arXiv subjects
Publications and source records attributed to Keshav Raj Acharya.
In this paper we study $2d$-dimensional canonical systems with positive semidefinite Hamiltonians $H(t)$ satisfying $\operatorname{tr} H(t) \equiv 1$. We first establish that every such system possesses at least one nontrivial $H$-integrable solution, ensuring the existence of solutions whose energy with respect to $H$ remains finite. We then show that the space of all $H$-integrable solutions has dimension at most $d$.
In many practical applications, signals and environments are time- varying, which makes fixed filters unreliable. Adaptive filtering, on the other hand, updates in real time to suppress noise, track nonstationary signals, and identify unknown systems. This paper investigates an adaptive filtering frame- work based on canonical systems with time-varying symmetric positive semi- definite Hamiltonian matrices. The proposed method adapts the Hamiltonian matrix using a gradient-based scheme designed to minimize the squared er- ror between the system output and a desired reference signal. We establish theoretical stability guarantees via Lyapunov analysis, ensuring boundedness of system trajectories and convergence of the error signal under suitable as- sumptions. Furthermore, we present numerical integration schemes preserving the underlying Hamiltonian structure and projective techniques to maintain positive semidefiniteness of the Hamiltonian matrix. Extensive simulations on synthetic nonstationary signals illustrate the effectiveness and robustness of the proposed adaptive filter.
This paper studies linear relations and their self-adjoint realizations arising from 2d-dimensional canonical systems, with a focus on how the symplectic structure interacts with boundary conditions. Understanding this interplay allows us to define self-adjoint realizations, which are crucial for analyzing the spectral properties of these systems. We prove that for each pair of Lagrangian boundary matrices Θ and B satisfying appropriate orthonormality conditions, the restricted relation TΘ,B is self-adjoint. Our approach relies on the symplectic geometry of boundary spaces and the isotropic structure of Lagrangian subspaces. We also discuss extensions to semi-infinite intervals using asymptotic boundary conditions. In the second part of the paper, we show how this framework applies to spectral problems from partial differential equations, including the stability of traveling waves and the linearization of the focusing nonlinear Schrodinger equation around soliton profiles. In particular, the self-adjoint structure with respect to the H-weighted inner product ensures the spectral properties needed for stability analysis using Evans function and transfer matrix methods. Applications to integrable systems, such as the stability of NLS bright solitons, are also presented.
This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the ω limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full multiplicity.
In this paper, we study compact linear relations and establish that the spectrum of compact linear relations consists of only the point spectrum and the zero set under suitable conditions.
We show that the half-line $m$ functions associated with the vector-valued Schrodinger operators are the elements in the Siegel upper half space. We introduce a metric on the space of $m$ functions associated to the vector-valued discrete Schrodinger operators. Then we show that the action of transfer matrices on these $m$ functions is distance decreasing.
We study the 2N-dimensional canonical systems and discuss some properties of its fundamental solution. We then discuss the Floquet theory of periodic canonical systems and observe the asymptotic behavior of its solution. Some important physical applications of the systems are also discussed: linear stability of periodic Hamiltonian systems, position-dependent effective mass, pseudo-periodic nonlinear water waves, and Dirac systems.
The Siegel upper half space, $\mathcal{S}_n$, the space of complex symmetric matrices, $Z$ with positive definite imaginary part, is the generalization of the complex upper half plane in higher dimensions. In this paper, we study a generalization of linear fractional transformations, $Φ_S$, where $S$ is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which $Φ_S(Z)$ is well defined. We also consider $\mathcal S_n$ and $\overline{\mathcal S}_n$ as metric spaces and discuss distance properties of the map $Φ_S$ from $\mathcal S_n$ to $\mathcal{S}_n$ and $\overline{\mathcal S}_n$ respectively.
The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on C. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H(x)=1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.
We prove the Remling's Theorem on canonical systems and discuss the connection between Jacobi and Schrödinger equation and canonical systems.