Retraction: "Algebraic approach to the Tavis–Cummings model with three modes of oscillation" [J. Math. Phys. 59, 073506 (2018)].

In particular, the main problem of the Bogoliubov transformation comes when, with the parameters HT <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow> r </mrow><mo stretchy="false"> </mo></mover></mrow></math> ht and HT <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow> </mrow><mo stretchy="false"> </mo></mover></mrow></math> ht , one tries to verify if the transformation is unitary. In fact, if one computes the commutators between HT <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow> A </mrow><mo stretchy="false"> </mo></mover></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msubsup><mrow><mover accent="true"><mrow> A </mrow><mo stretchy="false"> </mo></mover></mrow><mrow><mn>1</mn></mrow><mrow><mo>†</mo></mrow></msubsup></math> ht and HT <math display="inline" overflow="scroll" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover accent="true"><mrow> A </mrow><mo stretchy="false"> </mo></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msubsup><mrow><mover accent="true"><mrow> A </mrow><mo stretchy="false"> </mo></mover></mrow><mrow><mn>2</mn></mrow><mrow><mo>†</mo></mrow></msubsup></math> ht , an expression similar to Eq. (3) is obtained. [Extracted from the article]