EIGENFUNCTIONS OF 2d DIMENSIONAL CANONICAL SYSTEMS

ACHARYA, KESHAV RAJ; Ludu, Andrei

doi:10.21608/ejmaa.2025.352855.1309

EIGENFUNCTIONS OF 2d DIMENSIONAL CANONICAL SYSTEMS

Document Type : Regular research papers

Authors

KESHAV RAJ ACHARYA ¹
Andrei Ludu ²

¹ Department of Mathematics, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Daytona Beach, FL 32114, U.S.A

² Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL

10.21608/ejmaa.2025.352855.1309

Abstract

A 2d dimensional canonical systems is a system of 2d first order differential equation which can be written in the form Jy′ = zHy where J is a constant symplectic matrix, H is a positive semidefinite matrix and z is a complex number. In this article we show that the (normalized) eigenfunctions of 2d of such canonical systems with the boundary conditions (θ1, θ2)y(0) = (β1, β2)y(N) = 0
form an orthonormal basis for a Hilbert space L2(H, [0,N]) ⊖ Z, where Z is a closed subspace of L2(H, [0,N]), a Hilbert space of H-integrable functions. In addition, we also discuss the
application of this theory to a chain of particles governed by a time-dependent. More specifically We discuss an example of a Hamiltonian system for a one dimensional
chain of particles having time dependent effective masses and moving
in explicit time-dependent mutual and external interaction potentials. We present a few numerical examples of solutions featuring the existence of the orthogonal systems of solutions, in the range
of the numerical precision of the calculations Hamiltonian.

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