The Dirac equation with a confining potential

R. S.

Bhalerao

and

Budh

Ram

, “,” , – ().

Antonio S.

de Castro

, “,” , – ().

R. M.

Cavalcanti

, “,” , – ().

John R.

Hiller

, “,” , – ().

The solutions of the Dirac equation for the model of Ref. 1 are given in terms of the Hermite function $Hν$ or equivalently, the parabolic cylinder function $Dν.$ The index ν was apparently assumed to be a positive integer in Ref. 1, which is why only one bound state was found. But ν is in general a noninteger as shown in Refs. 2,3,4.

C. L.

 

Critchfield

, “,”  , – ();

C. L.

Critchfield

, “,” , – ().

Budh

Ram

, “,” , – ().

The terminology such as “Lorentz scalar” in the present context is not strictly legitimate. For example, the “scalar potential” $S(x)=g|x|$ is not a Lorentz scalar because $|x|$ is not invariant under the Lorentz transformations. However, we adopt this commonly used terminology.

F.

Cooper

,

A.

Khare

,

R.

Musto

, and

A.

Wipf

, “,” , – ().

Y.

Nogami

and

F. M.

Toyama

, “,” , – ().

In Eq. (4) $v$ has been eliminated in favor of χ or $u.$ Alternatively, we can eliminate $u$ in favor of $v$ by redefining χ as $χ=v(m+S−E+V)−1/2.$ This different choice of χ, which we do not use in this note, results in the same physics, that is, the same wave functions (in terms of $u$ and $v)$⁠, and the same energy eigenvalues. In this connection, note that Eq. (2) is invariant under the simultaneous substitutions of $u→v,$ $v→u,$ $E→−E,$ and $V→−V.$ If $V=0,$ there is a symmetry between positive and negative energy eigenvalues.

If $ν<1$ and if $(g−h)(m+E)<0,$ the last term of $W$ becomes singular and attractive. Then there can be a problem for $W$ in supporting bound states. We exclude such situations. Essentially the same remark applies to the $W(n)$ of Eq. (9).

F. A. B.

Coutinho

and

Y.

Nogami

, “,” , – ().

L. I. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1968), 3rd ed., Chap. 13.