Conventional perturbation theory, developed for Hermitian matrices, is inadequate for determining the eigenvalues and eigenvectors of bosonic Bogoliubov–de Gennes (BdG) Hamiltonians, whose dynamical structure leads to a non-Hermitian generalized eigenvalue problem. Here we extend the Hellmann-Feynman perturbation theory to bosonic BdG systems and formulate a unified framework applicable to both nondegenerate and nearly degenerate cases. The method provides systematic expansions for eigenvalues, eigenstates, and effective Hamiltonians, revealing that the nearly degenerate theory emerges naturally from the nondegenerate one. A key result of this work is that the perturbative framework gives access not only to the spectrum but also to geometric quantities. In particular, we obtain fully analytical expressions for the Berry curvature, including a previously overlooked contribution arising from the transformed basis. To demonstrate the robustness of the formalism, we apply it to both a ferromagnetic honeycomb lattice and a more complex triangular-lattice antiferromagnet described by a $6\times 6$ BdG Hamiltonian with multiple magnon bands and Dirac-point degeneracies. In both systems, the resulting effective $2\times 2$ Hamiltonians accurately capture the low-energy band structure and Berry curvature, in excellent agreement with full numerical diagonalization. These results establish a versatile and broadly applicable perturbation framework for bosonic BdG systems and provide new analytical insight into the geometric and topological properties of magnonic excitations.

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• Magnetism
• Spin waves
• Ferromagnets
• Magnetic systems
• Approximation methods for many-body systems
• Holstein-Primakoff method
• Many-body techniques
• Methods in magnetism
• Spin wave theory