Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$

where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The domain of the operator is $$ D(A) = \{u\in L^2(\mathbb{R}^n): A(x,D_x)u \in L^2(\mathbb{R}^n) \}. $$ Note that the operator is self-adjoint and positive. Hence, one can associate Sobolev spaces to the operator $A$ as follows $$ H^\alpha(A) := \{u\in L^2(\mathbb{R}^n): A^{\alpha/2}u \in L^2(\mathbb{R}^n) \} $$
for $\alpha \in \mathbb{R}$. These spaces are in fact equivalent to the following spaces $$ \{u\in L^2(\mathbb{R}^n): \langle x \rangle^\alpha \langle D_x \rangle^\alpha u \in L^2(\mathbb{R}^n) \}, \quad \langle D_x \rangle :=(1-\Delta_x)^{1/2}. $$

Now, the spectral theorem for pseudodifferential operators on $\mathbb{R}^n$ (see Theorem 4.2.9 in Nicola, F. and Rodino, L. (2011) Global Pseudo-differential Calculus on Euclidean Spaces. Birkhauser Basel.) suggests the existence of an orthonormal basis $\{e_i\}$ of $L^2(\mathbb{R}^n)$ and a nondecreasing sequence $\{\lambda_i\}$ of nonnegative real numbers diverging to $+\infty$ such that $Ae_i=\lambda_i^2e_i.$ This allows one to identify elements $v\in L^2(\mathbb{R}^n)$ with sequences $\{v_i\}\in \ell^2$, where $v_i=\langle v,e_i\rangle_{L^2}$.

Using the Spectral theorem one can define the following spaces: $$ \tilde H^{\alpha}(A) := \Big\{v\in L^2(\mathbb{R}^n): \sum_{i=0}^{+\infty} \lambda_i^{2\alpha}v_i^2 < +\infty \Big\}. $$

Are the spaces $H^{\alpha}(A)$ and $\tilde H^{\alpha}(A)$ equivalent? If not for all real $\alpha$, can we prove the equivalence when $\alpha$ is an even integer?

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$

where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The domain of the operator is $$ D(A) = \{u\in L^2(\mathbb{R}^n): A(x,D_x)u \in L^2(\mathbb{R}^n) \}. $$ Note that the operator is self-adjoint and positive. Hence, one can associate Sobolev spaces to the operator $A$ as follows $$ H^\alpha(A) := \{u\in L^2(\mathbb{R}^n): A^{\alpha/2}u \in L^2(\mathbb{R}^n) \} $$
for $\alpha \in \mathbb{R}$. These spaces are in fact equivalent to the following spaces $$ \{u\in L^2(\mathbb{R}^n): \langle x \rangle^\alpha \langle D_x \rangle^\alpha u \in L^2(\mathbb{R}^n) \}, \quad \langle D_x \rangle :=(1-\Delta_x)^{1/2}. $$

Now, the spectral theorem for pseudodifferential operators on $\mathbb{R}^n$ (see Theorem 4.2.9 in Nicola, F. and Rodino, L. (2011) Global pseudo-differential calculus on Euclidean spaces. Birkhaüser Basel.) suggests the existence of an orthonormal basis $\{e_i\}$ of $L^2(\mathbb{R}^n)$ and a nondecreasing sequence $\{\lambda_i\}$ of nonnegative real numbers diverging to $+\infty$ such that $Ae_i=\lambda_i^2e_i.$ This allows one to identify elements $v\in L^2(\mathbb{R}^n)$ with sequences $\{v_i\}\in \ell^2$, where $v_i=\langle v,e_i\rangle_{L^2}$.

Using the Spectral theorem one can define the following spaces: $$ \tilde H^{\alpha}(A) := \Big\{v\in L^2(\mathbb{R}^n): \sum_{i=0}^{+\infty} \lambda_i^{2\alpha}v_i^2 < +\infty \Big\}. $$

Are the spaces $H^{\alpha}(A)$ and $\tilde H^{\alpha}(A)$ equivalent? If not for all real $\alpha$, can we prove the equivalence when $\alpha$ is an even integer?

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$

where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The domain of the operator is $$ D(A) = \{u\in L^2(\mathbb{R}^n): A(x,D_x)u \in L^2(\mathbb{R}^n) \}. $$ Note that the operator is self-adjoint and positive. Hence, one can associate Sobolev spaces to the operator $A$ as follows $$ H^\alpha(A) := \{u\in L^2(\mathbb{R}^n): A^{\alpha/2}u \in L^2(\mathbb{R}^n) \} $$
for $\alpha \in \mathbb{R}$. These spaces are in fact equivalent to the following spaces $$ \{u\in L^2(\mathbb{R}^n): \langle x \rangle^\alpha \langle D_x \rangle^\alpha u \in L^2(\mathbb{R}^n) \}, \quad \langle D_x \rangle :=(1-\Delta_x)^{1/2}. $$

Now, the spectral theorem for pseudodifferential operators on $\mathbb{R}^n$ (see Theorem 4.2.9 in Nicola, F. and Rodino, L. (2011) Global Pseudo-differential Calculus on Euclidean Spaces. Birkhauser Basel.) suggests the existence of an orthonormal basis $\{e_i\}$ of $L^2(\mathbb{R}^n)$ and a nondecreasing sequence $\{\lambda_i\}$ of nonnegative real numbers diverging to $+\infty$ such that $Ae_i=\lambda_i^2e_i.$ This allows one to identify elements $v\in L^2(\mathbb{R}^n)$ with sequences $\{v_i\}\in \ell^2$, where $v_i=\langle v,e_i\rangle_{L^2}$.

Using the Spectral theorem one can define the following spaces: $$ \tilde H^{\alpha}(A) := \Big\{v\in L^2(\mathbb{R}^n): \sum_{i=0}^{+\infty} \lambda_i^{2\alpha}v_i^2 < +\infty \Big\}. $$

Are the spaces $H^{\alpha}(A)$ and $\tilde H^{\alpha}(A)$ equivalent?

Consider a generalised harmonic oscillator of the form $$ A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n, $$

where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The domain of the operator is $$ D(A) = \{u\in L^2(\mathbb{R}^n): A(x,D_x)u \in L^2(\mathbb{R}^n) \}. $$ Note that the operator is self-adjoint and positive. Hence, one can associate Sobolev spaces to the operator $A$ as follows $$ H^\alpha(A) := \{u\in L^2(\mathbb{R}^n): A^{\alpha/2}u \in L^2(\mathbb{R}^n) \} $$
for $\alpha \in \mathbb{R}$. These spaces are in fact equivalent to the following spaces $$ \{u\in L^2(\mathbb{R}^n): \langle x \rangle^\alpha \langle D_x \rangle^\alpha u \in L^2(\mathbb{R}^n) \}, \quad \langle D_x \rangle :=(1-\Delta_x)^{1/2}. $$

Now, the spectral theorem for pseudodifferential operators on $\mathbb{R}^n$ (see Theorem 4.2.9 in Nicola, F. and Rodino, L. (2011) Global Pseudo-differential Calculus on Euclidean Spaces. Birkhauser Basel.) suggests the existence of an orthonormal basis $\{e_i\}$ of $L^2(\mathbb{R}^n)$ and a nondecreasing sequence $\{\lambda_i\}$ of nonnegative real numbers diverging to $+\infty$ such that $Ae_i=\lambda_i^2e_i.$ This allows one to identify elements $v\in L^2(\mathbb{R}^n)$ with sequences $\{v_i\}\in \ell^2$, where $v_i=\langle v,e_i\rangle_{L^2}$.

Using the Spectral theorem one can define the following spaces: $$ \tilde H^{\alpha}(A) := \Big\{v\in L^2(\mathbb{R}^n): \sum_{i=0}^{+\infty} \lambda_i^{2\alpha}v_i^2 < +\infty \Big\}. $$

Are the spaces $H^{\alpha}(A)$ and $\tilde H^{\alpha}(A)$ equivalent? If not for all real $\alpha$, can we prove the equivalence when $\alpha$ is an even integer?