AndreLaestadius |MichaelBenedicks |MarkusPenz UniquecontinuationforthemagneticSchrödingerequation

F U L L P A P E R

Unique continuation for the magnetic Schrödinger equation

Andre Laestadius

| Michael Benedicks

| Markus Penz

1Department of Chemistry, Hylleraas Centre for Quantum Molecular Sciences, University of Oslo, Oslo, Norway

2Department of Mathematics, Uppsala University, Uppsala, Sweden

3Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany

Correspondence

Andre Laestadius, Department of Chemistry, Hylleraas Centre for Quantum Molecular Sciences, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.

Email: [email protected]

Funding information

Austrian Science Fund, Grant/Award Number:

J 4107-N27; H2020 European Research Council, Grant/Award Number: 639508;

Norges Forskningsråd, Grant/Award Number:

262695; Vetenskapsrådet, Grant/Award Number: 2016-05482

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.

K E Y W O R D S

Hohenberg-Kohn theorem, Kato class, magnetic Schrödinger equation, molecular Hamiltonian, unique-continuation property

1 | I N T R O D U C T I O N

Within the Schrödinger model for quantum systems of (interacting) electrons, in order to be able to describe interesting phenomena like the Zee- man effect, the quantum Hall effect, or the Hofstadter butterfly one has to include the effects of both an electric and a magnetic field. Hohenberg and Kohn showed for systems without magnetic fields that the one-body ground-state particle density determines the electric (scalar) potential up to a constant.^[1]Strictly speaking, the particle density determines at most one potential (modulo an additive constant) since some densities are not associated with any potential.^[2]The above correspondence between densities and potentials constitutes the theoretical foundation on which density functional theory (DFT)—a ubiquitous tool in quantum chemistry and materials science^[3,4]—is built.

In the presence of magnetic fields though, the approach of Hohenberg and Kohn to set up a universal density functional requires more than just the particle density due to the fact that an additional vector potential enters the system's Hamiltonian. Both the paramagnetic current density and the total (physical) current density have been suggested as basic variables alongside the particle density^[5–7]and the resulting framework is called current density functional theory (CDFT). For the theory that uses the paramagnetic current density, counterexamples to a Hohenberg- Kohn theorem are known,^[8,9]although a weaker version still holds. Note that even for degenerate systems the weaker version is enough to define a universal paramagnetic current density functional.^[10]Diener has presented an argument^[7]for establishing a full Hohenberg-Kohn theorem using the total current density. However, as first noted in Reference [11], the argument is at best incomplete.

For more detailed accounts on the existence of generalized Hohenberg-Kohn theorems within CDFT see References [9, 11, 12], and for related and positive results within the Maxwell-Schrödinger theory and quantum-electrodynamical DFT see References [13–15]. An interesting and recent develop- ment is also given in Reference [16] where the existence of generalized Hohenberg-Kohn theorems is further explored. A different route, where a Hohenberg-Kohn result comes for free by virtue of the convex-analytic properties of aregularizedenergy functional was taken in References [17, 18]. It was specifically implemented for CDFT in Reference [19] and can even be used to prove convergence of the associated Kohn-Sham iteration Scheme.^[20]

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2020 The Authors.International Journal of Quantum Chemistrypublished by Wiley Periodicals, Inc.

Int J Quantum Chem.2020;120:e26149. http://q-chem.org 1 of 12

https://doi.org/10.1002/qua.26149

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 con- tinuation, 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.^[21–26]In the context of CDFT the issue was first raised in Reference [9]. As far as DFT and CDFT are con- cerned, 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 increasingN), 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 Cor- ollary 8, that include the singular Coulomb potentials of atoms and molecules as a special case (Corollary 9), are formulated in terms of the Kato classKⁿ_locand its generalizationK^n,δ_loc, withn= 3 andn= 6 (Definition 3). Although we cannot answer the question of the existence of a gener- alized 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).

2 | U N I Q U E - C O N T I N U A T I O N P R O P E R T Y A N D T H E H O H E N B E R G - K O H N T H E O R E M

In the most simple setting of only one particle and without vector potential, it is known that the (unique) ground stateψinH¹(R³) can be chosen to be strictly positive, see Theorem 11.8 in Lieb-Loss.^[27]This means that we can setρ^1/2=ψ> 0 and the following relation to the scalar potential vmust hold (from the Schrödinger equation, here written as [−Δ+v]ψ=eψ)

v xð Þ=e+Δρð Þx¹⁼²

ρð Þx¹⁼² , x∈R³, ð1Þ

whereΔdenotes the Laplacian andeis the ground-state energy. Conversely, given a particle densityρwe can ask if a potentialvexists such that the givenρis 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 constraintΔρ^1/2≤Cρ^1/2andρ⁻¹∈L¹_locbesidesN-representability) provides sufficient conditions for one-particlev-representability, that is,vcan be computed fromρas given in Equation (1) andρis the ground-state density of thatv.

Returning to the generalN-electron case without magnetic field, we first recall the Hohenberg-Kohn theorem: Given two systems, ifρ1=ρ2

thenv1=v2+ constant, whereρ^k,k= 1,2, is the ground-state particle density of the corresponding system defined by the potentialvk. The proof of this result relies on the fact that ifψis a ground state of both systems, thenPN

k= 1ðv1ð Þxk −v2ð Þxk Þψ= constant×ψ. Ifψdoes not vanish on a set of positive (Lebesgue) measure we havev1=v2+ constant almost everywhere. The proof can then be completed by means of the variational prin- ciple, 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 ofNinteracting, nonrelativistic (spinless) particles subjected to both a scalar and a vector potential. The fundamental question then is, whether any eigenfunction of the corresponding Hamiltonian

HN=X^N

j= 1

ir_j+A x_j 2

+v x_j +X

l<j

u x _j,x_l

" #

ð2Þ

can be zero on a set of positive measure without being identically zero. This is a problem ofunique continuation.

Definition 1 We say that the Schrödinger equation HNψ= eψhas the unique-continuation property (UCP) from sets of positive (Lebesgue) measure if a solution that satisfiesψ= 0 on a set of positive measure is identically zero. Furthermore, the Schrödinger equation is said to have the strong UCP if when- everψvanishes to infinite order at some point x0, that is, for all m > 0

ð

jx−x0j≤rjψð Þxj²dx=Oð Þr^m ðr!0Þ,

thenψ is identically zero. Additionally, ifψ = 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 concludeψ6¼0 almost everywhere for any eigen- function of HN.

There exists a considerable amount of literature that treats the UCP for differential inequality |Δψ|≤|ξ¹||rψ| + |ξ²||ψ|.^[21–26] In particular if ξ1∈Lⁿ_locð ÞRⁿ andξ2∈Lⁿ⁼²_locð Þ, the corresponding differential equationRⁿ Δψ=ξ1rψ+ξ2ψhas the UCP from sets of positive measure.^[26]Note that suchL^p_locconstraints become more restrictive with increasing particle number, since the dimension of the configuration spacenenters in the con- ditions. Directly applied toHNψ=eψthis means that if a solutionψin the Sobolev spaceH^2N=_loc^{ðN+ 2Þ}R^3N

vanishes on a set of positive measure,

X^N

j= 1

½v xj +A xj ²+irjA xj

+X

l<j

u xj,xl

∈L^3N=2_loc R^3N ,

and each component ofAbelongs toL^3N_locR^3N

, thenψis 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 theo- rem^[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 exploi- ting 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 potentialsv∈L^p_loc R³,p> 2.

3 | P R E R E Q U I S I T E S

Let the Hamiltonian HN be as in Equation (2). The Schrödinger equation is then given by HNψ = eψ. We write HN = TA+V+U, where T_A=PN

j= 1irj+A x_j 2

,rj= _∂x^∂1 j

,_∂x^∂₂

j

,_∂x^∂₃

j

andx_j=x¹_j,x²_j,x³_j

∈R³are the coordinates of thejth electron. Here we use (irj+A(xj))²inTAinstead of (−irj+A(xj))²in order to follow the notation in Kurata.^[25]We use a slight variation of atomic unitsℏ= 2me= 1 andqe=−1, such that the Laplace operator appears without a factor 1/2.

The electric potentialVis a one-body potential given byV xð Þ=PN

j= 1v xj withv:R³!R. The two-particle interactionUbetween the electrons is modeled byU(x) =P

1≤j<l≤Nu(xj,xl), for some nonnegative functionuonR³×R³. We setW=V+U. Furthermore,A:R³!R³denotes the vec- tor potential, from which the magnetic field is obtained byB=r×A. With the notationA xð Þ= A xj

N

j= 1, the Schrödinger equation is rewritten as

−Δψ+ 2iArψ+ðWA−eÞψ= 0, ð3Þ

whereWA=W+j jA²+iðrAÞ.

A functionf∈L²_locð ÞRⁿ belongs to the Sobolev spaceH^k_locð ÞRⁿ iffhas weak derivatives up to orderkthat belong toL²_locð Þ. Let the set of infi-Rⁿ nitely differentiable functions with compact support onR^3Nbe denoted byC^∞₀R^3N

. We say thatψ∈H¹_locR^3N

is a solution of Equation (3) in the weak sense, which will be our standard notion for solutions from here on, if for allφ∈C^∞₀R^3N

ð

R^3Nrψrφdx+ 2i ð

R^3NA rð ψÞφdx+ ð

R^3NðWA−eÞψφdx= 0: ð4Þ

The present work takes off from the following result:

Theorem 2 (Theorem 1.2 in Regbaoui^[26]).LetN≥1. Assume thatWA∈L^3N=2_loc R^3N

and each component ofAis an element ofL^3N_locR^3N

.Then the Schrödinger equation has the UCP from sets of positive measure, that is, if a solutionψ∈H^2N=_loc^{ðN+ 2Þ}R^3N

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 withN= 1, since there is no two-particle interaction it suffices to assume thatv,|A|²andrAare elements of L³⁼²_loc R³ to obtain the UCP from sets of positive measure. With increasing particle number, however, the assumptions on the potentialsv,u, and Aare such that they rule out most types of singularities. On the other hand, the particle-number dependence that enters inH^2N=_loc^{ðN+ 2Þ}fulfills the inequality 2N/(N+ 2)≤2 for allN. Following Kurata^[25]theL^p_locconstraints, withpproportional toN, can be avoided.

Definition 3 A function f∈L¹_locð ÞRⁿ belongs to the Kato class Kⁿ_loc, n6¼2, if for every R > 0, lim

r!0 +η^Kð Þr;f = 0, where

η^Kð Þr;f = sup

jxj≤R

ð

Brð Þx

jf yð Þ j x−y j jⁿ⁻²dy:

Furthermore, f∈K^n,_loc^δKⁿ_loc,δ> 0, if for every R > 0

r!0 +lim sup

jxj≤R

ð

Brð Þx

jf yð Þ j x−y

j j^{n−2 +δ}dy= 0:

We writef=f+−f₋, wheref₋(f+) is the negative (positive) part offgiven byf₋(x) = max(−f(x), 0) (f+(x) = max(f(x), 0)). Lety∈R^3Nbe fixed and forx= (x1,…,xN)∈R^3N(cf. the notation in Kurata^[25])

a xð Þ=jA xð Þj²,

byð Þx =jx−yj²X^N

j= 1

B xj

², Q_yð Þx = 2Wð +ðx−yÞrWÞ₋: With the notation above, we formulate.

Assumption 1 Suppose

rA∈L²_locR³ ,

A^j∈L⁴_loc R³ A^jcomponent ofA , a,b_y,W,Q_y∈K^3N_loc for fixedy∈R^3N

,

and that for somer0> 0

ðr0 0

θyð Þr

r dr<∞ ð5Þ

holds, whereθ^y(r) =η^K(r; Qy) +η^K(r; by)^1/2.

Remark 3 Remark 1.2 in Kurata^[25]gives by,Qy∈K^3N,δ_loc , for someδ> 0, as a sufficient condition for Equation (5) to hold.

Remark 4 The main condition inAssumption 1is with respect to the Kato class Kⁿ_loc, n = 3 N being the dimensionality of the underlying configuration space. This condition isoptimalin the sense that the class cannot be enlarged tosmallerorders than n = 3 N, or the UCP will be lost. This follows from the inclusion L^p_locKⁿ_locfor 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_locwith 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-valuedVandU. In the sequel we use the notation |F| for the Frobenius norm (also called the Hilbert-Schmidt norm) of a matrix (Fj,l)j,l.

Theorem 4 Suppose Assumption 1. Ifψ∈H²_locR^3N

is a solution of (3) and vanishes to infinite order atx0∈R^3N,thenψis identically zero. Thus the Schrödinger equation has the strong UCP.

Lemma 5 (Corollary 1.1 in Kurata^[25]).Let n≥3, x0∈Rⁿbe fixed, x=x¹,…,xⁿ

∈Rⁿ,Ae=eA¹,…,Aeⁿ

:Rⁿ!Rⁿ,We:Rⁿ!R, F= Fj,l n j,l= 1 with F_j,l=∂Ae^j=∂x^l−∂eA^l=∂x^j,and suppose that

Ae^j∈L⁴_locð Þ,Rⁿ reA∈L²_locð Þ,Rⁿ jeAj²∈K_locⁿ , x−x0

j jj jF

ð Þ²∈Kⁿ_loc, ð6Þ

and

We∈Kⁿ_loc, 2We+ðx−x₀ÞrWe

−∈Kⁿ_loc: ð7Þ

Furthermore assume, for somer0> 0, Ðr0

0 η^K r; 2 We+ðx−x0ÞrWe

−

+η^Kr;ðjx−x0j jFjÞ²1=2 dr

r <∞ ð8Þ

holds. Then ifψ∈H²_locð ÞRⁿ satisfies

Xⁿ

j= 1

i∂

∂x^j+Ae^jð Þx

2

+W xeð Þ

!

ψ= 0 ð9Þ

and vanishes to infinite order atx0, it follows thatψis identically zero.

Proof of Theorem 4 We will show that Assumption 1 directly fulfills all the conditions of Lemma 5 that thus becomes applicable. Letn = 3 N and W = We −e.By Assumption 1, (7) is then fulfilled. The choiceà = AimpliesT_A=ir+eA2

and Equation (3) can be written as Equation (9).

Each component ofA∈L⁴_loc R³ yieldsAe^j∈L⁴_locR^3N

for j= 1,…, 3 N.FromreA=PN

k= 1rkA xð Þk andrA∈L²_loc R³,we obtainreA∈L²_locR^3N . Since a= |Ã|²,it holds thatjeAj²∈K^3N_loc.Moreover, the matrixFsatisfies

x−x₀ j jj jF

ð Þ²=jx−x₀j²X^3N

j,l= 1

F_j,l²= 2b_x0ð Þ,x ð10Þ

sinceFcontainsNrepeated blocks of sub matrices of the form

0 −B3 xj B2 xj

B3 xj 0 −B1 xj

−B2 xj B1xj 0 2

64

3 75:

This establishes Equation (6).

From Equation (5),We=W−eand Equation (10), we conclude that Equation (8) holds. Lemma 5 gives the strong UCP for Equation (3) and the

proof is complete. □

Remark 5 As stated in Remark 1.1 in Kurata,^[25]Lemma 5 and thus Theorem 4 also holds if inAssumption 1, K^3N_loc is replaced by K^3N_loc+F^p_locR^3N , 1 < p≤3 N/2. Here F^p_locR^3N

is the Fefferman-Phong class and in this case a solution must be an element of H²_locR^3N

\L^∞_locR^3N

, and there is an addi- tional condition onV₋.

4 | M A I N R E S U L T S

Theorem 4 above establishes the strong UCP under Assumption 1. If in addition the negative part ofvis locallyL^3/2(R³) summable we obtain the UCP from sets of positive measure:

Theorem 6 Suppose Assumption 1. If in additionv₋∈L³⁼²_loc R³ andψ∈H²_locR^3N

solves (3) and vanishes on a set of positive measure, thenψis identically zero. Consequently, the Schrödinger equation has the UCP from sets of positive measure.

Remark 6 The requirement u≥0 can be relaxed if one assumes that u(x1,x2) = u⁰(x1−x2) (see Lemma A.2 in Lammert^[30]).

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 v₋+j jA²+iðrAÞ∈L³⁼²_loc R³ gives the UCP from sets of positive measure.

Due to the particular form of the potentials, we can write

W xð Þ=X

j

v x_j +1 2

X

j6¼l

u x _j,x_l :

BecauseQyis defined as the negative part of the function 2W+ (x−y)rW, we have with the choicex₀=ðx₀,…,x₀Þ∈R^3N, for fixedx0∈R³, 0≤Q_x0ð Þx =Q_x

0ð Þx ≤X

j

q_1;x

0 x_j +X

j6¼l

q_2;x

0x_j,x_l

ð11Þ

for some functionsq_1;x0andq_2;x0. (See below the proof of Corollary 9, where this decomposition is done for the choice ofWcorresponding to the molecular case.) Furthermore,

bx₀ð Þ=x bx₀ð Þx =X^N

j,l= 1

x_j−x₀ ²jB xð Þ_lj²

can be split as

bx₀ð Þ=x X

j

b1;x₀ x_j +X

j6¼l

b2;x₀x_j,x_l :

We can now formulate our main result that includesHNmodeling atoms and molecules, and where the exponents in the integrability con- straints are independent of the particle numberN.

Corollary 8 ForN≥2 andx0∈R³fixed, suppose

rA∈L²_loc R³, A^j∈L⁴_locR³ , j jA²∈K³_loc,

b1;x₀∈K^3,_loc^δ, b2;x₀∈K^6,_loc^δ: ð12Þ

Further, letv₋∈L³⁼²_loc R³ ,v∈K³_loc, andu∈K⁶_loc, as well asQx₀satisfying(11)withq_1;x0∈K^3,δ_locandq_2;x0∈K^6,δ_loc. Then the Schrödinger equation(3)has the UCP from sets of positive measure.

In particular, the magnetic Schrödinger equation has the UCP from sets of positive measure forHNmodeling atoms and molecules in magnetic fields if just Equation (12) is fulfilled.

Corollary 9 ForN≥2, suppose the magnetic field is such that Equation (12) holds. Then with

v xð Þ₁ =−X^Mnuc

j= 1

Z_j

jx_nuc;j−x₁j, u xð₁,x₂Þ= 1 jx₁−x₂j,

wherexnuc;j∈R³andZj>0 are the positions and charges of theMnucnuclei, respectively, the UCP from sets of positive measure holds for the Schrödinger equation.

5 | A P P L I C A T I O N T O C D F T

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 functionψ, define the particle density and the paramagnetic current density according to

ρψð Þx =N ð

R^3ðN−1Þjψðx,x2,…,x_NÞj²dx₂…dx_N, j^p_ψð Þx =N Im

ð

R^3ðN−1Þψðx,x₂,…,x_NÞ rxψðx,x₂,…,x_NÞdx2…dx_N:

For a vector potentialAwe may compute the total current density by the sumj=j^p_ψ+ρψA. Now, fix the particle numberNas well as the two- particle interactionu(eg,u(x1,x2) = |x1−x2|⁻¹) and writeHN=H(v,A). Ifψis a ground state for somevandA, that is,H(v,A)ψ=eψ, whereeis the ground-state energy, thenρψ,j^p_ψ, andj=j^p_ψ+ρψAare called ground-state densities ofH(v,A). Whether the ground-state particle densityρand the total current densityjdeterminevandA(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 potentialsvandA.^[8])

Now, assume that two systems with HamiltoniansH(v1,A1) andH(v2,A2) have the same ground-state particle density (i.e.,ρ1=ρ2=ρ) and r×A1=r×A2=B. Suppose thatvk,Akfork= 1,2, andBfulfill Assumption 1 and the requirements given in Theorem 6. Since there exists a functionfsuch thatA1=A2−rf, the variational principle yields

e1≤hψ2,H vð1,A1Þψ2i≤e2+ ð

R²ðv1−v2Þρdx, whereψ²is the ground state ofH(v2,A2−rf). Switching the indices, we find that

e₁−e₂= ð

R²

v₁−v₂ ð Þρdx:

Consequently,ψ2is a ground state of bothH(v1,A1) andH(v2,A2−rf), which leads to H vð ₁,A₁Þ−Hðv2,A₂−rfÞ

½ ψ2=X^N

j= 1

v₁ x_j −v₂ x_j

ψ2=ðV₁−V₂Þψ2=ðe₁−e₂Þψ2:

However, Theorem 6 allows us to concludeψ26¼0 and it followsv1=v2+ constant.

Theorem 10 Assumer×A1=r×A2= Band thatvk, Akfork = 1,2,andBfulfill Assumption 1 and take the requirements of Theorem6for H(v1,A1)andH(v2,A2)to hold. If the ground-state particle densities satisfyρ¹=ρ²,thenv1= v2+ Calmost everywhere for some constantC.

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 forH(v1,A1)andH(v2,A2) and that the ground-state densities fulfill ρ=ρ1=ρ2, j = j1= j2.For systems withj^p= 0,it followsB1= B2(evenA1= A2holds) andv1= v2+ Cfor some constantC.

Proof. For systems withj^p= 0,j1=j2implies

ρA1=ρA2,

sinceρ1=ρ2=ρ. Theorem 6 givesρ> 0 almost everywhere and we may concludeA1=A2. Theorem 10 now gives the equalityv1=v2+Cfor

some constantC. □

6 | P R O O F S O F T H E M A I N R E S U L T S

Proof of Theorem 6 In the sequel letD = 3 N.By Assumption 1,the strong UCP holds for Equation (3) by Theorem4.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 settingA = 0here, and moreover we exploitu≥0instead of the assumptionu(x1,x2) = u⁰(x1−x2).)We start by showing the following inverse Poincaré inequality for solutions of the Schrödinger equation:

For an arbitrary pointx₀=x_0;jN

j= 1∈R^Dandr≤r0

ð

Brð Þx0jrψð Þxj²dx≤C r² ð

B2rð Þx0jψð Þxj²dx: ð13Þ

HereCis a positive constant that depends onr0> 0,v, andA(but is independent ofu≥0).

Chooseh∈C^∞₀ðB2rð Þx0Þthat satisfiesh(x) = 1 if |x−x0|≤r,h≤1 for |x−x0|≤2r, and |rh(x)|≤2r⁻¹. In the Schrödinger equation (4), we choose φ=h²ψand move all terms except one to the right hand side so that

ð

R^Djhrψj²dx=−2 ð

R^D

hðrψÞψrhdx−2i ð

R^DA rð ψÞh²ψdx+ ð

R^D

e−W_A

ð Þj jhψ²dx: ð14Þ

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 2khrψk2kψrhk2. Using the inequality 2ab≤a²/6 + 6b², we obtain an upper bound

1 6 ð

R^Djhrψj²dx+ 6kψrhk²₂: ð15Þ

To continue, letI1=−2iÐ

R^DA rð ψÞh²ψdx. The Cauchy-Schwarz inequality together with 2ab≤a²/6 + 6b²yield I₁≤1

6 ð

R^Djhrψj²dx+ 6 ð

R^D

j jA²j jhψ²dx: ð16Þ

For the last term of the right hand side in Equation (14), we use the definition ofWAand writee−W_A=e−W−j jA²−iðrAÞ. Thus ð

R^Dðe−WAÞj jhψ²dx= ð

R^De−W−j jA² hψ j j²dx−i

ð

R^DðrAÞj jhψ²dx and it follows fromW=V++U+−V₋≥−V₋that

ð

R^D

e−W_A

ð Þj jhψ²dx≤ ð

R^D

V₋+jej

ð Þj jhψ²dx+ ð

R^DjðrAÞkhψj²dx: ð17Þ

DefineΘ=V₋+jej+ 6j jA²+j rAj, from Equations (15), (16), and (17) we obtain an upper bound for the right hand side of Equation (14) given by

1 3 ð

R^Djhrψj²dx+ 6kψrhk²₂+ ð

R^DΘj jhψ²dx: ð18Þ

With the notationΘ1=v₋+N⁻¹jej + 6|A|²+ jr Aj, the inequalityΘð Þx ≤PN

j= 1Θ1 xj holds. Furthermore, we have ð

R^DΘð Þxj jhψ²dx≤X^N

j= 1

ð

R^DΘ1 x_j j jhψ²dx=I₂,

where the last equality definesI2.

By assumptionv₋∈L³⁼²_loc R³ ,j jA²∈L²_locR³ , andrA∈L²_loc R³ , and it follows thatΘ1∈L³⁼²_loc R³ . To bound the termI2from above, we closely follow Lammert^[30]and defineeρð Þx₁ =NÐ

R^3ðN−1Þj jhψ²dx₂dxN. ForM> 0 we letM⁰=kΘ1χB2rðx0;1ÞχfΘ1≥Mgk₃₌₂, where the characteristic func- tion of a setXis denotedχX. Hölder's inequality gives

I2= ð

Θ1<M

f gΘ1eρdx1+ ð

Θ1≥M

f gΘ1eρdx1≤M hk kψ ²₂+M⁰k keρ ₃,

and by a Sobolev inequality k keρ ₃≤Creρ¹⁼²²

2 . A direct computation of rð~ρ¹⁼²Þ, using the definition of ~ρ, shows that reρ¹⁼²

²

2≤Ð

R^Djrð Þhψ j²dx(see also the original argument of Lieb,^[31]Theorem 1.1). From |a+b|²≤2|a|²+ 2|b|², we get reρ¹⁼²

²

2≤2 ð

R^Djhrψj²dx+ 2kψrhk²₂: We chooseM> 0 such that 2CM⁰≤1/6 and then one has

I₂≤1 6 ð

R^D

hrψ j j²dx+1

6kψrhk²₂+M hk kψ ²₂: ð19Þ

Returning to Equation (18), we setC= 74 + 3Mrh ²₀i

=3 and use Equation (19) and |rh|≤2/rto conclude forr≤r0

1 2 ð

R^Djhrψj²dx≤37

6kψrhk²₂+M hk kψ ²₂≤C r² ð

B2rð Þx0j jψ²dx:

Hence Equation (13) holds.

Supposeψ∈H²_locvanishes on a setEof positive measure. Almost every point ofEis a density point. Letx0be such a density point and let Br=Br(x0). Givenε> 0 there is anr0=r0(ε) so that (cf. (3.11) in Regbaoui^[26])

jE\Brj

jB_rj ≥1−ε, jE^c\Brj

jB_rj ≤ε, for r≤r₀: ð20Þ

Lemma 3.3 in Regbaoui^[26](or Lemma 3.4 in Ladyzenskaya-Ural'tzeva^[34]) gives

ð

Br\E^cj jψ²dx≤Cr^D

jEjjB_r\E^cj^1=Dð

Br

j r ψ² jdx ð21Þ

for some constantC. Applying the Cauchy-Schwarz inequality to the right hand side of Equation (21), we obtain ð

Br

ψ

j j²dx≤Cr^2D

j jE²jBr\E^cj^2=Dð

Br

rψ j j²dx

for some new constantC. Since |E|≥|E\Br|, Equation (20), and the inverse Poincaré inequality (13) allow us to conclude that ð

Br

ψ

j j²dx≤C ε^2=D 1−ε ð Þ²r²

ð

Br

rψ

j j²dx≤C⁰ ε^2=D 1−ε ð Þ²

ð

B2r

ψ

j j²dx: ð22Þ

Introduce the functionf rð Þ=Ð

Brj jψ²dx, fix an integernand chooseε> 0 so thatC⁰ε^2/D/(1−ε)²= 2⁻ⁿ. Then Equation (22) can be written f(r)≤2⁻ⁿf(2r). By iteration

f rð Þ⁰ ≤2^−knf2^kr⁰

, r⁰≤2^1−kr₀

holds. For fixedrandkchosen such that 2^−kr0≤r≤2^1−kr0, it follows that

f rð Þ≤2^−knfð2r0Þ≤ r r₀

n

fð2r0Þ,

wherer0depends onn. Consequentlyfvanishes to infinite order, that is, for allmthere is anr0(m) such that

f rð Þ=ð

Br

ψ

j j²dx≤C_mr^m, r≤r₀ð Þm:

Thatψ= 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, sinceΘ1, by assumption, is an element ofL³⁼²_loc R³ . □ Proof of Corollary 8 We first demonstrate that the conditions of Corollary8fulfills Assumption 1.Due to the particular form of the potentials, we make use of the following: Letf1∈K^3,δ_locandf2∈K^6,δ_loc.Then bothPN

k= 1f1ð Þxk andP

k6¼lf2(xk, xl)are elements ofK^3N,δ_loc .Similar statements forKⁿcan be found in Simon^[35] (Example F) and Aizenman-Simon^[36](Theorem 1.4). We prove our claim by direct computations. Define I^δ₁ and I^δ₂ according to

I^δ₁ð Þ=x X^N

j= 1

ð

Brð Þ0

jf1 yj+xj

j y²₁++y²_N

^{3N−2 +δ}₂ dy₁dy_N,

I^δ₂ð Þ=x X

j6¼l

ð

Brð Þ0

jf2yj+xj,yl+xlj y²₁++y²_N

^{3N−2 +δ}₂ dy₁dy_N:

We next demonstrate that

I^δ₁ð Þx ≤C_N ð

Br; 3ð Þx

jf1ð Þ jy₁ y₁−x1

j j^{3−2 +δ}dy₁, ð23Þ

I^δ₂ð Þx ≤C_N ð

Br; 6ð Þx

jf₂ðy₁,y₂Þ j y₁,y₂

ð Þ−ðx1,x2Þ

j j^{6−2 +δ}dy₁dy₂, ð24Þ

where the second index in the given ball-setsBr; 3(x)R³andBr; 6(x)R⁶refers to the respective dimensionality.

To show Equation (23), setq= (y2,…,yN) and note that

I^δ₁ð Þx ≤N ð

Br; 3×Br; 3ðN−1Þ

jf₁ðy₁+x₁Þ j y²₁+q²

^{3N−2 +δ}₂ dy₁dq=C_N

ð

Br; 3

jf₁ðy₁+x₁Þ j ðr

0

q^{3ðN−1Þ−1}dq y²₁+q²

^{3N−2 +δ}₂

0

@

1 Ady₁

=C_N ð

Br; 3

jf₁ðy₁+x₁Þ j y₁ j j^{3N−2 +δ}

ðr 0

q^3N−4dq 1 +ðq=jy₁jÞ²

^{3N−2 +δ}₂

0 BB

@

1 CC Ady₁

≤C_NJ^δ₁ ð

Br; 3ð Þx1

jf₁ð Þ jy₁ y₁−x1

j j^{3−2 +δ}dy₁, where we have defined the integral

J^δ₁= ð∞

0

s^3N−4 1 +s²

ð Þ^{3N−2 +δ2} ds:

Now,J^δ₁is finite since

J^δ₁≤ð1 0

s^3N−4ds+ ð∞

1

s^−2−δds< +∞:

This establishes Equation (23). The proof of Equation (24) is similar and included for the sake of completeness. Setq= (y3,…,yN), then

I₁≤N ð

Br; 6×Br; 3ðN−2Þ

ju yð ₁+x₁,y₂+x₂Þ j y²₁+y²₂+q²

^{3N−2 +δ}₂ dy₁dy₂dq=C_N

ð

Br; 6

ju yð ₁+x₁,y₂+x₂Þ j ðr 0

q^{3ðN−2Þ−1}dq y²₁+y²₂+q²

^{3N−2 +δ}₂

0

@

1 Ady₁dy₂

=CN

ð

Br; 6

ju yð₁+x₁,y₂+x₂Þ j y₁,y₂

ð Þ

j j^{3N−2 +δ} ðr

0

q^3N−7dq 1 +ðq=jðy₁,y₂ÞjÞ²

^{3N−2 +δ}₂

0 BB

@

1 CC Ady₁dy₂

≤C_N ð

B_{r; 6}ðx₁,x₂Þ

ju yð ₁,y₂Þ j y₁,y₂ ð Þ−ðx1,x2Þ

j j^{6−2 +δ}dy₁dy₂ ð_∞

0

s^3N−7 1 +s² ð Þ^{3N−2 +δ2} ds:

Corollary 8 now follows from Theorem 6 (Equation (5) in Assumption 1 is fulfilled by Remark 3). □

Proof of Corollary 9 We first reduce the molecular case to the atomic one. Since the UCP from sets of positive measure is local, it can be applied to any open set in the domain individually. So instead of one singularity (theyof Assumption 1), we can treat an arbitrary (yet countable) number of singularities if they do not have an accumulation point. For this just choose an open cover{Uj}ofR³where each Ujcontains not more than one nucleus xnuc;j.It remains to show that allq_xnuc;j,bxnuc;j belong to the respective local Kato classes and we are done if we prove the results for atoms.

In the sequel we let v(x1) = −Z|x1−xnuc|⁻¹,xnuc∈R³,Z> 0, and u(x1,x2) = |x1−x2|⁻¹. In this casev₋∈L³⁼²_loc R³ and with the choice x_nuc=ðx_nuc,…,x_nucÞ∈R^D, we have withQ_xnucð Þ=x Q_x

nucð Þx the equality

Q_xnucð Þ=x X

j

−2Z jxj−xnucj+X

j6¼l

1

jxj−xlj +X

j

x_j−x_nuc

rj −Z jxj−xnucj

+1 2

X

j6¼l

x_j−x_nuc

rj+ðx_l−x_nucÞrl

1

jx_j−x_nuc

−ðx_l−x_nucÞ j

!

−

=ðV xð Þ+U xð ÞÞ₋≤2V₋ð Þx :

Thus, in this case we can chooseq_1,xnuc=v₋andq_2,xnuc= 0.

Furthermore, for 0 <δ< 1, we claim thatV,U∈K^D,_loc^δ. By the first part it suffices to showv∈K^3,_loc^δandu∈K^6,_loc^δ. Forv∈K^3,_loc^δ, we introduce polar coordinates with radiussand polar anglet. Then it holds thatdy= 2πs²sint dtdsandyx=−s|x|cost. Forf1(x) = |x|⁻¹it follows that

η^Kðr;f₁Þ= sup

jxj≤R

ðr 0

ð_π

0

2πs^1−δsintdt s²−2sjxjcost+j jx²

1=2

0 B@

1 CAds:

We integrate overt, use |s+ |x|−|s−|x|||≤2|x|, and the conclusion is obtained forv. In a similar fashion, foruwe establish that with f2(x) = |x1−x2|⁻¹

η^Kðr;f₂Þ≤C ðr

0

ðr s1

s₂ds₂ s²₁+s²₂ 2 +²_δ

0

@

1

As²₁ds₁≤Cr^1−δ

such that it followsu∈K^6,_loc^δ, since ð

Brð Þ0

1 jy₁−y₂j

1

j jy^{6−2 +δ}dy₁dy₂≤C ðr

0

ðr 0

ð_π

0

2πs²₁s²₂sintdt s²₁−2s1s2cost+s²₂

1=2

! ds1ds2

s²₁+s²₂ 2 +²_δ

≤C ðr

0

ðr 0

s1+s2−js1−s2j

ð Þ

s²₁+s²₂

2 +²_δ s₁s₂ds₁ds₂

≤C ðr

0

ðs1 0

s1s²₂ds2

s²₁+s²₂ 2 +²_δ+

ðr s₁

s²₁s2ds2

s²₁+s²₂ 2 +²_δ

2 4

3 5ds₁

≤C ðr

0

ðr s1

s₂ds₂ s²₁+s²₂

2 +²_δs²₁ds₁≤Cr^1−δ:

The atomic case is now a consequence of Corollary 8. □

7 | C O N C L U S I O N

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.

A C K N O W L E D G M E N T S

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 sugges- tions that improved the manuscript.

O R C I D

Andre Laestadius https://orcid.org/0000-0001-7391-0396

R E F E R E N C E S

[1] P. Hohenberg, W. Kohn,Phys. Rev.1964,136, B864.

[2] H. Englisch, R. Englisch,Physica A Stat. Mech. Appl.1983,121, 253.

[3] K. Burke,J. Chem. Phys.2012,136, 150901.

[4] R. O. Jones,Rev. Mod. Phys.2015,87, 897.

[5] G. Vignale, M. Rasolt,Phys. Rev. Lett.1987,59, 2360.

[6] G. Vignale, M. Rasolt,Phys. Rev. B1988,37, 10685.

[7] G. Diener,Condens. Matter1991,3, 9417.

[8] K. Capelle, G. Vignale,Phys. Rev. B2002,65, 113106.

[9] A. Laestadius, M. Benedicks,Int. J. Quantum Chem.2014,114, 782.

[10] A. Laestadius, E. I. Tellgren,Phys. Rev. A2018,97, 022514.

[11] E. I. Tellgren, S. Kvaal, E. Sagvolden, U. Ekström, A. M. Teale, T. Helgaker,Phys. Rev. A2012,86, 062506.

[12] E. I. Tellgren, A. Laestadius, T. Helgaker, S. Kvaal, A. M. Teale,J. Chem. Phys.2018,148, 024101.

[13] M. Ruggenthaler, arXiv:1509.01417 [quant-ph].2017 [14] E. I. Tellgren,Phys. Rev. A2018,97, 012504.

[15] L. Garrigue, arXiv:1901.03207 [math-ph].2019 [16] L. Garrigue,J. Stat. Phys.2019,177, 415.

[17] S. Kvaal, U. Ekström, A. M. Teale, T. Helgaker,J. Chem. Phys.2014,140, 18A518.

[18] A. Laestadius, M. Penz, E. I. Tellgren, M. Ruggenthaler, S. Kvaal, T. Helgaker,J. Chem. Phys.2018,149, 164103.

[19] A. Laestadius, M. Penz, E. I. Tellgren, M. Ruggenthaler, S. Kvaal, T. Helgaker,J. Chem. Theory Comput.2019,15, 4003.

[20] M. Penz, A. Laestadius, E. I. Tellgren, M. Ruggenthaler,Phys. Rev. Lett.2019,123, 037401.

[21] L. Hörmander,Commun. PDE1983,8, 21.

[22] B. Barcelo, C. E. Kenig, A. Ruiz, C. D. Sogge,Illinois J. Math.1988,32, 230.

[23] T. H. Wolff,Geom. Funct. Anal.1992,2, 225.

[24] K. Kurata,Commun. PDE.1993,18, 1161.

[25] K. Kurata,Proc. Amer. Math. Soc.1997,125, 853.

[26] R. Regbaoui, Unique Continuation from Sets of Positive Measure. inCarleman Estimates and Applications to Uniqueness and Control Theory, Birkhäuser, Switzerland, Basel2001, p. 179.

[27] E. H. Lieb, M. Loss,Analysis, American Mathematical Society, Providence, Rhode Island2001.

[28] A. Laestadius,J. Math. Chem.2014,52, 2581.

[29] L. Garrigue,Math. Phys. Anal. Geom.2018,21, 27.

[30] P. E. Lammert,J. Math. Phys.2018,59, 042110.

[31] E. H. Lieb,Int. J. Quantum Chem.1983,24, 243.

[32] H. Koch, D. Tataru,J. Reine Angew. Math.2002,542, 133.

[33] C. J. Grayce, R. A. Harris,Phys. Rev. A1994,50, 3089.

[34] O. A. Ladyzenskaya, N. N. Ural'tzeva,Linear and Quasilinear Elliptic Equations, Academic Press, New York1968.

[35] B. Simon, B. Amer,Math. Soc. (N.S.)1982,7, 447.

[36] M. Aizenman, B. Simon,Commun. Pure Appl. Math1982,35, 209.

How to cite this article:Laestadius A, Benedicks M, Penz M. Unique continuation for the magnetic Schrödinger equation.Int J Quantum Chem. 2020;120:e26149.https://doi.org/10.1002/qua.26149