content_image/356916.jpg content_image/356915.jpg content_image/356914.jpg content_image/356912.jpg content_image/356911.jpg
To our best knowledge in the case of l ̸ = 0 the solution of (3.6) is not known. For l = 0 the solution to (3.6) can be expressed by means of the Airy function Ai( x ) [20], namely 

<img src='content_image/356916.jpg'>

where C is constant. We point out that l = 0 was also the case discussed in [6], where the recurrence was identified satisfied by coefficients of the formal power series expansion for the solution to the spinless Salpeter equation corresponding to the massive relativistic harmonic oscillator. Clearly, l = 0 refers to the vanishing angular momentum, therefore we deal in this case with the quantization of the motion of a massless particle in the segment 0 ≤ r ≤ r$_{max}$ = √$_{2}$$_{E}$ κ discussed in the previous section corresponding to the condition J = 0 (third type of the motion). Furthermore, for l = 0 the ansatz (3.5) takes the form 

<img src='content_image/356915.jpg'>

Demanding that ˜ ψ$_{E}$ ( k ) is well defined for k = 0 we find χ (0) = 0 (compare [21] Eq. (32.11)), which leads to Ai ( − 2 E (2 cκ ℏ ) 2 3 ) = 0. This quantization condition means that the values of the energy E$_{n}$ , n = 1 , 2 , . . . , are given by zeros of the Airy function a$_{n}$ . We have 

<img src='content_image/356914.jpg'>

Using the fact that the functions Ai( x + a$_{n}$ ) / $_{Ai}$′$_{(}$$_{a}$$_{n}$$_{),}$ n = 1 , 2 , . . . $_{, where Ai}$′$_{(}$$_{x}$$_{) designates}$ the derivative of the Airy function Ai( x ), form an orthonormal basis on the interval [0 , ∞ ) [16], we find that the normalized solutions (3.5) to (3.4) in the Hilbert space L $^{2}$( R $^{3}$, d$^{3}$k ), with l = 0 can be written as 

<img src='content_image/356912.jpg'>

From (3.10) and (3.3) we finally obtain the normalized wave functions such that 

<img src='content_image/356911.jpg'>

where r = | x | . 

As in the case of the nonrelativistic harmonic oscillator with the probability density different from zero outside the turning points, the probability density ρ$_{n}$ ( r ) = | ψ$_{n}$ ( x ) | 2 does not vanish for r > r$_{max}$ ( E$_{n}$ ), where r$_{max}$ ( E$_{n}$ ) = √$_{2}$$_{E}$ n κ , n = 1 , 2 , . . . , . However, it 

