The spectrum of the Lichnerowicz Laplacian on \(\mathbb{C} P^n\) (Q2756231)

Let \((M^n,g)\) be a closed oriented Riemannian manifold, \(S^p M\) the space of symmetric \(p\)-forms, \(\delta_0: S^1M\to C^\infty(M)\), \(\delta_1: S^2 M\to S^1M\) the divergence operators and \(\Delta^1\), \(\Delta^2\) the corresponding Lichnerowicz Laplacians. The author calculates the spectrum of \(\Delta^1_{\mathbb{C} P^n}\), \(\Delta^2_{\mathbb{C} P^n}\) and proves the following theorems:NEWLINENEWLINENEWLINETheorem 1. The spectrum of \(\Delta^1_{\mathbb{C} P^n}\) is NEWLINE\[NEWLINE\sigma(\Delta^1_{\mathbb{C} P^n})= \{4k(n+ k)\mid k\geq 1\}\cup \{4(k+ 1)(k+ n)\mid k\geq 0\}NEWLINE\]NEWLINE with the multiplicities NEWLINE\[NEWLINE\text{mult}(4k(k+ n))= 4n(n+ 2k)\Biggl({(n+ 1)\cdot(n+ k-1)\over k!}\Biggr)^2,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{mult}(4(k+ 1)(k+ n))= {4((k+ n-1)!)^2(n- 1)k\over n!(n- 1)!((k+ 1)!)^2} (2k^2+ 3(n+ 1)k+ (n+ 1)^2).NEWLINE\]NEWLINE Theorem 2: NEWLINE\[NEWLINE\begin{multlined}\sigma(\Delta^2_{\mathbb{C} P^n})= \{4k(k+ n)\mid k\geq 0\}\cup \{4(k+ 1)(k+ n)\mid k\geq 1\}\cup\\ \{4(k+ 1)(k+ n-1)\mid k\geq 0\}\cup \{4(k^2+ nk+ n+ 1)\mid k\geq 0\}.\end{multlined}NEWLINE\]NEWLINE The proofs use the explicit expressions for \(\Delta^p_{\mathbb{C} P^n}\), \(\Delta^p_{S^{2n+1}}\), the O'Neill formulas and the knowledge of \(\sigma(\Delta^p_{S^k})\).