Unique continuation for the magnetic Schrödinger equation

Abstract The unique‐continuation property from sets of positive measure is here proven for the many‐body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one‐body or two‐body functions, typical for Hamiltonians in many‐body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique‐continuation property plays an important role in density‐functional theories, which underpins its relevance in quantum chemistry.

The current work arises as a natural ingredient for proving a generalized Hohenberg-Kohn theorem in total (physical) CDFT. It addresses the property that a solution of the magnetic Schrödinger equation cannot vanish on a set of positive measure, a property called unique continuation, see Definition 1. Unique continuation is also a fundamental property for solutions of the magnetic Schrödinger equation in its own right and has been well-studied. [21][22][23][24][25][26] In the context of CDFT the issue was first raised in Reference [9]. As far as DFT and CDFT are concerned, it is useful to have the assumptions guaranteeing the unique-continuation property as particle-number independent as possible (at least avoid increasing integrability constraints with increasing N), which is a difficult task. In the present work we obtain results that are adapted to the many-body Schrödinger equation and that furthermore include vector potentials, building on the results of Kurata [25] and Regbaoui. [26] This means that the specific structure of the potentials is beneficially taken into account. The main results, Theorem 6 and Corollary 8, that include the singular Coulomb potentials of atoms and molecules as a special case (Corollary 9), are formulated in terms of the Kato class K n loc and its generalization K n,δ loc , with n = 3 and n = 6 (Definition 3). Although we cannot answer the question of the existence of a generalized Hohenberg-Kohn theorem for the total current in CDFT, we exemplify the use of the unique-continuation property in a limited special case (Corollary 11).

| UNIQUE-CONTINUATION PROPERTY AND THE HOHENBERG-KOHN THEOREM
In the most simple setting of only one particle and without vector potential, it is known that the (unique) ground state ψ in H 1 (R 3 ) can be chosen to be strictly positive, see Theorem 11.8 in Lieb-Loss. [27] This means that we can set ρ 1/2 = ψ > 0 and the following relation to the scalar potential v must hold (from the Schrödinger equation, here written as [ where Δ denotes the Laplacian and e is the ground-state energy. Conversely, given a particle density ρ we can ask if a potential v exists such that the given ρ is the ground-state density of that potential. For the one-particle case, this problem has been studied by Englisch and Englisch [2] and was answered in the negative even for well-behaved densities (N-representable densities). Corollary 3 in Reference [28] (including the additional constraint Δρ 1/2 ≤ Cρ 1/2 and ρ −1 ∈ L 1 loc besides N-representability) provides sufficient conditions for one-particle v-representability, that is, v can be computed from ρ as given in Equation (1) and ρ is the ground-state density of that v.
Returning to the general N-electron case without magnetic field, we first recall the Hohenberg-Kohn theorem: Given two systems, if ρ 1 = ρ 2 then v 1 = v 2 + constant, where ρ k , k = 1,2, is the ground-state particle density of the corresponding system defined by the potential v k . The proof of this result relies on the fact that if ψ is a ground state of both systems, then If ψ does not vanish on a set of positive (Lebesgue) measure we have v 1 = v 2 + constant almost everywhere. The proof can then be completed by means of the variational principle, using the Hohenberg-Kohn argument by reductio ad absurdum. [1] (Note that a strict inequality in the variational principle is not needed, see, eg, Reference [29] and that the results also hold for systems with degeneracy. [2] ) In this article we address the more general case of N interacting, nonrelativistic (spinless) particles subjected to both a scalar and a vector potential. The fundamental question then is, whether any eigenfunction of the corresponding Hamiltonian can be zero on a set of positive measure without being identically zero. This is a problem of unique continuation.

Definition 1
We say that the Schrödinger equation H N ψ = eψ has the unique-continuation property (UCP) from sets of positive (Lebesgue) measure if a solution that satisfies ψ = 0 on a set of positive measure is identically zero. Furthermore, the Schrödinger equation is said to have the strong UCP if whenever ψ vanishes to infinite order at some point x 0 , that is, for all m > 0 then ψ is identically zero. Additionally, if ψ = 0 on a non-empty open set implies that ψ is identically zero, then the Schrödinger equation has the weak UCP.

Remark 1
The strong UCP implies the weak UCP. The UCP from sets of positive measure allows us to conclude ψ 6 ¼ 0 almost everywhere for any eigenfunction of H N .
There exists a considerable amount of literature that treats the UCP for differential inequality |Δψ| ≤ |ξ 1 ||rψ| + |ξ 2 ||ψ|. [21][22][23][24][25][26] In particular if ξ 1 ∈ L n loc R n ð Þ and ξ 2 ∈ L n=2 loc R n ð Þ, the corresponding differential equation Δψ = ξ 1 Árψ + ξ 2 ψ has the UCP from sets of positive measure. [26] Note that such L p loc constraints become more restrictive with increasing particle number, since the dimension of the configuration space n enters in the conditions. Directly applied to H N ψ = eψ this means that if a solution ψ in the Sobolev space H vanishes on a set of positive measure, and each component of A belongs to L 3N loc R 3N À Á , then ψ is identically zero.
Such results are used by Lammert, [30] particularly in his Theorem 5.1, to give a mathematically precise proof of the Hohenberg-Kohn theorem [1] in DFT including the UCP as remarked by Lieb. [31] Yet he does not consider magnetic fields and the constraints are very susceptible to the particle number. A recent effort by Garrigue [29] removed the dependence on particle numbers for the constraints on the scalar potential by exploiting their specific shape in the context of many-body (molecular) Hamiltonians. Reference [29] also contains a rigorous proof of the Hohenberg-Kohn theorem including all the mathematical details for potentials v ∈ L p loc R 3 À Á , p > 2.

| PREREQUISITES
Let the Hamiltonian H N be as in Equation (2). The Schrödinger equation is then given by H N ψ = eψ. We write are the coordinates of the jth electron. Here we use (ir j + A(x j )) 2 in T A instead of (−ir j + A(x j )) 2 in order to follow the notation in Kurata. [25] We use a slight variation of atomic units ℏ = 2m e = 1 and q e = −1, such that the Laplace operator appears without a factor 1/2.
The electric potential V is a one-body potential given by V The two-particle interaction U between the electrons is modeled by U(x) = P 1 ≤ j < l ≤ N u(x j , x l ), for some nonnegative function u on R 3 × R 3 . We set W = V + U. Furthermore, A : R 3 ! R 3 denotes the vector potential, from which the magnetic field is obtained by B = r × A. With the notation A x ð Þ = A x j À Á À Á N j = 1 , the Schrödinger equation is rewritten as where W A = W + A j j 2 + i rÁA ð Þ.
A function f ∈ L 2 loc R n ð Þ belongs to the Sobolev space H k loc R n ð Þ if f has weak derivatives up to order k that belong to L 2 loc R n ð Þ. Let the set of infinitely differentiable functions with compact support on R 3N be denoted by C ∞ 0 R 3N À Á . We say that ψ ∈ H 1 loc R 3N À Á is a solution of Equation (3) in the weak sense, which will be our standard notion for solutions from here on, if for all The present work takes off from the following result: Theorem 2 (Theorem 1.2 in Regbaoui [26] ).
Then the Schrödinger equation has the UCP from sets of positive measure, that is, if a solution ψ ∈ H vanishes on a set of positive measure then it is identically zero.
Remark 2 See also Theorem 1.1 in Regbaoui [26] for the strong UCP and Wolff [23] for the weak UCP.
If one employs Theorem 2 with N = 1, since there is no two-particle interaction it suffices to assume that v,|A| 2 and rÁ A are elements of L 3=2 loc R 3 À Á to obtain the UCP from sets of positive measure. With increasing particle number, however, the assumptions on the potentials v, u, and A are such that they rule out most types of singularities. On the other hand, the particle-number dependence that enters in H 2N= N + 2 ð Þ loc fulfills the inequality 2 N/(N + 2) ≤ 2 for all N. Following Kurata [25] the L p loc constraints, with p proportional to N, can be avoided.
Definition 3 A function f ∈ L 1 loc R n ð Þ belongs to the Kato class K n loc , n 6 ¼ 2, if for every R > 0, lim ). Let y ∈ R 3N be fixed and for x = (x 1 , …, x N ) ∈ R 3N (cf. the notation in Kurata [25] ) With the notation above, we formulate.

Remark 4
The main condition in Assumption 1 is with respect to the Kato class K n loc , n = 3 N being the dimensionality of the underlying configuration space. This condition is optimal in the sense that the class cannot be enlarged to smaller orders than n = 3 N, or the UCP will be lost. This follows from the inclusion L p loc & K n loc for all p > n/2 and a sharp counterexample provided in Reference [32] for a potential in L p , p < n/2. So if the order of the Kato class would be any m < n then it also includes L p loc with m/2 < p < n/2 and that is ruled out by the given counterexample.
The following is obtained by adapting Corollary 1.1 in Kurata [25] (denoted Lemma 5 below). For the sake of simplicity, and since it is enough for our purposes here, we make the restrictions to real-valued V and U. In the sequel we use the notation |F| for the Frobenius norm (also called the Hilbert-Schmidt norm) of a matrix (F j,l ) j,l .
is a solution of (3) and vanishes to infinite order at x 0 ∈ R 3N , then ψ is identically zero. Thus the Schrödinger equation has the strong UCP.
Lemma 5 (Corollary 1.1 in Kurata [25] ). Let n ≥ 3, x 0 ∈ R n be fixed, and Furthermore assume, for some r 0 > 0, holds. Then if ψ ∈ H 2 loc R n ð Þ satisfies and vanishes to infinite order at x 0 , it follows that ψ is identically zero.
Proof of Theorem 4 We will show that Assumption 1 directly fulfills all the conditions of Lemma 5 that thus becomes applicable. Let n = 3 N and e W = W − e. By Assumption 1, (7) is then fulfilled. The choice Ã = A implies T A = ir + e A 2 and Equation (3) can be written as Equation (9).
Since a = |Ã| 2 , it holds that j e Aj 2 ∈ K 3N loc . Moreover, the matrix F satisfies since F contains N repeated blocks of sub matrices of the form This establishes Equation (6).
If the strong UCP can be obtained under other assumptions than Assumption 1, the following corollary can be used to obtain the UCP from sets of positive measure.
Corollary 7 Suppose the strong UCP for the Schrödinger equation (not necessarily by means of Assumption 1), then the constraint gives the UCP from sets of positive measure.
Due to the particular form of the potentials, we can write Because Q y is defined as the negative part of the function 2W + (x − y)ÁrW, we have with the choice for some functions q 1;x0 and q 2;x0 . (See below the proof of Corollary 9, where this decomposition is done for the choice of W corresponding to the molecular case.) Furthermore, We can now formulate our main result that includes H N modeling atoms and molecules, and where the exponents in the integrability constraints are independent of the particle number N.
In particular, the magnetic Schrödinger equation has the UCP from sets of positive measure for H N modeling atoms and molecules in magnetic fields if just Equation (12) is fulfilled.
where x nuc;j ∈ R 3 and Z j > 0 are the positions and charges of the M nuc nuclei, respectively, the UCP from sets of positive measure holds for the Schrödinger equation.

| APPLICATION TO CDFT
In the presence of a magnetic field, no equivalence of a (general) Hohenberg-Kohn result exists at present. [9,11] However, we shall now address how the UCP from sets of positive measure for the magnetic Schrödinger equation plays an important role in the argument for restricted Hohenberg-Kohn theorems in CDFT and the nonuniversal variant magnetic-field density-functional theory (BDFT) of Grayce and Harris. [33] Given a wave function ψ, define the particle density and the paramagnetic current density according to For a vector potential A we may compute the total current density by the sum j = j p ψ + ρ ψ A. Now, fix the particle number N as well as the twoparticle interaction u (eg, u(x 1 , x 2 ) = |x 1 − x 2 | −1 ) and write H N = H(v,A). If ψ is a ground state for some v and A, that is, H(v,A)ψ = eψ, where e is the ground-state energy, then ρ ψ , j p ψ , and j = j p ψ + ρ ψ A are called ground-state densities of H(v,A). Whether the ground-state particle density ρ and the total current density j determine v and A (up to a gauge transformation) is still an open question in the general case. [9,11] (For the ground-state density pair (ρ,j p ) it is well-known that this density pair does not determine the potentials v and A. [8] ) Now, assume that two systems with Hamiltonians H(v 1 ,A 1 ) and H(v 2 ,A 2 ) have the same ground-state particle density (i.e., ρ 1 = ρ 2 = ρ) and r × A 1 = r × A 2 = B. Suppose that v k , A k for k = 1,2, and B fulfill Assumption 1 and the requirements given in Theorem 6. Since there exists a function f such that A 1 = A 2 − rf, the variational principle yields where ψ 2 is the ground state of H(v 2 , A 2 − rf ). Switching the indices, we find that Consequently, ψ 2 is a ground state of both H(v 1 , A 1 ) and H(v 2 , A 2 − rf ), which leads to However, Theorem 6 allows us to conclude ψ 2 6 ¼ 0 and it follows v 1 = v 2 + constant.
Theorem 10 Assume r × A 1 = r × A 2 = B and that v k , A k for k = 1,2, and B fulfill Assumption 1 and take the requirements of Theorem 6 for H(v 1 ,A 1 ) and H(v 2 ,A 2 ) to hold. If the ground-state particle densities satisfy ρ 1 = ρ 2 , then v 1 = v 2 + C almost everywhere for some constant C.
Remark 7 Theorem 10 is the Hohenberg-Kohn theorem for BDFT, first established by Grayce and Harris [33] but missing the UCP argument (see also Reference [9]).

Theorem 10 can be used to obtain
Corollary 11 Assume Assumption 1 and the requirements of Theorem 6 for H(v 1 ,A 1 ) and H(v 2 ,A 2 ) and that the ground-state densities fulfill ρ = ρ 1 = ρ 2 , j = j 1 = j 2 . For systems with j p = 0, it follows B 1 = B 2 (even A 1 = A 2 holds) and v 1 = v 2 + C for some constant C.

| PROOFS OF THE MAIN RESULTS
Proof of Theorem 6 In the sequel let D = 3 N. By Assumption 1, the strong UCP holds for Equation (3) by Theorem 4. Next, we follow the proof of Theorem 1.2 given after Lemma 3.3 in Regbaoui [26] and Lemma A.2 in Lammert. [30] (Lemma A.2 corresponds to setting A = 0 here, and moreover we exploit u ≥ 0 instead of the assumption u(x 1 ,x 2 ) = u 0 (x 1 − x 2 ).) We start by showing the following inverse Poincaré inequality for solutions of the Schrödinger equation: For an arbitrary point Here C is a positive constant that depends on r 0 > 0, v, and A (but is independent of u ≥ 0).
In the Schrödinger equation (4), we choose φ = h 2 ψ and move all terms except one to the right hand side so that We now bound each of the terms of the right hand side in Equation (14).
It is immediate that the first term is less or equal to 2khrψk 2 kψrhk 2 . Using the inequality 2ab ≤ a 2 /6 + 6b 2 , we obtain an upper bound To continue, let I 1 = − 2i Ð R D AÁ rψ ð Þh 2 ψ dx. The Cauchy-Schwarz inequality together with 2ab ≤ a 2 /6 + 6b 2 yield For the last term of the right hand side in Equation (14), we use the definition of W A and write e− Þhψ j j 2 dx and it follows from Define Θ = V − + j e j + 6 A j j 2 + j rÁA j, from Equations (15), (16), and (17) we obtain an upper bound for the right hand side of Equation (14) given by With the notation Θ 1 = v − + N −1 j e j + 6|A| 2 + jr Á Aj, the inequality Θ x ð Þ ≤ P N j = 1 Θ 1 x j À Á holds. Furthermore, we have where the last equality defines I 2 .
Suppose ψ ∈ H 2 loc vanishes on a set E of positive measure. Almost every point of E is a density point. Let x 0 be such a density point and let B r = B r (x 0 ). Given ε > 0 there is an r 0 = r 0 (ε) so that (cf. (3.11) in Regbaoui [26] ) Lemma 3.3 in Regbaoui [26] (or Lemma 3.4 in Ladyzenskaya-Ural'tzeva [34] ) gives for some constant C. Applying the Cauchy-Schwarz inequality to the right hand side of Equation (21), we obtain for some new constant C. Since |E| ≥ |E \ B r |, Equation (20), and the inverse Poincaré inequality (13) allow us to conclude that Introduce the function f r ð Þ = Ð Br ψ j j 2 dx, fix an integer n and choose ε > 0 so that C 0 ε 2/D /(1 − ε) 2 = 2 −n . Then Equation (22) can be written f(r) ≤ 2 −n f(2r). By iteration f r 0 ð Þ≤ 2 − kn f 2 k r 0 , r 0 ≤ 2 1−k r 0 holds. For fixed r and k chosen such that 2 −k r 0 ≤ r ≤ 2 1 − k r 0 , it follows that where r 0 depends on n. Consequently f vanishes to infinite order, that is, for all m there is an r 0 (m) such that That ψ = 0 follows now by the strong UCP given by Theorem 4. □ Proof of Corollary 7 This is a consequence of the proof of Theorem 6, since Θ 1 , by assumption, is an element of L 3=2 loc R 3 À Á . □

Proof of Corollary 8
We first demonstrate that the conditions of Corollary 8 fulfills Assumption 1. Due to the particular form of the potentials, we make use of the following: Let f 1 ∈ K 3,δ loc and f 2 ∈ K 6,δ loc . Then both P N k = 1 f 1 x k ð Þ and P k 6 ¼ l f 2 (x k , x l ) are elements of K 3N,δ loc . Similar statements for K n can be found in Simon [35] (Example F) and Aizenman-Simon [36] (Theorem 1.4). We prove our claim by direct computations. Define I δ 1 and I δ 2 according to We next demonstrate that where the second index in the given ball-sets B r; 3 (x) & R 3 and B r; 6 (x) & R 6 refers to the respective dimensionality.
To show Equation (23), set q = (y 2 , …, y N ) and note that where we have defined the integral Now, J δ 1 is finite since This establishes Equation (23). The proof of Equation (24) is similar and included for the sake of completeness. Set q = (y 3 ,…,y N ), then In the sequel we let v(x 1 ) = − Z|x 1 − x nuc | −1 , x nuc ∈ R 3 , Z > 0, and u(x 1 , x 2 ) = |x 1 − x 2 | −1 . In this case v − ∈ L 3=2 loc R 3 À Á and with the choice Thus, in this case we can choose q 1,xnuc = v − and q 2,xnuc = 0.
Furthermore, for 0 < δ < 1, we claim that V, U ∈ K D,δ loc . By the first part it suffices to show v ∈ K 3,δ loc and u ∈ K 6,δ loc . For v ∈ K 3,δ loc , we introduce polar coordinates with radius s and polar angle t. Then it holds that dy = 2πs 2 sint dtds and y Á x = −s|x|cost. For f 1 (x) = |x| −1 it follows that We integrate over t, use |s + |x| − |s − |x|||≤2|x|, and the conclusion is obtained for v. In a similar fashion, for u we establish that with The atomic case is now a consequence of Corollary 8. □

| CONCLUSION
In this work we were able to show the unique-continuation property from sets of positive measures for the important case of the many-body magnetic Schrödinger equation for classes of potentials that are independent of the particle number. This is crucial in order to not artificially restrict the permitted potentials in large systems. We further specifically addressed molecular Hamiltonians, thus covering most cases that usually arise in physics.

ACKNOWLEDGMENTS
AL is grateful for the hospitality received at the Max Planck Institute for the Structure and Dynamics of Matter while visiting MP. The authors want to thank Louis Garrigue for useful discussion that led to the inclusion of the molecular case, and Fabian Faulstich for comments and suggestions that improved the manuscript.