An asymptotic approximation to the solutions and eigenvalues of a boundary value problem for a singularly perturbed relativistic analog of the Schrödinger equation (Q5951200)

When studying properties of the bound states of elementary particles for the case in which relativistic effects are essential, it is preferable to use quasipotential equations obtained in the one time statement of the two-body problem in quantum field theory. This approach has the advantage of being close to the formalism of nonrelativistic quantum mechanics while taking into account relativistic effects. The probabilistic interpretation of the wave function is preserved in quasipotential models. In this paper, a quasipotential equation of the form \[ L_{2m}(\psi_{2m})+ \lambda_{2m}\psi_{2m}= 0\tag{1} \] is considered in the relativistic configuration space for the radial wave functions. The operator \(L_{2m}\) is described by \[ L_{2m}= \sum^{2m}_{p=1} {2(-1)^{p+1}\over (2p)!} \varepsilon^{2p-2} {d^{2p}\over dr^{2p}}- {l(l+ 1)\over r(r+ i\varepsilon)} \sum^{2m}_{p=0} {i^p\over p!} \varepsilon^p{d^p\over dr^p}- v(r) \] with \(\varepsilon> 0\) a small parameter, \(v(r)\) is a quasipotential, \(l\) is the angular momentum, \(\psi_{2m}\) is the solution to the differential equation of order \(2m\). Supplementing equation (1) with the boundary conditions \[ D^i\psi_{2m}(0)= D^i\psi_{2m}(r_0)= 0,\quad i= 0,1,\dots, m-1,\quad D^i= d^i/dr^i,\tag{2} \] a spectral boundary value problem on the closed interval \([0,r_0]\) is obtained. Here, the behavior of the eigenfunctions \(\psi_{2m}\) and eigenvalues \(\lambda_{2m}\) to problem (1)--(2) is investigated in two cases: first, when the small parameter \(\varepsilon\) tends to zero for a given order \(2m\) of the differential equation, and second, for the case in which the order \(2m\) of the equation varies for a given \(\varepsilon\ll 1\).