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I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" imply "bounded"?

For the statement, see for example:

arXiv:1803.04993 below eq.(2.7)

arXiv:0710.5373 below eq.(2.62)

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" imply "bounded"?

For the statement, see for example:

https://arxiv.org/abs/1803.04993 below eq.(2.7)

https://arxiv.org/abs/0710.5373 below eq.(2.62)

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" imply "bounded"?

For the statement, see for example: arXiv:1803.04993 below eq.(2.7)

arXiv:0710.5373 below eq.(2.62)

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" imply "bounded"?

For the statement, see for example:

arXiv:1803.04993 below eq.(2.7)

arXiv:0710.5373 below eq.(2.62)

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" implies "bounded"?

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane specifically?

Does this have something to do with infinity or does "holomorphic" imply "bounded"?

For the statement, see for example: arXiv:1803.04993 below eq.(2.7)

arXiv:0710.5373 below eq.(2.62)