# Eigenfunction Behavior and Adaptive Finite Element Approximations of Nonlinear Eigenvalue Problems in Quantum Physics

In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that the eigenfunction cannot be a polynomial on any open set, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing result.

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## 1 Introduction

In this paper, we investigate the eigenfunction behavior and adaptive finite element approximations of the following nonlinear eigenvalue problem: find such that

 (1)

where , , , is a given function, , and maps a nonnegative function to some function on . We observe that Schrödinger-Newton equation modeling the quantum state reduction [19, 24], Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensates (BEC) [2, 34] and Thomas-Fermi-von Weizsäcker (TFvW) type equations and Kohn-Sham equations appearing in electronic density functional theory [3, 6, 14, 21, 22] are typical examples of (1).

We understand that it is significant to solve eigenvalue problem (1) accurately and efficiently. And we note that the a priori knowledge of their eigenfunctions is very helpful in designing and analysis of numerical methods. To improve the approximation accuracy and reduce the computational cost in solving the eigenvalue problem, we see from the regularity of eigenfunction [18, 33] that adaptive finite element approaches should be employed (see also [5, 7, 10, 11, 13, 15, 23, 30] and references cited therein). We observe that the adaptive finite element analysis of nonlinear eigenvalue problem (1) in [5, 7, 8] requires that the initial mesh size is small enough. However, our numerical experiments show that the small initial mesh size requirement is unnecessary [5, 7, 8]. In this paper, we study the adaptive finite element approximations when the initial mesh is not fine, for which we need to apply an eigenfunction behavior that is also investigated.

We see that the unique continuation property is significant in the context of partial differential equations (see, e.g.,

[20, 27, 32] and references cited therein). After looking into the behavior of eigenfunction of (1), we find that the eigenfunction cannot be a polynomial on any open subset, which may be reviewed as a refinement of the classic unique continuation property and is indeed a key in our adaptive finite element analysis. Taking into account the eigenfunction behavior, we are indeed able to prove the convergence of adaptive finite element approximations without the requirement of small initial mesh size.

The rest of this paper is organized as follows. In the next section, we describe some basic notation and review the adaptive finite element method for solving eigenvalue problem (1). Then we show some polynomial property which is crucial in our adaptive finite element analysis. In Section 3, we obtain that any eigenfunction of problem (1) cannot be a polynomial on any open subset of under some assumptions, which may be reviewed as an extension and refinement of the classic unique continuation property. In Section 4, based on the non-polynomial behavior of eigenfunctions, we study the convergence of the adaptive finite element method. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing result.

## 2 Preliminaries

Let be a polyhedral bounded domain. Let and is an -tuple. We denote , define for any , and use notation

 ∂iϕ=∂ϕ∂ξi, i=1,2,3.

For any , we denote if the first non-zero element of is greater than and if or . For convenience, we define

 PQ+(Ω)=⎧⎪⎨⎪⎩∑α∈I⊂Q3+aαxα:x∈Ω, |I|<∞ and aα∈R,∀α∈I⎫⎪⎬⎪⎭,

where means the cardinality of . We shall use the notation

for any . We call with a monomial. Denote the degree of monomial . We shall let the degree of polynomial be . For any , define as the max degree of terms of , which is called the degree of . We shall also denote for any and and for any . Let be the set of real polynomials on with degrees not greater than . It is clear that . The standard notation for Sobolev space and their associated norms shall also be used [1]. We write

 G⊂⊂Ω

if and is compact. denotes the space of function satisfying that for any open set , . We use to denote a class of functions satisfying some growth conditions:

 P(s,(c1,c2))={f:∃ a1,a2∈R% such that c1ts+a1≤f(t)≤c2ts+a2∀t≥0}

with and .

### 2.1 Quantum eigenvalue problem

We consider nonlinear eigenvalue problem (1) when has a form of

 (2) V=−M∑j=1fjgj

with for some , is divided into two parts:

 (3) N(ρ)=N1(ρ)+N2(ρ),

where is defined by

 (4) N1(t)=K∑i=1pi(t)hi(t)lnqi(t)

with , polynomials satisfying and , and is given by a convolution integral

 (5) N2(ρ)=α∫Ωρ(y)|⋅−y|dy

with some constant .

The energy functional associated with (1) is

 E(Φ)=∫Ω(κN∑i=1|∇ϕi|2+VρΦ+E(ρΦ))+α2D(ρΦ,ρΦ)

for , where , is defined by

 E(s)=∫s0N1(t)dt,

and is a bilinear form as follows

 D(f,g)=∫Ω∫Ωf(x)g(y)|x−y|dxdy.

For any , we denote

We see that (1) includes the GPE, the Schrödinger-Newton equation, the TFvW type equation, and the Khon-Sham equation (see Remark 3.2, Example 3.3, Example 3.4, and Example 3.5 for more details).

Let be a subspace of :

 Q={Φ∈H:ΦTΦ=IN×N},

where . The ground state charge density of system (1) is obtained by solving minimization problem

 (6) inf{E(Φ):Φ∈Q}.

We see that any minimizer of (6) satisfies

 (7)

where is a Hamiltonian operator defined by

 ⟨HΦu,v⟩=κ(∇u,∇v)+(VρΦ+N(ρΦ)u,v)∀u,v∈H10(Ω)

and

 Λ=(λij)Ni,j=1=(∫ΩϕiHΦϕj)Ni,j=1

is the Lagrange multiplier. We call a ground state of (7) and define the set of ground states by

 Θ={(Λ,Φ)∈RN×N×Q:E(Φ)=minΨ∈QE(Ψ) and (Λ,Φ) solves (???)}.

We define the set of states of (7) by

 W={(Λ,Φ)∈RN×N×H:(Λ,Φ) solves (???)}.

Since electron density and operator are invariant under any unitary transform, we may diagonalize Lagrange multipliers and arrive at

 (8)

which is equivalent to (7) and a weak form of (1).

### 2.2 An adaptive finite element method

Let be the diameter of and be a shape regular family of nested conforming meshes over with size : there exists a constant such that

 hτρτ≤γ∗∀τ∈Th,

where is the diameter of , is the diameter of the biggest ball contained in , and . Let denote the set of interior faces of . We shall also use a slightly abused of notation that denotes the mesh size function defined by

 h(x)=hτ,x∈τ∀τ∈Th.

Let be the corresponding finite element space consisting of continuous piecewise polynomials over of degrees no greater than and

 Sh0(Ω)=Sh∩H10(Ω).

Let .

Consider the finite element approximation of (6):

 (9) inf{E(Φh):Φh∈Vh∩Q}.

We see that any minimizer of (9) solves Euler-Lagrange equation

 (10)

with Lagrange multiplier

 Λh=(λij,h)Ni,j=1=(∫Ωϕi,hHΦhϕj,h)Ni,j=1

when the energy functional is differentiable. Define the set of finite dimensional ground state solutions:

 Θh={(Λh,Φh)∈RN×N×(Q∩Vh):E(Φh)=minΨ∈Q∩VhE(Ψ),(Λh,Φh) solves (???)}.

With using the unitary transformation, we have the following discrete Kohn-Sham equation

 (11)

We recall that the adaptive finite element method is to repeat the following procedure [5]:

 Solve→Estimate→Mark→Refine.

For convenience, we shall replace subscript (or ) by an iteration counter of the adaptive method afterwards.

Given an initial triangulation so that the dimension of is greater than or equal to . The above procedure generates a sequence of nested triangulations . Given an iteration counter , procedure “Solve” is to get the discrete solution over . Procedure “Estimate” determines the element indicators for all elements . In this step, a posteriori error estimators play an critical role. Then, element indicators are used by procedure “Mark” to create a subset of marked elements . To maintain mesh conformity, we usually partition a few more elements in procedure “Refine”.

Given a triangulation and the corresponding finite element solution , we define finite element residual and jump by

 Rτ(Φh)=(HΦhϕi,h−N∑j=1λjiϕj,h)Ni=1in τ∈Th, Je(Φh)=(je(ϕi,h))Ni=1,je(ϕi,h)=κ∇ϕi,h|τ1⋅→n1+κ∇ϕi,h|τ2⋅→n2,

where is the common face of elements and with unit outward normals and , respectively. For , we define the local error indicator as follows:

 η2h(Φh,τ)=h2τ∥Rτ(Φh)∥20,Ω+∑e∈Eh,e⊂∂τhe∥Je(Φh)∥20,e.

Depending on the a posteriori error indicators , procedure “Mark” gives a strategy to create a subset of elements of . Here, we consider “maximum strategy” which only requires that the set of marked elements contains at least one element of holding the largest value estimator. Namely, there exists at least one element such that

 ηk(Φk,τmaxk)=maxτ∈Tkηk(Φk,τ).

The adaptive finite element algorithm for solving (8) is stated as follows [5, 7, 8]:

We observe that there are a number of works on analyzing adaptive finite element methods in literature. We refer to [4, 10, 11, 16, 17] and references cited therein for linear eigenvalue problems and to [5, 7, 8] for nonlinear cases when the initial mesh is fine enough. We see that under the so-called Non-Degeneracy assumption111No eigenfunction is equal to a polynomial of degree on an open subset of , where denotes the polynomial degree of the finite element bases being used., [15] proved convergence of an adaptive finite element method starting from any initial mesh for some linear elliptic eigenvalue problem.

### 2.3 A polynomial theory

In our analysis, we need the following basic results, which are motivated by [35]. Let be a prime number and

 δ=a1t+a2t2+⋯+ak−1tk−1,

where . Then there exist real polynomials

 {pj(t1,t2,…,tk):j=2,…,k}

such that is a polynomial of degree with respect to and

 pj(λt1,λt2,…,λtk−1,tk)=λjpj(t1,t2,…,tk−1,tk) ∀λ∈R, δk+k∑j=2pj(a1,a2,…,ak−1,tk)δk−j=0.

The proof of Lemma 2.3 is given in Appendix Eigenfunction Behavior and Adaptive Finite Element Approximations of Nonlinear Eigenvalue Problems in Quantum Physics thanks: This work was partially supported by the National Science Foundation of China under grants 91730302 and 11671389 and the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences under grant QYZDJ-SSW-SYS010. . Suppose is a prime. Then for any positive integer , there exist polynomials

 {Hn,j(t1,t2,…,tn):j=0,1,2,…,kn−1}

with real coefficients satisfying

1. are homogeneous:

 Hn,j(λt1,λt2,…,λtn)=λjHn,j(t1,t2,…,tn) ∀λ∈R,

and is a monic polynomial of degree with respect to each variable ;

2. if , then

 kn−1∑j=0Hn,j(tk1,tk2,…,tkn)δk(kn−1−j)=0.

We prove the conclusion by induction on . Obviously, Lemma 2.3 is true when . Assume Lemma 2.3 is true for . We show that Lemma 2.3 is true for . Let

 n+1∑j=1tj=δ.

It follows from the induction hypothesis and

 n∑j=1tj=δ−tn+1

that there exist polynomials

 {Hn,j(s1,s2,…,sn):j=0,1,2,…,kn−1}

with real coefficients satisfying that are homogeneous, is a monic polynomial of degree with respect to each variable , and

 kn−1∑j=0Hn,j(tk1,tk2,…,tkn)(δ−tn+1)k(kn−1−j)=0.

We obtain from Newton binomial theory that

 k−1∑i=1aiδk(−tn+1δ)i=a,

where

 a =−Hn,kn−1(tk1,…,tkn)−δkn−(−tn+1)kn−kn−1−1∑ℓ=1(knkℓ)δkn−kℓ(−tn+1)kℓ −kn−1−1∑j=1Hn,j(tk1,…,tkn)kn−1−j∑ℓ=0(kn−kjkℓ)δkn−kj−kℓ(−tn+1)kℓ, ai =kn−1−1∑j=0Hn,j(tk1,…,tkn)kn−1−j∑ℓ=1(kn−kjkℓ−k+i)δkn−kj−kℓ(−tn+1)kℓ−k, i=1,2,…,k−1.

Since is a prime, there exist satisfying Lemma 2.3, namely,

 0=ak+k∑j=2pj(a1δk,a2δk,…,ak−1δk,(−tn+1)k/δk)ak−j,

or

 0=ak+k∑j=2pj(a1,a2,…,ak−1,(−tn+1)k/δk)δkjak−j.

We conclude that Lemma 2.3 is true when is replaced by . This completes the proof.

Since every integer greater than can be written as a product of one or more primes, we arrive at Let and be two positive integers. Then there exists a homogeneous polynomial with real coefficients satisfying

1. the degree of with respect to each variable is the same, and is a monic polynomial with respect to ;

2. if , then

 P(tk1,tk2,…,tkn,δk)=0.

We mention that the coefficients of the polynomial in Proposition 2.3 can be integers and there exists a real homogeneous polynomial such that any zero of

 n∑i=1k√ti=δ

is an zero of

 P(t1,t2,…,tn,δk)=0.

If and , then for any open set , there exists such that . We see from the definition of that

 p=n∑i=1aα(i)xα(i),

where , , and is the max index. Hence we can choose positive integer such that all components of are integers.

Assume in . Then there exists a homogeneous polynomial satisfying the conclusion of Proposition 2.3 and

 P(xkα(1),xkα(2),…,xkα(n))=0,% in G.

Set . Then is a polynomial with positive which is a contradiction to . This completes the proof.

Let be an positive integer and be an open subset of . Let for some positive integer and . If in with in for some and

 degpj0>degpj∀j∈{1,2,…,n}∖{j0},

then there exists such that

 n∑j=1pj(x0)lnqj(x0)≠0.

## 3 Behavior of eigenfunction

In this section, we investigate the non-polynomial behavior of eigenfunctions of (1), which will be applied to analyze convergence of their so-called adaptive finite element approximations. We may refer to [18, 33] for the regularity behavior of eigenfunctions that indeed result in applying adaptive finite element computations.

We first recall the unique continuation property.

Equation (1) has a unique continuation property if every solution in that vanishes on an open set of vanishes identically.

To look into if (1) has a unique continuation property, we may apply the following conclusion, which can be found in [32]. Assume and such that . If vanishes on an open set of , then is identically zero on .

If and with , then (1) has a unique continuation property.

It follows from [9, 12] that , which together with Sobolev imbedding theorem leads to .

Not that Young’s inequality and Sobolev imbedding theorem imply

 ∥N2(ρ)∥0,∞,Ω≤C∥ρ∥0,Ω≤CN∑i=1∥ϕi∥20,4,Ω≤C∥Φ∥21,Ω<∞.

We have that and , where . Thus we arrive at the conclusion by using Lemma 3.

###### Remark 3.1

We may see from the proof of Theorem 3 that if is replaced by and any solution of is in , then has a unique continuation property.

Let and be defined by (2) and (3)-(5) with , respectively. If is a non-constant function and

 (12) degp1−degh1>max{0, max1≤j≤M(degfj−deggj)/2, max2≤i≤K(degpi−deghi)},

then for any solution of (1), there exists an eigenfunction being not a non-zero polynomial on any open set . If in addition, and with , then there exists an eigenfunction being not the polynomial on any open set . Assume that all eigenfunctions are polynomials on some open set : for some positive integer . Without loss of generality, let . We have and see from (1) that

 (13) −κΔϕ1−M∑j=1fjgjϕ1+K∑i=1pi(ρ)hi(ρ)ϕ1lnqi(ρ)=λ1ϕ1, in G.

If , then we see from (12) that

 (degp1−degh1)degρ+degϕ1 >degΔϕ1, (degp1−degh1)degρ+degϕ1 >deg(fjgjϕ1), j=1,2,…,N, (degp1−degh1)degρ+degϕ1 >(degpi−deghi)degρ+degϕ1, i=2,3,…,K, (degp1−degh1)degρ+degϕ1 >degϕ1.

Since are polynomials implying , we obtain from Lemma 2.3 that

 −κΔϕ1(x0)−M∑j=1fj(x0)gj(x0)ϕ1(x0)+K∑i=1pi(ρ)(x0)hi(ρ)(x0)ϕ1(x0)lnqi(ρ)(x0)≠λ1ϕ1(x0)

for some , which is a contradiction to (13). Thus we arrive at that on . Since , we have that are constants on . If for all , then

 M∑j=1fjgj=K∑i=1pi(ρ)hi(ρ)lnqi(ρ)−λ1,in G,

with constant , which is impossible. Hence for some .

If in addition, and with , then Theorem 3 implies that in for some , which is a contradiction to . This completes the proof.

###### Remark 3.2

Note that Theorem 3 may be also true even if

 degp1−degh1=max1≤j≤M(degfj−deggj)/2.

For instance, no eigenfunction of GPE [2, 34]

 (−12Δ+V+β|ϕ|2)ϕ=λϕ

with a harmonic trap potential

 V(x)=γ1ξ21+γ2ξ22+γ3ξ23, γ1,γ2,γ3>0

can be a polynomial on any open set , where .

Let and be defined by (2) and (3)-(5) with , respectively. Suppose is not a positive constant function. If either

 max1≤i≤K(degpi−deghi)≤1 and max1≤j≤M(degfj−deggj)<4,

or and

 (14) degp1−degh1>max{2, max1≤j≤M(degfj−deggj)/2, max2≤i≤K(degpi−deghi)},

then for any solution of (1), there exists an eigenfunction being not the non-zero polynomial on any open set . If in addition, and