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The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time-invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set. [ABSTRACT FROM AUTHOR]

This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of the second-order moments of the system variables. Small-gain-theorem bounds are obtained for the back-action of the observer on the covariance dynamics of the plant in terms of the plant–observer coupling. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant–observer coupling matrix in order to mitigate the back-action effect. For the discounted mean square optimal CQF problem with penalized back-action, we establish the first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and Lie-algebraic techniques, this set of equations is represented in a more explicit form for equally dimensioned plant and observer. For a class of such observers with autonomous estimation error dynamics, we obtain a solution of the CQF problem and outline a homotopy method. The computation of the performance criteria and the observer synthesis are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Let $ J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n} $. A matrix $ A \in \mathbb{R}^{2n\times 2n} $ is said to be Hamiltonian if $ (AJ)^{\top} = AJ $. In this paper, we first consider the following generalized inverse eigenvalue problem (GIEP): Given a pair of matrices $ (\Lambda, X) $ in the form $ \Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p} $ and $ X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p} $, where diagonal elements of $ \Lambda $ are all distinct with rank$ (X) = p $, and both $ \Lambda $ and $ X $ are closed under complex conjugation in the sense that $ \lambda_{2i} = \bar{\lambda}_{2i-1}\in \mathbb{C}, $ $ {\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n} $ for $ i = 1, \cdots, l, $ and $ \lambda_{j}\in \mathbb{R}, $ $ {\bf{x}}_{j}\in \mathbb{R}^{2n} $ for $ j = 2l+1, \cdots, p. $ Find Hamiltonian matrices $ A $ and $ B $ such that $ AX\Lambda = BX. $ Then, we consider the associated optimal approximation problem (OAP): Given $ \tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n} $. Find $ (\hat{A}, \hat{B}) \in \mathbb{S_{E}} $ such that $ \|\hat{A}-\tilde{A}\|^2+\|\hat{B}-\tilde{B}\|^2 = {\min_{(A, B)\in \mathbb{S_{E}}}}\left(\|A-\tilde{A}\|^2+\|B-\tilde{B}\|^2\right), $ where $ \mathbb{S_{E}} $ is the solution set of Problem GIEP. By using the QR-decomposition, we deduce the representation of the general solution of Problem GIEP. Also, we obtain the unique optimal approximation solution $ (\hat{A}, \hat{B}) $ of Problem OAP.

An algorithm is proposed for solving the problem of analytical constructing of an optimal fractional-order regulator (OFOR) in the general case. By inscribing the extended functional, the corresponding fractional order Euler-Lagrange equation is determined. Then, using the Mittag-Leffler function, a fundamental solution to the corresponding Hamiltonian system is constructed. It is shown that to obtain an analogue of the analytical construction of AM Letov's regulators, the order of the fractional derivatives must be a rational number, the denominator and numerator of which are odd numbers. Numerical illustrative examples are provided.

[Display omitted] • Energies of the H 3 C− anion ν 2 mode were found at 1) CCSD(T)/CBS(T,Q,5) and 2) CCSD(T)/s,d,t,q-Aug-cc-pVTZ levels of theory • The inversion splittings of the ν 2 mode converge to the different limits for the 1) and 2) levels of theory • The inversion splittings of the umbrella mode of the H 3 S radical were found at the CCSD(T)/CBS(D,T,Q) level of theory • An original set of two vibrational coordinates was used to simplify calculations Splitting of the ground state and some excited symmetric bending vibrational states due to inversion tunneling in the H 3 C− anion and H 3 Si radical is analyzed by numerically solving the vibrational Schrödinger equation of restricted (2D) dimensionality. We used the following two vibrational coordinates for the H 3 X structure (X = C, Si): the distance of the X atom from the plane of a regular triangle formed by three hydrogen atoms (1) and a symmetry coordinate composed of three distances between chemically non-bonded hydrogen atoms (2). The kinetic energy operator in this case takes the simplest form. The 2D potential energy surface (PES) in the given coordinates was calculated for H 3 C− at the CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, CCSD(T)/aug-cc-pV5Z, CCSD(T)/CBS(TQ5), CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/ t -aug-cc-pVTZ, and CCSD(T)/q-aug-cc-pVTZ levels of theory, based on recommendations from recently published work [M.C. Bowman, B. Zhang, W.J. Morgan, H.F. Schaefer III, Mol.Phys., 117 (2019) 1069–1077]. The same 2D PES for the H 3 Si radical was calculated at the CCSD(T)/aug-cc-pVDZ CCSD(T)/aug-cc-pVTZ, CCSD(T)/aug-cc-pVQZ, and CCSD(T)/CBS(D,T,Q) as well as at the CCSD(T)/d-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVTZ, CCSD(T)/un-aug-cc-pVQZ levels of theory. The tunneling splittings for the D 3 C− anion and D 3 Si radical were calculated as well. The results of calculations demonstrate good agreement with available experimental data on umbrella vibration frequencies and inversion splittings for the title molecules. [ABSTRACT FROM AUTHOR]

This paper presents a methodology to describe quantum mechanical states of charge qubits, realized as coupled quantum dots occupied by single electrons, using equivalent electrical circuits. We explain how to construct all equations starting from low-level simulations of wave functions and interpret the relationship between the parameters appearing in the resulting model and the parameters of the physical system. An efficient methodology to obtain 2D wave functions is presented. Results obtained in this study using a circuit simulator correspond to quantum dot arrays fabricated in a CMOS technology and recently published in the literature. The generalization of quantum mechanical equations, their conversion to equivalent circuits, and numerical examples are discussed.

It is well-known that the symplectic Lanczos method is an efficient tool for computing a few eigenvalues of large and sparse Hamiltonian matrices. A variety of block Krylov subspace methods were introduced by Lopez and Simoncini to compute an approximation of exp(M)V for a given large square Hamiltonian matrix M and a tall and skinny matrix V that preserves the geometric property of V. For the same purpose, in this paper, we have proposed a new method based on a global version of the symplectic Lanczos algorithm, called the global J-Lanczos method (GJ-Lanczos). To the best of our knowledge, this is probably the first adaptation of the symplectic Lanczos method in the global case. Numerical examples are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Certain known facts concerning spectral peculiarities of products of symmetric, skew-symmetric, and Hermitian matrices are extended to paired products of matrices belonging to some special classes. [ABSTRACT FROM AUTHOR]

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number. [ABSTRACT FROM AUTHOR]

Abstract Let J∈Rn×n $J \in{\mathbb{R}}^{n\times n}$ be a normal matrix such that J2=−In $J^{2}=-I_{n}$, where In $I_{n}$ is an n-by-n identity matrix. In (S. Gigola, L. Lebtahi, N. Thome in Appl. Math. Lett. 48:36–40, 2015) it was introduced that a matrix A∈Cn×n $A \in{\mathbb {C}}^{n\times n}$ is referred to as normal J-Hamiltonian if and only if (AJ)∗=AJ ${(AJ)}^{*}=AJ$ and AA∗=A∗A $AA^{*}=A^{*}A$. Furthermore, the necessary and sufficient conditions for the inverse eigenvalue problem of such matrices to be solvable were given. We present some alternative conditions to those given in the aforementioned paper for normal skew J-Hamiltonian matrices. By using Moore–Penrose generalized inverse and generalized singular value decomposition, the necessary and sufficient conditions of its solvability are obtained and a solvable general representation is presented.

This note studies linear quadratic (LQ) mean field social optimization problems with non-uniform agents and indefinite state weight, and analyzes the resulting linear system of ordinary differential equations (ODEs) subject to a partial initial condition and a growth condition. We apply a subspace decomposition method for existence analysis and computation. This is accomplished by carefully choosing the unspecified initial condition in the ODE system such that the modes exceeding the admissible growth rate are not activated. We further extend this method to LQ mean field games. [ABSTRACT FROM AUTHOR]

In this paper we find a symmetric solution of the matrix Riccati equation: XCX + XBT + BX + A = 0, where A, B, C, X are real square matrices of order n, A and C are symmetric. This equation appears when one seeks for the normal form of a quadratic Hamiltonian with the help of a canonical transformation. We consider the case where zero eigenvalues of the corresponding Hamiltonian matrix V = (BT C -A -B), form blocks of even order in the matrix of Jordan normal form (with the exception, per- haps, of one pair, forming two blocks of the first order). Matrix Riccati equation arises in optimal control problems. Solution method relies on finding appropriate eigenvectors and generalized eigenvectors of the Hamiltonian matrix V . We prove that the vectors satisfy the symplectic conditions. We obtain sufficient conditions for existence of a solution. [ABSTRACT FROM AUTHOR]

A simple proof of Williamson’s theorem is given. This theorem states that a real symmetric positive definite matrix A of even order can be brought to diagonal form Λ by a symplectic congruence transformation. The diagonal entries of Λ are called symplectic eigenvalues of A. The problem of calculating these values is also discussed. [ABSTRACT FROM AUTHOR]