and r$_{max}$ defined by (2.14) and (2.15) we find where Jc κ 2 ( 3 κ 2 2 E ) 3 2 = 1 refers to the motion in a circle, 0 < Jc κ 2 ( 3 κ 2 2 E ) 3 2 < 1 corresponds to the motion between two circles, and Jc κ 2 ( 3 κ 2 2 E ) 3 2 = 0, i.e. J = 0 is the condition for the motion in the segment. In relativistic quantum mechanics the massless harmonic oscillator defined by the Hamil- tonian (2.1) is described by a massless version of the spinless Salpeter equation where Δ$_{x}$ = ( ∂ ∂ $_{x}$) $^{2}$. Therefore the eigenvalue equation for the Hamiltonian ˆ Hψ$_{E}$ = Eψ$_{E}$ takes the form of the pseudodifferential equation Performing the Fourier transformation we get from (3.2) the following equation: where Δ$_{k}$ = ( ∂ ∂ $_{k}$) $^{2}$. Finally, switching over to the spherical coordinates k = ( k sin α cos β, k sin α sin β, k cos α ), where k = | k | , and making the ansatz where Y m l ( α, β ) are the spherical functions, we obtain the “radial equation” ( − d 2 dk 2 + l ( l + 1) k 2 + 2 c ( κ ℏ ) $_{2}$k ) χ ( k ) = 2 E ( κ ℏ ) $_{2}$χ ( k ) . (3.6) ˜ ψ$_{E}$ ( k ) = χ ( k ) k Y m l ( α, β ) , (3.5) ( − Δ$_{k}$ + 2 c ( κ ℏ ) 2 | k | ) ˜ ψ$_{E}$ ( k ) = 2 E ( κ ℏ ) 2 ˜ ψ$_{E}$ ( k ) , (3.4) ψ ( x ) = 1 (2 π ℏ ) 3 2 ∫ d $^{3}$k e i k · x ℏ ˜ ψ ( k ) , (3.3) ( c ℏ √ − Δ$_{x}$ + κ 2 2 x 2 ) ψ$_{E}$ ( x ) = Eψ$_{E}$ ( x ) . (3.2) i ℏ ∂ ∂t ψ ( x , t ) = ( c ℏ √ − Δ$_{x}$ + κ 2 2 x 2 ) ψ ( x , t ) , (3.1) 10 III. QUANTUM MECHANICS OF THE RELATIVISTIC MASSLESS HAR- MONIC OSCILLATOR 0 ≤ Jc κ 2 ( 3 κ 2 2 E )$^{3}$ 2 ≤ 1 , (2.25)