2021/11/26 by F. Kecita, A. Bounames, M. Maamache +1
Mathematics · Physics and Astronomy · #Algorithm #Hamiltonian (control theory) #Hermitian matrix #Mathematics #Nonlinear Waves and Solitons #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #quant-ph
paper · pdf · doi:10.1088/1402-4896/ac3dbd
published as Phys. Scr. 96 (2021) 125265 · 12 pages
openalex publication_date 2021/11/26 · arxiv created 2021/12/27 · arxiv updated 2021/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="italic">PT</mml:mi> </mml:math> −inner product, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> is a obtained from the charge conjugation operator <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> </mml:math> through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="italic">PT</mml:mi> </mml:math> normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.