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Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le -C<0$ bounded away by some constant. Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

[click me.][1]

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. we get that $$p(x,\xi,z)=g(x,\xi,z)(h(x,\xi)-z)$$

for some local functions $h,g.$

But I am not really sure what kind of iteration may be meant here? Which kind of symbolic iteration allows us to go from this equation to the identity above. Apparently $P$ is the symbol associated with $h$ but how do we construct $A$?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know. [1]: https://math.berkeley.edu/~zworski/qmr.pdf

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le -C<0$ bounded away by some constant. Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

click me.

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. we get that $$p(x,\xi,z)=g(x,\xi,z)(h(x,\xi)-z)$$

for some local functions $h,g.$

But I am not really sure what kind of iteration may be meant here? Which kind of symbolic iteration allows us to go from this equation to the identity above. Apparently $P$ is the symbol associated with $h$ but how do we construct $A$?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know.

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le -C<0$ bounded away by some constant. Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

[click me.][1]

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. there is a function $g$ such that a level-set of the symbol $\{p=\lambda\}$ is parametrized as $p(x,\xi,g(x,\xi)).$ But I am not really sure what kind of iteration may be meant here?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know. [1]: https://math.berkeley.edu/~zworski/qmr.pdf

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le -C<0$ bounded away by some constant. Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

[click me.][1]

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. we get that $$p(x,\xi,z)=g(x,\xi,z)(h(x,\xi)-z)$$

for some local functions $h,g.$

But I am not really sure what kind of iteration may be meant here? Which kind of symbolic iteration allows us to go from this equation to the identity above. Apparently $P$ is the symbol associated with $h$ but how do we construct $A$?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know. [1]: https://math.berkeley.edu/~zworski/qmr.pdf

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le C<0.$ Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

[click me.][1]

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. there is a function $g$ such that a level-set of the symbol $\{p=\lambda\}$ is parametrized as $p(x,\xi,g(x,\xi)).$ But I am not really sure what kind of iteration may be meant here?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know. [1]: https://math.berkeley.edu/~zworski/qmr.pdf

Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this parameter is not the semiclassical parameter which is $h$.

Moreover, $P$ is supposed to be self-ajoint, of principal type, elliptic and satisfy $\partial_z p(z)\le -C<0$ bounded away by some constant. Here $p$ is the symbol of $P.$

Then, in the link it is claimed that we can write $P(z)= A^*(z)(P-z)A(z)$ where $P,A \in \Psi_h^0(\mathbb{R}):$

[click me.][1]

In other words, what is claimed here is that by an application of the implicit function theorem and some iteration we can transform any PDO that may depend non-linearly on $z$ into a form that depends on $z$ in the trivial way $P-z.$

What I understand so far is that the condition $\partial_zp(x,\xi,z)<0$ allows us to apply the implicit function theorem, i.e. there is a function $g$ such that a level-set of the symbol $\{p=\lambda\}$ is parametrized as $p(x,\xi,g(x,\xi)).$ But I am not really sure what kind of iteration may be meant here?

In other words, does anybody understand how those operators $A$ and $P$ are constructed then?

If you have any questions, please let me know. [1]: https://math.berkeley.edu/~zworski/qmr.pdf