Hermite polynomials and Fibonacci oscillators

Abstract

We compute the (q 1 , q 2 )-deformed Hermite polynomials by replacing the quantum harmonic oscillator problem to Fibonacci oscillators. We do this by applying the (q 1 , q 2 )-extension of Jackson derivative. The deformed energy spectrum is also found in terms of these parameters. We conclude that the deformation is more effective in higher excited states. We conjecture that this achievement may find applications in the inclusion of disorder and impurity in quantum systems. The ordinary quantum mechanics is easily recovered as q 1 = 1 and q 2 → 1 or vice versa.

The Generalized Fibonacci Oscillator as an Open Quantum System

“…FF is a member of GNAM-PA-INdAM Italy. FF first met Michael Schürmann at the conference Quantum Probability and Applications III held in Oberwolfach, January [25][26][27][28][29][30][31] 1987, organized by their respective advisors Professors Luigi Accardi and Wilhelm von Waldenfels [30]. Over most of these years he has had the pleasure of meeting him at the annual QP conferences, that nowadays reached the number 42, visiting him in Greifswald, exchange views and follow reports on his scientific work.…”

“…Recently, the inclusion of two distinct deformation parameters r, q has been proposed to allow more flexibility while retaining the good properties and the possibility of finding explicit formulas as in the case of single parameter deformations (see [6,14,20,25] and the references therein). The two parameters deformed commutation relations become aa † − ra † a = q N , aa † − qa † a = r N where N is the number operator (see Section 2 for precise definitions) in their one-mode Fock space representation.…”

“…In this way one finds a quantum system with Hamiltonian H S = a † a which is a two parameter deformation of the harmonic oscillator whose spectrum {(r n −q n )/(r−q)} n≥0 is a generalized Fibonacci sequence (that turns out to be the well-known Fibonacci sequence for r = 1 + √ 5 /2, q = 1 − √ 5 /2) and therefore is called Fibonacci Hamiltonian. Two parameters Hermite polynomials have been computed and the energy spectrum has been studied showing that the deformation is more effective in highly excited states (see [20,25]). Deformed Fock spaces and deformed Gaussian processes have been analyzed in connection with the single-parameter deformation of the full Fock space of free probability [6].…”

The Generalized Fibonacci Oscillator as an Open Quantum System

“…Recently, the inclusion of two distinct deformation parameters r, q has been proposed to allow more flexibility while retaining the good properties and the possibility of finding explicit formulas as in the case of single parameter deformations (see [6,14,20,24] and the references therein). The two parameters deformed commutation relations become aa † − ra † a = q N , aa † − qa † a = r N where N is the number operator (see Section 2 for precise definitions) in their one-mode Fock space representation.…”

“…In this way one finds a quantum system with Hamiltonian H S = a † a which is a two parameter deformation of the harmonic oscillator whose spectrum { (r n − q n )/(r − q) } n≥0 is a generalized Fibonacci sequence (that turns out to be the well-known Fibonacci sequence for r = (1 + √ 5)/2, q = (1 − √ 5)/2) and therefore is called Fibonacci Hamiltonian. Two parameters Hermite polynomials have been computed and the energy spectrum has been studied showing that the deformation is more effective in highly excited states (see [20,24]). Deformed Fock spaces and deformed Gaussian processes have been analyzed in connection with the single-parameter deformation of the full Fock space of free probability ( [6]).…”