A note on the eigenvalues of the Hamiltonian of the harmonic oscillator perturbed by the potential \(\lambda x^2/(1+gx^2)\) (Q1373523)

The author uses a perturbation argument for the Fredholm determinant of the Birman-Schwinger kernel of the operator \(H_0-{\lambda\over g(1+gx^2)}\), where \[ H_0= -{d^2\over dx^2}+ x^2, \] to determine the ground state energy and the first excited level energy of the Hamiltonian \[ H_0- {\lambda x^2\over(1+ gx^2)}= H_0-{\lambda\over g(1+ gx^2)}+ {\lambda\over g} \] in terms of smooth functions of the parameters \(\lambda\) and \(\eta= g^{-1/2}\) in a neighbourhood of \((0,0)\).