Nonadditive quantum mechanics as a Sturm–Liouville problem

Abstract

The modified Schrödinger equation obtained by Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] is shown to be a Sturm–Liouville problem. This demonstration guarantees that Hamiltonian eigenvalues obtained in this formalism are real. It also allows us to show that, regardless of the non-Hermitian characteristic of the Hamiltonian operator in the Hilbert space, its time evolution remains unitary.

κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

“…We consider that the techniques employed in this work could stimulate the seek of other generalizations of classical and quantum mechanical aspects, as has been reported in recent research studies by means of the q-algebra. 7,[48][49][50][51][52][53][54][55][56]…”

“…Similarly, as was done in Ref. 54, we consider as a quantitative measure of the error the function defined by R(N)…”

“…Complementarily, it has been found that the mathematical foundations of the PDM systems rely on the assumption of the noncommutativity between the mass operator m(x) and the linear momentum operatorp, thus giving place to the ordering problem for the kinetic energy operator. 4,[40][41][42][43][44][45][46][47] In addition, the development of generalized translation operators motivated the introduction of a positiondependent linear momentum for characterizing a particle with a PDM 7,[48][49][50][51][52][53][54][55][56] that can be related to a generalized algebraic structure (called q-algebra 57 ) inherited from the mathematical background of nonextensive statistics. 58 Concerning these formal structures, the κ-deformed statistics, originated from the κ-exponential and κ-logarithm functions, allows us to develop an algebraic structure, called κ-algebra, [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74] with similar properties to those of the q-algebra.…”

$κ$-Deformed quantum and classical mechanics for a system with position-dependent effective mass

“…Concerning the Sturm-Liouville problem, Braga et al 54 have introduced a Fourier series in terms of deformed trigonometric functions that emerge from the formalism studied in Ref. 48.…”

“…Complementarily, it has been found that the mathematical foundations of the PDM systems rely on the assumption of the noncommutativity between the mass operator m(x) and the linear momentum operator p, thus giving place to the ordering problem for the kinetic energy operator, 4,[40][41][42][43][44][45][46][47] . In addition, the development of generalized translation operators motivated the introduction of a position-dependent linear momentum for characterizing a particle with a PDM 7,[48][49][50][51][52][53][54][55][56] that can be related to a generalized algebraic structure (called q-algebra 57 ) inherited from the mathematical background of nonextensive statistics. 58 Concerning these formal structures, the κ-deformed statistics, originated from the κ-exponential and κ-logarithm functions, allows to develop an algebraic structure, called κ-algebra, [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74] with similar properties to the those of the q-algebra.…”