| Summary: | For further insight into the perturbation technique within the framework of the asymptotic iteration method (PAIM), we suggest this method to be used as an alternative method to the traditional well-known perturbation techniques. We show by means of very simple algebraic manipulations that PAIM can be directly applied to obtain the symbolic expectation value of any perturbed potential piece without using the eigenfunction of the unperturbed problem. One of the fundamental advantages of PAIM is its ability to extract a reference unperturbed potential piece or pieces from the total Hamiltonian which can be solved exactly within AIM. After all, one can easily compute the symbolic expectation values of the remaining potential pieces. As an example, the present scheme is applied to the semi-relativistic wave equation with the harmonic-oscillator potential implemented with the Fermi–Breit potential terms. In particular, the non-trivial symbolic expectation values of the Dirac delta function, and the momentum-dependent orbit–orbit coupling terms are successfully calculated. Results are then used, as an illustration, to compute the semi-relativistic s-wave heavy-light meson masses. We obtain good agreement with experimental data for the meson mass splittings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mover accent="true"><mi>d</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mover accent="true"><mi>s</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>.
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