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This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. In general, for any $m$, Weyl's spectral estimates imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. (Hat-tip to Marc Palm!) In particular this shows that for $m\geq 2$ and a generic metric we have

$$0<\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.

This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. In general, for any $m$, Weyl's spectral estimates imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. (Hat-tip to Marc Palm!) In particular this shows that for $m\geq 2$ and a generic metric we have

$$0<\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.

This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. In general, for any $m$, Weyl's spectral estimates imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. (Hat-tip to Marc Palm!) In particular this shows that for $m\geq 2$ and a generic metric we have

$$\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.

This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. In general, for any $m$, Weyl's spectral estimates imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. (Hat-tip to Marc Palm!) In particular this shows that for $m\geq 2$ and a generic metric we have

$$0<\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.

This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. For a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. Weyl's spectral estimates then imply that

$$\lambda_k \sim C_m k^{\frac{2}{m}}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. In particluar this shows that for $m\geq 2$ and a generic metric we have

$$\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.

This is mostly enhancing Nik Weaver's comments. Suppose that $M$ is compact of dimension $m$. If $m\geq 2$, then for a generic metric $g$ on $M$ the eigenvalues $\lambda_k$ of the Laplacian $\Delta_g$ are simple. In general, for any $m$, Weyl's spectral estimates imply that

$$\lambda_k \sim C_m \left(\frac{k}{{\rm vol}_g(M)}\right)^{\frac{2}{m}}\;\;\mbox{as $k\to\infty$}, $$

where $C_m$ is an explicit universal constant that depends only on $m$. (Hat-tip to Marc Palm!) In particular this shows that for $m\geq 2$ and a generic metric we have

$$\lambda_{k+1}-\lambda_k =O(1). $$

Now Nik Weaver's argument shows that there exists $r_0>0$ such for any $\xi\in [0,\infty)\setminus {\rm spec}\;(\Delta)$ we have $\Vert(R(\xi)\Vert\geq r_0$.