Unbounded Self-adjoint Operators Research Articles

Design of Quantum Gates Using Quantum Scattering Theory

Given two Hamiltonians \(H_0,H_1=H_0+V\) corresponding respectively to a free particle Hamiltonian and the free particle Hamiltonian plus its interaction energy with a scattering centre, we can compute the S-matrix, i.e., the matrix that determines the quantum mechanical amplitude of the process involving a particle coming from a free particle state at an infinite distance from the scattering centre at time \(t=-\infty\), entering into an initial scattered state, interacting with the scattering centre, entering into a final scattered state, and then escaping away to infinity at time \(t\rightarrow\infty\) to a final free particle state. Such amplitudes were first studied by Lippmann and Schwinger (Steven Weinberg, The Quantum Theory of Fields, Vol. 1, W.O. Amrein, Hilbert Space Methods in Quantum Mechanics) and then analyzed in a mathematically precise way by [T. Kato], [W. Amrein], and [K.B. Sinha]. In this paper, we first derive the Lippmann-Schwinger equation for the input and output scattered states in terms of the input and output free particle states using intuitive arguments originally due to Lippmann and Schwinger, and then, using rigorous operator theoretic arguments revolving around the spectral theorem for unbounded self-adjoint operators in a Hilbert space, we obtain an elegant formula for the scattering matrix elements in terms of the interaction potential and the free particle Hamiltonian between two free particle states corresponding to the same measure. We then discuss a computationally more efficient method based on the Dyson series expansion (Steven Weinberg, The Quantum Theory of Fields, Vol. 1), which works even in the case when the interaction Hamiltonian is time varying, in contrast to the Lippmann-Schwinger method, for calculating the S-matrix. We assume that the scattering potential, which can be time varying, can be controlled by incorporating control parameters and explain how the resulting unitary S-matrix can be made to approximate a given unitary gate in infinite-dimensional Hilbert space by optimizing the Hilbert-Schmidt/Frobenius norm distance between the given gate and the S-matrix gate, in which the latter is approximated by a truncated Dyson series. Of course, the S-matrix given by the Dyson series is a Taylor series functional expansion in the interaction potential and hence is a highly nonlinear function of the control parameters, and hence we propose numerical methods for such optimization. We then observe that if the control parameters are taken as random variables with joint moments, then the TPCP map involving statistical averaging of the adjoint action of the S-matrix acting on an initial mixed state w.r.t. the probability distribution of the control parameters will result in a final mixed state that is linear in the statistical moments of the control parameters, and hence optimization w.r.t. these moments can be easily carried out. Our method of TPCP map design is to optimize the sum of Frobenius distance squares between a sequence of desired output mixed states and the S-matrix based formula for the TPCP map acting on the corresponding input mixed states w.r.t. the statistical moments of the control parameters. We then generalize this method to include the case of TPCP maps obtained from the Hudson-Parthasarathy quantum noisy Schrödinger equation. In summary, we simulate the S-matrix gate for independent realizations of the control parameters having a probability distribution defined by their joint moments obtained by the above optimization procedure and apply the adjoint action of these independent S-matrices on a given input state, and take their ensemble average to obtain a good approximation to the action of the desired TPCP map on any given input state. This suggests that TPCP maps of arbitrarily large size can be realized using scattering experiments in the laboratory.

Functions of perturbed noncommuting unbounded selfadjoint operators

Let f f be a function on R 2 \mathbb {R}^2 in the inhomogeneous Besov space {\text \textit {\Russian {B}}}_{\infty ,1}^{1}(\mathbb {R}^2). For a pair ( A , B ) (A,B) of not necessarily bounded and not necessarily commuting self-adjoint operators, the function f ( A , B ) f(A,B) of A A and B B is introduced as a densely defined linear operator. It is shown that if 1 ≤ p ≤ 2 1\le p\le 2 , ( A 1 , B 1 ) (A_1,B_1) and ( A 2 , B 2 ) (A_2,B_2) are pairs of not necessarily bounded and not necessarily commuting selfadjoint operators such that both A 1 − A 2 A_1-A_2 and B 1 − B 2 B_1-B_2 belong to the Schatten–von Neumann class S p {\boldsymbol {S}}_p and f\in {\text \textit {\Russian {B}}} _{\infty ,1}^{1}(\mathbb {R}^2), then the following Lipschitz type estimate holds: \begin{equation*} \|f(A_1,B_1)-f(A_2,B_2)\|_{{\boldsymbol {S}}_p} \le \operatorname {const}\|f\|_{\text \textit {\Russian {B}}_{\infty ,1}^{1}}\max \big \{\|A_1-A_2\|_{{\boldsymbol {S}}_p},\|B_1-B_2\|_{{\boldsymbol {S}}_p}\big \}. \end{equation*}

Spectral theorem approach to commutative C⁎-algebras generated by Toeplitz operators on the unit ball: Quasi-elliptic related cases

We consider commutative C⁎-algebras of Toeplitz operators in the weighted Bergman space on the unit ball in Cn. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.

On an inverse problem for a linearized system of Navier–Stokes equations with a final overdetermination condition

Abstract The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved.

Inverse learning in Hilbert scales

We study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.

An algorithm for identifying eigenvectors exhibiting strong spatial localization

We introduce an approach for exploring eigenvector localization phenomena for a class of (unbounded) selfadjoint operators. More specifically, given a target region and a tolerance, the algorithm identifies candidate eigenpairs for which the eigenvector is expected to be localized in the target region to within that tolerance. Theoretical results, together with detailed numerical illustrations of them, are provided that support our algorithm. A partial realization of the algorithm is described and tested, providing a proof of concept for the approach.

Functions of Pairs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation

Functions of Pairs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation

Commutative Algebras of Toeplitz Operators on the Bergman Space Revisited: Spectral Theorem Approach

For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols.

Two sharp inequalities for operators in a Hilbert space

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.

On Murray-von Neumann algebras—I: topological, order-theoretic and analytical aspects

For a countably decomposable finite von Neumann algebra \({\mathscr {R}}\), we show that any choice of a faithful normal tracial state on \({\mathscr {R}}\) engenders the same measure topology on \({\mathscr {R}}\) in the sense of Nelson (J Funct Anal 15:103–116, 1974). Consequently it is justified to speak of ‘the’ measure topology of \({\mathscr {R}}\). Having made this observation, we extend the notion of measure topology to general finite von Neumann algebras and denominate it the \({\mathfrak {m}}\)-topology. We note that the procedure of \({\mathfrak {m}}\)-completion yields Murray-von Neumann algebras in a functorial manner and provides them with an intrinsic description as unital ordered complex topological \(*\)-algebras. This enables the study of abstract Murray-von Neumann algebras avoiding reference to a Hilbert space. Furthermore, it makes apparent the appropriate notion of Murray-von Neumann subalgebras, and the intrinsic nature of the spectrum and point spectrum of elements, independent of their ambient Murray-von Neumann algebra. In this context, we show the well-definedness of the Borel function calculus for normal elements and use it along with approximation techniques in the \({\mathfrak {m}}\)-topology to transfer many standard operator inequalities involving bounded self-adjoint operators to the setting of (unbounded) self-adjoint operators in Murray-von Neumann algebras. On the algebraic side, Murray-von Neumann algebras have been described as the Ore localization of finite von Neumann algebras with respect to their corresponding multiplicative subset of non-zero-divisors. Our discussion reveals that, in addition, there are fundamental topological, order-theoretic and analytical facets to their description.

Nonlinear dispersive equations: classical and new frameworks

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrodinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.

Perturbation series for Jacobi matrices and the quantum Rabi model

We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.

Strong Commutativity of Unbounded Self-adjoint Operators on a Separable Hilbert space

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

Laplacians on Smooth Distributions as C*-Algebra Multipliers

In this paper we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold, initiated in a previous paper. Under assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded regular self-adjoint operator in some Hilbert module over the foliation $C^*$-algebra.

The Fuglede theorem and some intertwining relations

In this paper, we show new versions of the Fuglede theorem in an unbounded setting. A related counterexample is also presented. In the second part of the paper, we give a pair of a closed and a self-adjoint (unbounded) operator which is not intertwined by any (bounded or closed) operator except the zero operator.

An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

The existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form aA+bB (a,bin {{mathbb {R}}}) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of f(aA+bB) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions f: {{mathbb {R}}}rightarrow {{mathbb {C}}}. In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.

Non-Archimedean analysis and a wave-type pseudodifferential equation on finite adèles

In this work the ring of finite adeles $${\mathbb {A}}_f$$ of the rational numbers $${\mathbb {Q}}$$ is obtained as a completion of $${\mathbb {Q}}$$ with respect to a certain non-Archimedean metric related to the second Chebyshev function, which allows us to represent any finite adele as a convergent series, generalizing m-adic analysis. This polyadic analysis allows us to introduce a novel pseudodifferential operator $$D^{\alpha }$$ on $$L^2({\mathbb {A}}_f)$$ of fractional differentiation, similar to the Vladimirov operator on the p-adic numbers. The operator $$D^{\alpha }$$ is a positive selfadjoint unbounded operator whose spectrum $$\sigma (D^{\alpha })$$ is essential and it consists of a countable number of eigenvalues, which converges to zero, and zero. Moreover, a sort of multiresolution analysis on $${\mathbb {A}}_f$$ provides us with a wavelet basis which is an orthonormal basis of eigenfunctions of $$D^{\alpha }$$ as well. The Cauchy problem of a wave-type pseudodifferential equation $$\begin{aligned} u_{tt}(x,t)+D^{\alpha }_x u(x,t) =F(x,t), \qquad (x \in {\mathbb {A}}_f), \end{aligned}$$ with appropriate initial conditions $$u(x,0)=f(x), u_t(x,0)=g(x),$$ and external force F(x, t), is solved separating variables and using the Fourier expansion of functions in $$L^2({\mathbb {A}}_f)$$ , with respect to the wavelet basis.

Multidimensional spectral order for selfadjoint operators

The notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order ≼ is preserved by transformations represented by spectral integrals of separately increasing Borel functions on Rκ. In particular, the κ-dimensional spectral order is the restriction of product of κ spectral orders for selfadjoint operators. In the context of positive κ-tuples of pairwise commuting selfadjoint operator, the relation A≼B holds if and only if Aα⩽Bα for every κ-tuples of nonnegative integer numbers α.