Bose-like oscillator in fractional-dimensional space

The system of a fractional-dimensional Bose-like oscillator of one degree of freedom whose canonical variables satisfy the general Wigner commutation relations is investigated. Momentum-position uncertainty relations are obtained. For states without definite parity the well known one-dimensional Heisenberg uncertainty principle is retained, but for odd and even states momentum-position uncertainty inequalities depending on the dimension of the space are obtained. The motions of both the free particle and the harmonic oscillator in a fractional-dimensional space are studied through the probability density function. The existence of compression (spread) of the probability density for dimensions D < 1 (D > 1) is shown. Fractional-dimensional Bose-like operators are also deduced. They together with the reflection operator form an R-deformed Heisenberg algebra with a deformation parameter depending on the dimension of the space.