-Deformed Graphene: First Results
2016
We discuss some mathematical aspects of two particular deformed versions of the Dirac Hamiltonian for graphene close to the Dirac points, one involving \(\mathscr {D}\)-pseudo bosons and the other supersymmetric quantum mechanics. In particular, in connection with \(\mathscr {D}\)-pseudo bosons, we show how biorthogonal sets arise, and we discuss when these sets are bases for the Hilbert space where the model is defined, and when they are not. For the SUSY extension of the model we show how this can be achieved and which results can be obtained.
A non self-adjoint model on a two dimensional noncommutative space with unbound metric
2013
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\L…
Two-Parameters Pseudo-Bosons
2010
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.
From self-adjoint to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls
2013
Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the hamiltonian $H$ changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of $H$ produces two deep consequences, not well understood in the literature: first of all, the original orthonormal basis of $H$ splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space $\Hil$. Secondly, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extensio…
Pseudo-fermions in an electronic loss-gain circuit
2013
In some recent papers a loss-gain electronic circuit has been introduced and analyzed within the context of PT-quantum mechanics. In this paper we show that this circuit can be analyzed using the formalism of the so-called pseudo-fermions. In particular we discuss the time behavior of the circuit, and we construct two biorthogonal bases associated to the Liouville matrix $\Lc$ used in the treatment of the dynamics. We relate these bases to $\Lc$ and $\Lc^\dagger$, and we also show that a self-adjoint Liouville-like operator could be introduced in the game. Finally, we describe the time evolution of the circuit in an {\em Heisenberg-like} representation, driven by a non self-adjoint hamilton…