Rigged Hilbert Space Research Articles

Moyal deformation of the classical arrival time

The quantum time of arrival (TOA) problem requires the statistics of measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time TC(q,p), usually in operator form T̂. In this paper, we consider the problem anew within the phase space formulation of quantum mechanics. The resulting quantum image is a real-valued and time-reversal symmetric function TM(q,p) in formal series of ℏ2 with the classical arrival time as the leading term. It is obtained directly from the Moyal bracket relation with the system Hamiltonian and is hence interpreted as a Moyal deformation of the classical TOA. We investigate its properties and discuss how it bypasses the known obstructions to quantization by showing the isomorphism between TM(q,p) and the rigged Hilbert space TOA operator constructed in Pablico and Galapon [Eur. Phys. J. Plus 138, 153 (2023)], which always satisfy the time-energy canonical commutation relation for arbitrary analytic potentials. We then examine TOA problems for a free particle and a quartic oscillator potential as examples.

Complex scaling method applied to the study of the Swanson Hamiltonian in the broken PT-symmetry phase

In this work, we study the non-PT symmetry phase of the Swanson Hamiltonian in the framework of the Complex Scaling Method. By constructing a bi-orthogonality relation, we apply the formalism of the response function to analyse the time evolution of different initial wave packages. The Wigner Functions, mean value of operators, and the probabilities of survival and persistence for the different wave packages are evaluated as a function of time. We analyse in detail the time evolution in the neighbourhood of Exceptional Points. We derive a continuity equation for the system. We compare the results obtained using the Complex Scaling Method to the ones obtained by working in a Rigged Hilbert Space.

Neural network classification of eigenmodes in the magnetohydrodynamic spectroscopy code Legolas

A neural network is employed to address a non-binary classification problem of plasma instabilities in astrophysical jets, calculated with the Legolas code. The trained models exhibit reliable performance in the identification of the two instability types supported by these jets. We also discuss the generation of artificial data and refinement of predictions in general eigenfunction classification problems.

Bounds on eigenfunctions of quantum cat maps

We study ℓ ∞ norms of ℓ 2-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000, Communications in Mathematical Physics, 211, 659–686)) we show that there exists a sequence of eigenfunctions u with . For general eigenfunctions we show the upper bound . Here the semiclassical parameter is . Our upper bound is analogous to the one proved by Bérard in P Bérard (1977, Mathematische Zeitschrift, 155, 249-276) for compact Riemannian manifolds without conjugate points.

The eigenvalues and eigenfunctions of the toroidal dipole operator

The toroidal dipole represents the lowest order in the family of toroidal multipoles. They appear in physics at all scales, from particle to condensed matter physics, and metamaterials. Nevertheless, in small systems they have to be investigated in the context of quantum mechanics and quantum operators are introduced for the projections of the toroidal dipole on the Cartesian axes. Here we give analytical expressions for the eigenvalues and generalized eigenfunctions of –the z-axis projection of the toroidal dipole operator–in a system consisting of a particle confined in a thin film bent into a torus shape. The eigenfunctions are not square integrable, so they do not belong to the Hilbert space of wave functions, but they can be interpreted in the formalism of rigged Hilbert spaces as kernels of distributions. We find the quantization rules for the eigenvalues, which are essential for describing measurements of . As an example, we estimate the splitting of the energy levels due to the interaction of the toroidal dipole with an external current.

Rigged Hilbert space approach for non-Hermitian systems with positive definite metric

We investigate Dirac’s bra–ket formalism based on a rigged Hilbert space for a non-Hermitian quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by a positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra–kets by the generalized eigenvectors of a quasi-Hermitian operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermitian and non-Hermitian systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.

Mathematical Models for Unstable Quantum Systems and Gamow States.

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.

Relativistic free-motion time-of-arrival operator for massive spin-0 particles with positive energy

A relativistic version of the Aharonov-Bohm time-of-arrival operator for spin-0 particles was constructed by Razavi [Il Nuovo Cimento B 63, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the time-of-arrival problem that goes beyond Razavi's original results. This allows us to use time-of-arrival eigenfunctions that exhibit unitary arrival to construct time-of-arrival distributions. The expectation value is also calculated and shown that particles can arrive earlier or later than expected classically. Last, the constructed time-of-arrival distribution and expectation value are shown to be consistent with special relativity.

Parameter-dependent filtering of Gaussian processes in Hilbert spaces

The filtering problem for non-Markovian Gaussian processes on rigged Hilbert spaces is considered. Continuous dependence of the filter and observation error on parameters which may be present both in the signal and observation processes is proved. The general results are applied to signals governed by stochastic heat equations driven by distributed or pointwise fractional noise. The observation process may be a noisy observation of the signal at given points in the domain, the position of which may depend on the parameter.

Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions

This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, Kp,q, that contain Hp,q and Ep,q as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of Kp,q. We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type Kp,q. By extending these Hilbert spaces, we obtain representations of Kp,q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform.

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted Lp-path spaces is proved. In particular, as special cases the classical Caputo derivative and other fractional derivatives appearing in applications are included. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of typeddt(k⁎u)(t)+A(t,u(t))=f(t),0<t<T, with (in general nonlinear) operators A(t,⋅) satisfying general weak monotonicity conditions. Here k is a non-increasing locally Lebesgue-integrable nonnegative function on [0,∞) with lims→∞k(s)=0. Analogous results for the case, where f is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators and the time-fractional (stochastic) p-Laplace equation are covered.

Change of representation and the rigged Hilbert space formalism in quantum mechanics

Generalized eigenvectors are key tools in the theory of rigged Hilbert spaces. Let H be a Hilbert space and let Φ be a dense subspace of H. Let A be a densely defined linear operator in H such that Φ ⊂ DA and AΦ ⊂ Φ. The generalized eigenvectors of A are the eigenvectors of the algebraic dual of A |Φ. In the case where Φ is endowed with a topology τ finer than the norm topology inherited from H, generalized eigenvectors that are τ-continuous may be of great interest. We discuss conditions which ensure the existence of representations associated to generalized eigenvectors of A. As an application, we review and refine Böhm's study of the algebra of the quantum harmonic oscillator.

Quantum Mechanics and Its Evolving Formulations.

In this paper, we discuss the time evolution of the quantum mechanics formalism. Starting from the heroic beginnings of Heisenberg and Schrödinger, we cover successively the rigorous Hilbert space formulation of von Neumann, the practical bra-ket formalism of Dirac, and the more recent rigged Hilbert space approach.

On eigenproblem for inverted harmonic oscillators

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is ${\mathbb C}$. The spectrum of the differential operator $-{\frac{d}{dx^2}}-{\omega}^{2}{x^2}$ is continuous and has physical significance only for the states which are in $L^{2}(\mathbb R)$ and correspond to real eigenvalues. To identify them we use two approaches. First we define a unitary operator between $L^{2}(\mathbb R)$ and $L^{2}$ for two copies of $\mathbb R$. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator $-i{\frac{d}{dx}}$. This shows that the (generalized) spectrum of the inverted harmonic operator is real. The second approach uses rigged Hilbert spaces.

Anisotropic Optical Shock Waves in Isotropic Media with Giant Nonlocal Nonlinearity.

Dispersive shock waves in thermal optical media are nonlinear phenomena whose intrinsic irreversibility is described by time asymmetric quantum mechanics. Recent studies demonstrated that the nonlocal wave breaking evolves in an exponentially decaying dynamics ruled by the reversed harmonic oscillator, namely, the simplest irreversible quantum system in the rigged Hilbert spaces. The generalization of this theory to more complex scenarios is still an open question. In this work, we use a thermal third-order medium with an unprecedented giant Kerr coefficient, the m-cresol/nylon mixed solution, to access an extremely nonlinear, highly nonlocal regime and realize anisotropic shock waves with internal gaps. We compare our experimental observations to results obtained under similar conditions but in hemoglobin solutions from human red blood cells, and found that the gap formation strongly depends on the nonlinearity strength. We prove that a superposition of Gamow vectors in an adhoc rigged Hilbert space, that is, a tensorial product between the reversed and the standard harmonic oscillators spaces, describes the beam propagation beyond the shock point. The anisotropy turns out from the interaction of trapping and antitrapping potentials. Our work furnishes the description of novel intriguing shock phenomena mediated by extreme nonlinearities.

Structural analysis of an L-infinity variational problem and relations to distance functions

In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the $L^2$-subdifferential of ${\cal J}$ and characterize the distance function as unique non-negative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as asymptotic profile of the gradient flow of ${\cal J}$ and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of ${\cal J}$. Finally, we transfer many of these results to a discrete version of the functional defined on a finite weighted graph. Here, we analyze properties of distance functions on graphs and their gradients. The main difference between the continuum and discrete setting is that the distance function is not the unique non-negative eigenfunction on a graph.

Dynamic Evolution Equations for Cores of Linear Crystal Defects in Colliding Solids

A discrete model in a generalized rectangular-pulse space is proposed for dislocation cores in a crystal with its undeformed perfect structure taken as a Hilbert space of Schrodinger wave functions and with the core of a dislocation as a rigged Hilbert space of step functions of opposite sign separated by a time interval. The model suggests Vlasov equations for cation and electron distribution functions and equations for an intermittent field and their solutions. It is shown that the particle dispersion law is complex, and its real part is nonlinear and quadratic.

Groups, Jacobi functions, and rigged Hilbert spaces

This paper is a contribution to the study of the relations between special functions, Lie algebras, and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations of their algebra of symmetry, which induce discrete and continuous bases coexisting on a rigged Hilbert space supporting the representation. Meaningful operators are shown to be continuous on the spaces of test vectors and the dual. Here, the chosen special functions, called “algebraic Jacobi functions,” are related to the Jacobi polynomials, and the Lie algebra is su(2, 2). These functions with m and q fixed also exhibit a su(1, 1)-symmetry. Different discrete and continuous bases are introduced. An extension in the spirit of the associated Legendre polynomials and the spherical harmonics is presented introducing the “Jacobi harmonics” that are a generalization of the spherical harmonics to the three-dimensional hypersphere S3.

Operators on anti-dual pairs: Generalized Schur complement

The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the Lebesgue-type decomposition. To demonstrate how this operator approach works in application, we derive the corresponding results for operators acting on rigged Hilbert spaces, and for representable functionals of ⁎-algebras.

Operators on anti-dual pairs: Lebesgue decomposition of positive operators

In this paper we introduce and study absolute continuity and singularity of positive operators acting on anti-dual pairs. We establish a general theorem that can be considered as a common generalization of various earlier Lebesgue-type decompositions. Different algebraic and topological characterizations of absolute continuity and singularity are supplied and also a complete description of uniqueness of the decomposition is provided. We apply the developed decomposition theory to some concrete objects including operators acting in a rigged Hilbert space, Hermitian forms, representable functionals, and additive set functions.