Symmetry conserving 1/N expansion and extended Holstein-Primakoff mapping for anharmonic O(N+1) oscillator model

Symmetry conserving 1/N expansion method combined with the Holstein-Primakoff mapping for bosonic systems is applied to the quantum anharmonic O(N+1) oscillator model which imitates essential features of the quantum-field linear sigma model. The Holstein-Primakoff mapping earlier extended to incorporate both single- and double-boson images is used. In the leading and next-to-leading order the existence of N-Goldstone modes and the validity of the Ward identity are proved. For matrix elements of the pion-pion interaction we show the equivalence of the results obtained via a diagrammatic technique and the extended Holstein-Primakoff mapping