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Trying to Understand the lower bound on the Schrodinger Operator of the Hydrogen atom. Using the kato-rellich theorem. My education has been in physics and i am slowly adding to my mathematics toolset.I am yet to study fuctional analysis exrensively the following problem is in my way to understanding the following theorem.

Could someone explain this to me.

For refrence This is a step in theorem 9.38 Quantum Theory for Mathematicians. Authors: Brian C. Hall. Series Title: Graduate Texts in Mathematics. DOI: https://doi.org/10.1007/978-1-4614-7116-5.

Given that $$ \mathbf{DOM}(\Delta) = \{ \psi \in \mathbf{L^2}(\mathbb{R^n})\space | \space |\mathbf{k^2}\hat{\psi}(k) \in \mathbf{L^2}(\mathbb{R^n}) \} $$

Define $ \Delta $ as

$$\Delta \psi = -\pmb{F^{-1}}(\mathbf{k^2}\hat{\psi}(\mathbf(k) ) $$

Show that for any $\epsilon \gt 0 $ there exists a constant $C_{\epsilon} $ such that $$ \lvert \psi \rvert \leq {C_\epsilon} \lVert \psi \rVert + \epsilon\lVert \Delta\psi \rVert $$

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  • $\begingroup$ I have been reading to get insight into my problem, now i understand that this requires understanding of Lp norms and then can be proved using plancherels theorem and hödlers inequality. $\endgroup$
    – Gedankenhooman
    Commented Jan 27 at 22:51
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    $\begingroup$ I think your question is answered in section 6.2 of Intermediate Spectral Theory and Quantum Dynamics by C. R. de Oliveira. See also Lemma 10.46 in Spectral Theory and Quantum Mechanics by V. Moretti. $\endgroup$
    – Tobias Fünke
    Commented Jan 28 at 12:48

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