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Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle \psi(x) +V(x)\psi(x)= E\psi(x)$, where $E$ is an eigenvalue of the Schroedinger Equation. Out of this equation arises the variation problem as follows: define the functional $\mathcal E(\psi) = T_{\psi} + V_{\psi}$ (the kinetic and potential energies, respectively). Then, $T_{\psi} = \int_{\mathbb R^n} \mid \nabla \psi(x) \mid ^2 dx$ and $V_{\psi} = \int_{\mathbb R^n} V(x) \mid \psi (x) \mid ^2 dx$. The idea here is that we want to find a minimizing function $\psi_0$ of the functional $\mathcal E(\psi)$ under the constraint $\|\psi\|_2 =1$. To first show that such a minimizer exists, we must show that $\mathcal E(\psi)$ is bounded from below.

A little thought shows us that we want the Potential Energy,$V_{\psi}$, (which can be negative), to be dominated by $T_{\psi}$ so that we can make sure that $\mathcal E(\psi)$ is bounded below. To assure this, we appeal to the generalized Sobolev inequality that promise us our desired bounds. Let us appeal to the following Sobolev inequality ( from now on restrict ourselves to case $n \geq 3$: $\| f \|_2 ^2 \geq S_n \|f\|_2 ^2$ where $S_n = n(n-2)/4 \mid \mathcal S^n \mid ^{2/n}$. So, we can say that $ T_\psi \geq S_n {\int_{\mathbb R^n} \mid \psi \mid ^ (2n/(n-2)) dx}^{(n-2)/n}$

Appealing to Hölder's inequality, we have for any potential $V \in L^{n/2} (\mathbb R^n)$ (and as I said above, restricting ourselves to the case $n\geq 3$, $T_{\psi} \geq S_n \langle \psi, V \psi \rangle \| V \|_{n/2} ^{-1}$. We immediately deduce from this fact that whenever $\| V \|_{n/2} \leq S_n$, then $T_\psi \geq V_\psi$. Since $T_{\psi}$ is never negative and dominates $V_\psi$, we found a lower bound for the fucntional: $\mathcal E(\psi) = T_\psi +V_\psi \geq 0$.

Now here is where the problem arises:

Let us extend this to potential $V \in L^{n/2} (\mathbb R^n) + L^{\infty} (\mathbb R^n)$. So suppose $V(x)=v(x) + w(x)$, and $v\in L^{n/2} (\mathbb R^n)$ and $w \in L^{\infty} (\mathbb R^n)$. To find the lower bound for the functional $\mathcal E(\psi)$ in this case, we must first how that there is some constant $\lambda$ such that $h(x)=\min (v(x)-\lambda,0)\leq 0$ satisfies $\| h\|_{n/2} \leq \frac 1 2 S_n$. Is this supposed to be simple to do? I have very little experience with Sobolev inequalities; does a proof of the above stated fact depend on them explicitly? The cool thing about proving this fact is that we can again find another lower bound for $\mathcal E(\psi) \leq \lambda - \|w\|_{\infty}$. Even better, we can confirm out physical intutions and bound the kinetic energy in terms of the total energy.

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I know that my question is a bit long, but I wanted to give some motivation for where this question came up for me. There may be people on this site who may not know much about physics, but who may be familiar with the analysis necessary to help me. – r.g. Dec 4 '11 at 6:04

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