2021/05/11 by E. Celeghini, M. Gadella, M. A. del Olmo · 1 citation
Physics and Astronomy · Mathematics · #Quantum Mechanics and Non-Hermitian Physics #Mathematical Analysis and Transform Methods #Algebraic and Geometric Analysis #Orthonormal basis #Mathematics #Hilbert space #Hermite polynomials #Square-integrable function #Fourier series #Fourier transform #Pure mathematics #Unit circle #Operator (biology) #Unitary state #Operator theory #Series (stratigraphy) #Discrete Fourier transform (general) #Space (punctuation) #Mathematical analysis #Fractional Fourier transform #Fourier analysis #Physics #Quantum mechanics
paper · pdf · doi:10.3390/sym13050853
openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.