We prove a priori bounds for all derivatives of non-relativistic Coulombic eigenfunctions ψ, involving negative powers of the distance to the singularities of the many-body potential. We use these to derive bounds for all derivatives of the corresponding one-electron densities ρ, involving negative powers of the distance from the nuclei. The results are both natural and optimal, as seen from the ground state of Hydrogen.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC-2111-390814868
Funding source: European Research Council
Award Identifier / Grant number: 202859
Funding source: Det Frie Forskningsråd
Award Identifier / Grant number: DFF-4181-00221
Funding statement: This article may be reproduced in its entirety for non-commercial purposes. Søren Fournais was partially supported by a Sapere Aude Grant from the Independent Research Fund Denmark, Grant number DFF-4181-00221, and by the European Research Council, ERC grant agreement 202859. Thomas Østergaard Sørensen was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2111-390814868).
A Some new a priori estimates
We start by recalling that estimate.
Theorem A.1 ([28, Theorem 1.2]).
Let ψ be as in (1.2). For all , there exists a constant such that
for all .
The proof of (A.1) is based on an “Ansatz” (see also (A.8) below) for the solution of the eigenvalue equation, and then on using elliptic regularity on the resulting equation. The objective of this Appendix is the following strengthening of Theorem A.1:
Let H be the operator in (1.1). For all and there exists (depending also on ) such that if , , then
for all .
Define, for ,
Note that there exists a constant such that (for Σ, see (1.3))
Next, let (for V, see (1.1))
Since for all , there exists a constant such that
then, (using that ), satisfies the equation
Note that since , we have that . This follows from (A.4) and Hardy’s inequality (note that any second order derivative of behaves like and ).
it follows from Theorem C.4 that for all .
with . (Of course, the constant C changes every time.)
Hence, we reach satisfying in maximally steps (that is, M is smaller equal this number). As above, the radius of the smaller ball decreases each time (above, from to ). However, splitting the original difference in equally large parts (we use Theorem C.3 times), we get: For all there exists a constant such that
Now use Theorem C.1 (ii) (Morrey’s Theorem): With (so ), to get, for some ,
Using (A.13), and that , this implies that, for all ,
for some .
Hence (since ), (A.2) follows, but with instead of ψ. It remains to recall that
As a consequence of Proposition A.2, we get the following, which is of independent interest:
in the sense that, for all , the directional derivative (exists and) satisfies
Furthermore, for all and there exists a constant such that
Note that, for all , ,
Now, for all ,
To prove (A.17), notice that
and use (A.15) with .
To prove (A.18), we differentiate and estimate, to get that
At this point we can use (A.2) and finish the estimate as above.
We argue in a similar fashion as above, with . Since for all , , we get from Fubini’s Theorem and (A.2) that
Here we also used that . Now, for all ,
B A partition of unity
In this appendix we gather various facts about a particular partition of unity (on ), needed when studying the electron density ρ; see Section 6.
We denote by the set of all smooth functions on Ω which are bounded together with all their derivatives.
Let , , , both monotone, with
The partition of unity depends on an index , where , with the sets to be described below. Here
and the corresponding function in the partition of unity is (with ),
For each , the set consists of all elements of the form with disjoint, and (possibly or ). The corresponding function is (with )
where the sum is over a subset of X.
To ease notation, let, for ,
Note that, by (B.1), for all j and ,
Using (B.2), we have (again, with )
The term in (B.7) with equals
The term in (B.7) with equals
and multiply out, to get
As before, the term with (that is, ) equals
and the term with (that is, ) equals
For the rest of the terms in (B.9), we have , and so , that is, . For each of these terms (with , disjoint), insert a factor of
and proceed as above, using (B.6) with , to write as a sum (over with , ) of terms of the form
Again, the terms with or have (see (B.4)) the correct form (namely, with , , , , and, respectively, with , , ). Furthermore, for all other terms in (B.10), we have , hence, , that is, . Continuing like this, we get a sum of terms of the form in (B.4), with the size of diminishing at each step, until (which occurs for ). Then the above two possibilities – or – are the only two, and we are done. ∎
The localization functions above are constructed in order to have the following lemma, bounding certain terms in the Coulomb-potential by , on the support of .
To prove (B.11), note that, since , for all the stated indices j we have
for some , . Hence, by (B.1),
which proves (B.11).
To prove (B.12), note that, for , we have
Hence, by (B.1), , and so for these j.
On the other hand, for ,
Finally, to prove (B.13), note that for the stated indices j, we have , and for the stated indices k, we have, for some ,
which proves (B.13). ∎
This last argument is the reason why we need in in the -factor, and at most in in the -factors, in (B.4).
The next lemma uses the previous one, to control derivatives with respect to of (a slightly changed version of) the localization functions .
Then, for all , there exists a constant such that
Furthermore, if , then there exist and a constant such that, for all and all ,
First note that, by Leibniz’ rule, to prove (B.15) it suffices to prove that for all , there exists a constant such that
for f any of the functions
Secondly, note that in each case, for any ,
with or 2, and either or . On () we have, in all cases (see (B.1)),
Hence, if , then for any , on ,
On the other hand, if , then for any , again on ,
To prove that (B.16) or (B.17) hold when , notice that in this case at least one of the functions in the product in (B.14) (that is, in (B.19)) gets differentiated (that is, ). For this one, do as above, but use additionally (B.21) to get, for all ,
C Needed a priori estimates
In this final appendix section we collect needed results from the literature. We start by Sobolev embedding.
Let be open and bounded, and let .
Assume Ω satisfies an interior cone condition. Then, for any with , we have the continuous embedding
Moreover, there exists a constant such that
Assume Ω is locally Lipschitz. Then for , we have the continuous embedding
Moreover, there exists a constant such that
Next, we list some results on elliptic regularity.
The following theorem is adapted from [22, Theorem 8.32] by choosing and for .
Theorem C.2 ([22, Theorem 8.32]).
Let , and let be a weak solution of
in a bounded domain , with , with
Then, for any subdomain , we have
for where .
The following theorem is adapted from [22, Theorem 9.11] by choosing for .
Theorem C.3 ([22, Theorem 9.11]).
Let Ω be an open set in and suppose that , , is a strong solution of the equation
in Ω with , , with
Then for any subdomain ,
where C depends on , and Ω.
Theorem C.4 ([23, Lemma 188.8.131.52]).
Let Ω be an open and bounded set in , let , and let be a strong solution of the equation
in Ω with . Then .
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