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Analysis of periodic
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Citation: HUNSICKER, E., NISTOR, V. and SOFO, J.O., 2008. Analysis of
periodic Schrodinger operators: regularity and approximation of eigenfunctions.
Journal of Mathematical Physics, 49 (8), 083501.
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Analysis of periodic Schrödinger operators: Regularity and approximation of
eigenfunctions
Eugenie Hunsicker, Victor Nistor, and Jorge O. Sofo
Citation: Journal of Mathematical Physics 49, 083501 (2008); doi: 10.1063/1.2957940
View online: http://dx.doi.org/10.1063/1.2957940
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JOURNAL OF MATHEMATICAL PHYSICS 49, 083501 共2008兲
Analysis of periodic Schrödinger operators: Regularity
and approximation of eigenfunctions
Eugenie Hunsicker,1,a兲 Victor Nistor,2 and Jorge O. Sofo3
1
Department of Mathematical Sciences, Loughborough University, Loughborough,
Leicestershire LE11 3TU, United Kingdom
2
Department of Mathematics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA
3
Department of Physics, Pennsylvania State University, University Park,
Pennsylvania 16802, USA
共Received 30 January 2008; accepted 24 June 2008; published online 4 August 2008兲
Let V be a real valued potential that is smooth everywhere on R3, except at a
periodic, discrete set S of points, where it has singularities of the Coulomb-type
Z / r. We assume that the potential V is periodic with period lattice L. We study the
spectrum of the Schrödinger operator H = −⌬ + V acting on the space of Bloch
waves with arbitrary, but fixed, wavevector k. Let T ª R3 / L. Let u be an eigenfunction of H with eigenvalue ␭ and let ⑀ ⬎ 0 be arbitrarily small. We show that the
classical regularity of the eigenfunction u is u 苸 H5/2−⑀共T兲 in the usual Sobolev
m
spaces, and u 苸 K3/2−
⑀共T \ S兲 in the weighted Sobolev spaces. The regularity index
m can be as large as desired, which is crucial for numerical methods. For any
choice of the Bloch wavevector k, we also show that H has compact resolvent and
hence a complete eigenfunction expansion. The case of the hydrogen atom suggests
that our regularity results are optimal. We present two applications to the numerical
approximation of eigenvalues: using wave functions and using piecewise
polynomials. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2957940兴
I. INTRODUCTION AND STATEMENT OF MAIN RESULTS
Let V be a potential that is a smooth periodic function on R3 except at a discrete set S, where
it has Coulomb-type singularities, that is, for any p 苸 S, the function 兩x − p兩V共x兲 is smooth in a
neighborhood of p. We are interested in studying the spectrum of Schrödinger operators H = −⌬
+ V acting on the space of Bloch waves with Bloch wavevector k 共see below兲. This question is
interesting for the study of the nonrelativistic Born–Oppenheimer approximation of the
Schrödinger operator for electrons moving in a lattice of atoms. Modeling electrons moving in a
lattice of atoms is part of the implementation of the “density-functional theory” codes in quantum
chemistry.1–7 We are especially interested in the regularity and approximability of the eigenfunctions u and eigenvalues ␭ of H.
Let L ª 兵n1v1 + n2v2 + n3v3其 ⯝ Z3 be the lattice of periods of V, where 共v j兲 is a basis of R3 and
n j 苸 Z. Let k 苸 R3. Let us denote by v · w the inner product of two vectors v , w 苸 R3. Then a Bloch
wave with Bloch wavevector k is a measurable function ␺k satisfying the twisted periodicity
condition ␺k共r + R兲 = eık·R␺k共r兲 for all R 苸 L. Every Block wave ␺k with Bloch wavevector k can
be written in the form
␺k共r兲 = eık·ruk共r兲,
共1兲
where uk is a truly periodic function with respect to the lattice L 共so uk is a Bloch wave with zero
Bloch wavevector兲.
a兲
Electronic mail: [email protected].
0022-2488/2008/49共8兲/083501/21/$23.00
49, 083501-1
© 2008 American Institute of Physics
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083501-2
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Hunsicker, Nistor, and Sofo
The function uk of Eq. 共1兲 identifies with a function on the three dimensional torus T
ª R3 / L, which in turn can be identified with the product 共S1兲3 of three circles S1. Let k
= 共k1 , k2 , k3兲 苸 R3 and define
3
Hk ª − 兺 共⳵ j + ık j兲2 + V.
共2兲
j=1
Then H␺k共r兲 = eık·rHkuk共r兲. This shows that the study of H on the space of Bloch waves with
Bloch wavevector k is equivalent to the study of Hk on the space of periodic functions on R3 or,
equivalently, to the study of Hk on suitable function spaces on T.
Let us denote by Lⴱ ª 兵w 苸 R3 , w · v 苸 Z , ∀ v 苸 L其 ⯝ Z3 the dual lattice of L. Then to find all
eigenvalues and eigenfunctions for the Schrödinger operator H, it is enough to solve this problem
for operators Hk corresponding to vectors k in a fundamental domain for Lⴱ, for instance, to the
first Brillouin zone. However, this is not an essential point for our purposes in this paper.
Our results are formulated using both the usual Sobolev spaces Hm共T兲 and the weighted
Sobolev spaces Km
a 共T \ S兲 of periodic functions. Let m 苸 Z+ ª 兵0 , 1 , 2 , . . .其. Then
再
冎
Hm共T兲 ª v:T → C, 兺 共1 + 兩␰兩兲2m兩vˆ 共␰兲兩2 ⬍ ⬁ = 兵v, ⳵␣v 苸 L2共T兲, ∀ 兩␣兩 ⱕ m其,
␰
共3兲
where ␰ 苸 Lⴱ and ˆf 共␰兲 = 兰Teı␰·x f共x兲dx is the 共unnormalized兲 Fourier transform. Note that we can
extend this definition to all m 苸 R if we take v 苸 C⬁共T兲⬘, the space of distributions on T.
Let us denote by S the set of points in T ⯝ 共S1兲3 where V has singularities. Let a 苸 R and
␳ : T → 关0 , 1兲 be a continuous function such that ␳共x兲 = 兩x − p兩 for x close to p 苸 S and such that ␳ is
smooth and ⬎0 on T \ S. The function ␳ will play an important role in what follows. We shall say
that a function F : T → R is smooth in polar coordinates around p 苸 S if F共␳x⬘兲 is a smooth
function of 共␳ , x⬘兲 苸 关0 , ⬁兲 ⫻ S2 in a generalized polar coordinate system 共␳ , x⬘兲 defined close to p.
A basic condition that we shall impose on a potential V is that ␳V be a smooth function in polar
coordinates near each singular point p 苸 S. A potential that satisfies this condition will be said to
satisfy Assumption 1. A typical Coulomb potential will thus satisfy Assumption 1.
The function ␳ is also needed in the definition of our weighted Sobolev spaces. Thus the mth
weighted Sobolev space with index a of periodic functions on R3 is given by
兩␣兩−a ␣
⳵ v 苸 L2共T兲, ∀ 兩␣兩 ⱕ m其.
Km
a 共T \ S兲 ª 兵v:T \ S → C, ␳
共4兲
The difference between the two classes of Sobolev spaces is thus only the appearance of the
weight function ␳, although this makes comparisons between the two spaces complicated, except
when m = 0. We discuss the relationship between these spaces further in Sec. V.
In this first paper of a series, we investigate the immediate consequences of the theory of
singular functions and totally characteristic differential operators for the study of Hk. In spite of
the obvious connections between singular functions and totally characteristic differential operators
on one side and Schrödinger operators of the form Hk on the other side, this approach seems not
to have been pursued before. The first results of this paper are a pair of regularity estimates. The
first theorem is a regularity result for the eigenfunctions of Hk itself.
Theorem I.1: Let us assume that ␳V is a smooth function of ␳ and x⬘ 苸 S2 in polar coordinates near each singular point p 苸 S , that is, V satisfies Assumption 1. Let us fix k 苸 R3 , m
苸 Z+ , and ⑀ ⬎ 0. Let Hku = ␭u, u 苸 L2共T兲, be an eigenvalue of Hk. Then
m
u 苸 H5/2−⑀共T兲 艚 K3/2−
⑀共T \ S兲.
共5兲
Moreover, for any p 苸 S, the eigenfunction u has a complete Taylor-type expansion in ␳ near p
with coefficients in smooth functions on the 2-sphere S2, where the degree zero term ␾0 is a
constant function. More precisely,
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Periodic Schrödinger operators
J. Math. Phys. 49, 083501 共2008兲
N
⬁
5/2+N−⑀
u共p + ␳x⬘兲 − 兺 ␳k␾k共x⬘兲 苸 K5/2−N−
共V p兲
⑀共V p \ S兲 傺 H
共6兲
k=0
in a small neighborhood V p of p , for suitable ␾k 苸 C⬁共S2兲 ; that is, u is smooth in polar coordinates 共␳ , x⬘兲 near each p.
Let H⬘ be the hydrogen atom Schrödinger operator. Then the ground state u0 of H⬘ is given by
m
3
u0 = Ce−c␳ for suitable constants C and c. Consequently, u0 苸 H5/2−⑀共R3兲 艚 K3/2−
⑀共R 兲 for any ⑀
m
5/2
3
3
⬎ 0, but u0苸
” H 共R 兲 and u0苸
” K3/2共R 兲. This suggests that our results are optimal. The above
theorem can also be used to rigorously justify the usual expansion of the eigenfunctions of the
hydrogen atom in terms of spherical harmonics. It is the generalization of this expansion in terms
of spherical harmonics for more general potentials that are not radially symmetric. We anticipate
that this result can be used to obtain a generalization of Kato’s “cusp theorem.”8
In practice, to estimate the eigenvalues and their eigenfunctions for many electron atoms or
molecules numerically, it is common to use the Hartree–Fock method or the density-functional
method.9 The Hartree–Fock method involves some recursively defined potentials in equations
similar to those considered in Theorem I.1. However, the recursively defined potentials are not as
regular as the potential in the original equation. Our second regularity result concerns eigenfunctions of Schrödinger operators with these slightly less regular potentials arising in the iterations of
the Hartree–Fock and density-functional methods. Before we can state it, we need to make some
definitions.
Let us denote by
W⬁,⬁共T \ S兲 ª 兵v 苸 C⬁共T \ S兲, ␳兩␣兩⳵␣v 苸 L⬁共T \ S兲, ∀ ␣ 苸 Z+3其,
共7兲
where S 傺 T is the set of points where the potential V may have singularities, as before. Note that
Coulomb-type potentials satisfy ␳V 苸 W⬁,⬁共T \ S兲. This property of V is referred to in this paper by
saying that V satisfies Assumption 2. Clearly, a potential satisfying Assumption 1 will also satisfy
Assumption 2.
Now we can state our regularity results for the eigenfunctions of Schrödinger operators as
above, but with Coulomb-type potentials instead of the more regular potentials considered in
Theorem I.1. The next theorem also guarantees that the potentials that arise in the Hartree–Fock
method are always of this form.
Theorem I.2: Let us assume that ␳V 苸 W⬁,⬁共T \ S兲 , that is, the potential V satisfies Assumption 2. Let us fix k 苸 R3 , m 苸 Z+ , and ⑀ ⬎ 0. Let Hku = ␭u , u 苸 L2共T兲 , be an eigenvalue of Hk.
Then
m
⬁,⬁
共T \ S兲.
u 苸 H2共T兲 艚 K3/2−
⑀共T \ S兲 艚 W
共8兲
⬁,⬁
Further, for any such u, ⌬ 兩u兩 苸 W 共T \ S兲.
Again, in the case of the hydrogen Schrödinger operator on R3, the fact that the domain is H2
and that eigenfunctions have H5/2−⑀ regularity are well known 共see Ref. 10, for example兲.
Note that in the Hartree–Fock and density-functional methods, one constructs iteratively a
sequence of potentials Vn as follows. We first find the first occupied energy levels and eigenfunctions corresponding to the potential Vn. Then, to the Coulomb potential, we add all terms of the
form ⌬−1兩u兩2, with u ranging through the determined set of eigenfunctions of the Schrödinger
operator for Vn. If ␳Vn 苸 W⬁,⬁共T \ S兲, then Theorem I.2 guarantees that Vn+1 will satisfy the same
assumption. By induction, all potentials appearing the Hartree–Fock and density-functional methods will satisfy this assumption. The same theorem then gives that all the eigenvalues of these
m
operators satisfy the regularity assumption u 苸 H2共T兲 艚 K3/2−
⑀共T \ S兲 needed in our first approximation result. At this point, we have only that the original eigenfunctions, that is, those for
potentials satisfying Assumption 1 below, will have the regularity required for our second approximation result.
In the proof of the above two theorems, we use the following lemma, which implies that the
operators Hk have many eigenvalues.
−1
2
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083501-4
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Hunsicker, Nistor, and Sofo
Theorem I.3: Assume that ␳V 苸 W⬁,⬁共T \ S兲. Then the Hamiltonians Hk define unbounded,
self-adjoint operators on L2共T兲 with domain H2共T兲 and with compact resolvent. In particular,
L2共T兲 has an orthonormal basis consisting of eigenfunctions of Hk.
Our second set of main results is a pair of approximation theorems which, by Theorems I.1
and I.2, apply to the eigenfunctions we want to compute. Theorem I.1 provides a priori estimates
for the eigenfunctions of Hk, which can then be used to approximate them using one of the
standard discretization techniques.11–15 The fact that in the usual Sobolev spaces we obtain limited
regularity 关only H5/2−⑀共T兲兴 is a bad thing for the standard finite element approximation.16,17 In
particular, we expect to obtain pollution effects when quadratic elements on quasiuniform meshes
are used. This is also likely to slow down any implementation using plane waves 共or trigonometric
polynomials兲.
On the other hand, the fact that we get unlimited regularity in the weighted Sobolev spaces
m
K3/2−
⑀共T \ S兲 means that, for 3 / 2 − ⑀ ⬎ 1, we can use the standard techniques developed in Refs.
18–23 to define a sequence of meshes that is graded toward the singular points to recover the
optimal approximation property. Our first approximation theorem uses such an approach.
For any mesh 共or tetrahedralization兲 T of T, let us denote by S共T , m兲 the usual finite element
space based on polynomials of order m. More precisely, S共T , m兲 will consist of those continuous
functions on T that on each tetrahedron T of T coincide with a polynomial of degree m. For any
fixed mesh T, we shall denote by uI,T,m 苸 S共T , m兲 the degree m Lagrange interpolant of u. We shall
construct a sequence of meshes 共or tetrahedralizations兲 Tn with the following quasioptimal approximation property with respect to the spaces Sn = S共Tn , m兲.
m+1
2
Theorem I.4: Let us assume that ␳V 苸 W⬁,⬁共T \ S兲. Let u 苸 K3/2−
⑀共T \ S兲 艚 H 共T兲 , where ⑀
1
苸 共0 , 2 兲 is fixed. Then there exists sequence Tn of tetrahedralizations of T ,
储u − uI,Tn,m储K1共T\S兲 ⱕ C dim共Sn兲−m/3共储u储Km+1
1
3/2−⑀
共T\S兲
+ max 兩u共p兲兩兲,
p苸S
共9兲
where m is the degree of polynomials we used in the approximation and Sn = S共Tn , m兲 is an
increasing sequence of spaces of dimensions ⬃23n.
Another approach, closer to physical intuition, is to approximate using elements of spaces
generated by plane waves. To implement this, first, let SN be the vector space generated by the
wave functions eı␰·r satisfying ␰ 苸 Lⴱ and 兩␰兩 ⱕ N. Then dim SN ⬃ N3. The first result in our second
approximation theorem uses these spaces. However, we can improve the exponent in the theorem
if we enlarge the space by including “orbital functions.”24 This method is analogous to that of
including singular functions in the finite element spaces on polygons.16
To do this, first we need some notation. Let r p be the distance function to p 苸 S. 共So ␳ = r p
close to p.兲 Let YN,a be the space spanned by spherical harmonics corresponding to eigenvalues ␭
of the Laplace operator on S2 satisfying 兩␭兩 ⱕ Na, a ⬎ 0. Finally, let ␹ p be a smooth cutoff function
that is equal to 1 in a neighborhood of p but is equal to 0 in a neighborhood of any other point
p 苸 S. The orbital functions with which we will enrich our space SN will be of the form w
= 兺 p,j,kr pjY k␹ p for Y 苸 YN,a and 1 ⱕ j ⱕ l, and will form a space denoted Ws. So define SN,a,l
⬘ to be
the vector space of functions of the form v + w, where v 苸 SN and w 苸 Ws is as above. By Weyl’s
theorem for S2, we have that the number of eigenvalues 共with multiplicity兲 less than ␭ is asymptotically given by C␭. For this reason, we will choose to work with a ⱕ 3 here so that for
sufficiently large N, we get dim共SN,a,l
⬘ 兲 ⱕ CN3, which is the same estimate we get for dim共SN兲.
Theorem I.5: Assume that ␳V is a smooth function of ␳ and x⬘ 苸 S2 in the neighborhood of
each point p 苸 S (that is, V satisfies Assumption 1). Let u be an eigenfunction of Hk. Also, let PNu
and PN⬘ u denote the projections of u onto SN and SN,3,l
⬘ , respectively, in the H1共T兲 -norm. Then
there exists C ⬎ 0 such that
储u − PNu储H1共T兲 ⱕ C dim共SN兲−共3−2⑀兲/6储u储H5/2−⑀共T兲 ,
but, in general, no exponent less than − 21 will satisfy this relation. On the other hand, there exists
a continuous norm p on AE共T \ S兲 艚 H5/2−⑀共T兲 such that, for a ⱕ 3 ,
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Periodic Schrödinger operators
⬘ 兲−共l+3/2−⑀兲/3 p共u兲.
储u − PN⬘ u储H1共T兲 ⱕ Ca dim共SN,a,l
The space AE共T \ S兲 here consists of functions with good expansions near S, and the norm p
will be described in the proof later. A complete discussion of this result would take us too far afield
from the contents of this paper, so below we will prove only the case when l = 1. Further results
along these lines will be included in a future paper.
A different kind of attempt to improve the approximation properties of the eigenvalues, using
finite differences and elementary methods, was made by Modine et al.25 Many authors have
studied the regularity properties of the eigenfunctions of Schrödinger operators, including Refs. 8
and 26–28.
Let us describe now the contents of the paper. In Sec. II we review the necessary results on the
regularity of the Laplace operator in weighted Sobolev spaces. Our approach is based on the
algebra of b-pseudodifferential operators. In Sec. III we prove Theorems I.1 and I.3, as well as the
necessary mapping and Fredholm of properties of Hk and of other related operators. In Sec. VI, we
prove Theorem I.2 and we give an application to the Hartree–Fock method. Finally, in Sec. V we
introduce our sequence of meshes and prove Theorems I.4 and I.5, including the necessary intermediate approximation results.
We hope to implement the results of this paper into currently used DFT codes.6
II. PRELIMINARIES ON DIFFERENTIAL OPERATORS
We have found it convenient in this paper to use the b-calculus of pseudodifferential operators
developed by Melrose29 in his book. 共See also Refs. 30–32 and the references therein. See also
Ref. 33 for another application to a problem inspired from physics.兲 The b-calculus can be used to
provide a rigorous mathematical foundation to the study of differential equations containing terms
of the forms ␳⳵␳ and ␳2⳵␳2, which are well known to occur in the determination of the eigenfunctions of the Schrödinger operator associated with the hydrogen atom. 共Generalizations can be
handled as well.兲 Thus in this section we will summarize the definitions and results from this
theory that we will use to prove our analytic results and refer the reader to that excellent book for
further details. We will use the theory to prove most of Theorem I.1. Since applying the theory in
this case is essentially the same as applying it in the case of the Laplace operator, we will also take
the opportunity to formulate the needed consequences for the Laplace operator.
A. Definition of b-differential operators
¯ be a smooth manifold with boundary ⳵ M
¯ . We shall denote by M ª M
¯ \ ⳵M
¯ its interior.
Let M
¯
Let ␳ be a smooth function on M on M that in a neighborhood of the boundary equals the distance
¯ and such that ␳ ⬎ 0 on M. We start by being more specific about the types of differential
to ⳵ M
equations that the b-calculus is designed to study. These involve a specific class of differential
¯ called b -differential operators.
operators on the interior M of M
¯ be a smooth manifold with boundary. A b -differential operator of
Definition II.1: Let M
¯ is a differential operator of degree m on M that, near ⳵ M
¯ , has
degree m on the interior M of M
the form
m
P = 兺 A j共␳兲共␳⳵␳兲 j ,
共10兲
j=0
¯ of degree ⱕ共m − j兲 for all ␳ , and this family is smooth
where A j共␳兲 is a differential operator on ⳵ M
¯.
up to ␳ = 0 , i.e., up to ⳵ M
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B. Examples
¯ = 兵兩x兩 ⱕ 1其, the closed unit ball in R3. If r
An example closely related to our applications is M
denotes the distance function to the origin, then we choose ␳ = 1 − r close to the boundary and we
smooth it near the origin. Yet another example, even more closely related to our work, is when
¯ = 关0 , ⬁兲 ⫻ S2, with 兵0其 ⫻ S2
M = R3 \ 兵0其, that is, the space with one point removed such that M
covering the origin. In this second example, we take the same function ␳ as we considered in Sec.
I. Let ⳵ j, j = 1 , 2 , 3, be the three partial derivatives on R3, which we restrict to M ª R3 \ 兵0其.
Recall that we have introduced the three dimensional torus T = R3 / L, where L ª 兵n1v1
+ n2v2 + n3v3其 is the lattice of orbits of the potential V. Also, recall that S 傺 T is the set of points
where V may have singularities. There are two ways we can complete the open manifold T \ S to
a compact manifold. One, clearly, is by putting back the set S to recover T. In this paper, instead
of this, we will add a spherical boundary around each point of S, thus “stretching out” these
¯ , where the boundary ⳵ M
¯ is a disjoint
punctures. We call the resulting manifold with boundary M
S
S
set of 2-spheres, one around each point in S. Formally, this space is the disjoint union
¯ ª 共T \ S兲 艛 共S2 ⫻ S兲.
M
S
共11兲
¯ is defined to be isometric to the product S2 ⫻ 关0 , ⑀兲, ⑀
Close to the boundary, our manifold M
S
¯ corresponds to taking spherical coordinates
⬎ 0. Essentially, the construction of the manifold M
S
around each of the singular points. So again, we may use for ␳ in the b-calculus the function ␳
defined in Sec. I, that is, a smoothed distance to the set of singular points S.
When we change the coordinates from rectangular coordinates xi with weighted rectangular
differentials ␳⳵ j to spherical coordinates and spherical differentials, we obtain exactly the coordinates ␳, ␪, and ␾ standard for spherical coordinates, and the differentials ␳⳵␳, ⳵␪, and ⳵␾, which are
¯ . Thus, for instance, we can restate
the building blocks of the set of b-differential operators on M
S
m
⬁,⬁
our definitions of W 共T \ S兲 and Ka 共T \ S兲, a 苸 N, to involve b-differential operators of degree a
rather than strings of differentials ␳兩␣兩⳵␣, where 兩␣兩 = a.
Let P = 兺兩␣兩ⱕma␣共x兲⳵␣, ⳵␣ ª 兿 j⳵␣j j, a␣ 苸 C⬁共Rn兲 be a differential operator of order m on Rn.
Here we use the notation 兩␣兩 = 兺nj=1␣ j for ␣ 苸 Z+n . Recall that the principal symbol of P is the
smooth function
␴共P兲共␰兲 =
兺 兿j ␰␣j .
j
兩␣兩ⱕm
共12兲
The definition of the principal symbol extends to an arbitrary differential operator P on a
smooth manifold M, so that the principal symbol ␴共P兲 becomes a smooth function on TⴱM, the
cotangent bundle of M. A b-differential operator P = 兺mj=0A j共␳兲共␳⳵␳兲 j is called b -elliptic if, and
only if, the principal symbol of the related operator
m
P⬘ = 兺 A j共␳兲⳵␳j
j=0
is elliptic up to ␳ = 0.
Let us now see how the Laplace operator fits into this setting. The differentials ⳵ j ª ⳵ / ⳵x j
descend to differentials on the quotient T ª R3 / L. Let ␳ : T → 关0 , 1兴 be equal to the distance to p,
close to p 苸 S, and be smooth and strictly positive otherwise 共that is, ␳ is a version of the
smoothed distance to S that we also used earlier兲. Then operators ␳m⳵␣ are typical examples of
¯ , provided that 兩␣兩 ⱕ m. These are the main examples of
b-differential operators on M
S
b-differential operators that will concern us. The following result is well known.
Lemma II.2: D ª −␳2⌬ is a b-differential operator.
Proof: Let ⌬S2 denote the 共negative definite兲 Laplace operator 共or, more precisely, the
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Periodic Schrödinger operators
Laplace–Beltrami operator兲 on the round 2-sphere. Then, in polar coordinates near one of the
singular points p 苸 S, we have
⌬ = ␳−2共共␳⳵␳兲2 + ␳⳵␳ + ⌬S2兲.
共13兲
䊐
This completes the proof.
C. Indicial family and regularity results
A very important invariant associated with a b-differential operator P is its indicial family
ˆP共␶兲 defined by
m
Pˆ共␶兲 ª 兺 A j共0兲␶ j
j=0
m
if
P = 兺 A j共␳兲共␳⳵␳兲 j .
共14兲
j=0
The operator Pˆ共␶兲 is the Mellin transform of 兺mj=0A j共0兲共␳⳵␳兲 j. The Mellin transform M共f兲共s兲
= 兰⬁0 x−s−1 f共x兲dx is a generalization of the Fourier transform, which justifies the notation Pˆ共␶兲 for
the indicial family of P. For example,
ˆ 共␶兲 = ␶2 + ␶ + ⌬ 2 .
−D
S
共15兲
Note that in some texts ␶ is replaced with ı␶ in the definition of indicial operators.
With the help of the indicial family of a b-differential operator P, we can study the Fredholm
and regularity properties of P with respect to the weighted b-Sobolev spaces defined by the
following.
¯ with boundary and if ␳ measures distance
Definition II.3: If M is the interior of a manifold M
to the boundary, we define
再
¯
Hm共M兲 = Hm
b 共M 兲 = u:M → C,
冕
冎
兩Pu兩2dvolb ⬍ ⬁ ,
M
共16兲
for all b-differential operators P of degree ⱕm , where near ⳵ M, dvolb = 共d␳ / ␳兲dvol⳵M¯ and, away
¯ , dvol is any smooth volume form. Then the weighted b-Sobolev spaces are the spaces
from ⳵ M
b
¯ 兲 = 兵u:M → C,u = ␳␥v, v 苸 Hm共M兲其.
␳␥Hmb 共M
b
共17兲
¯
We will use later the obvious fact that multiplication by ␳␣ is an isomorphism from ␳␥Hm
b 共M 兲
m ¯
␥ m
␥
to ␳ Hb 共M 兲. We endow the space ␳ Hb 共M兲 with the metric 储␳ u储␳␥Hm共M兲 = 储u储Hm共M兲. It is then
b
b
known, but not obvious, that the b-Sobolev spaces defined above can be identified with the
Sobolev spaces defined in Sec. I by
␥+␣
a−3/2 m
Km
Hb 共T \ S兲.
a 共T \ S兲 = ␳
共18兲
It is important to have available both definitions of these weighted Sobolev spaces, as some of
their properties are easier to prove in one formulation, whereas others are easier to prove in the
other formulation.
These weighted Sobolev spaces have a number of other well known nice properties, which
can be found in Melrose’s29 book 共see also Refs. 34–37兲. The first lemma states that b-differential
operators define bounded maps between the weighted b-Sobolev spaces ␳␥Hsb共M兲.
Lemma II.4: If P is a b -differential operator of degree m , then P : ␳␥Hsb共M兲 → ␳␥Hs−m
b 共M兲 is
a bounded map for all ␥ and a.
The next lemma gives the Sobolev embedding and Rellich compactness properties for maps
between these Sobolev spaces.
Lemma II.5: The inclusion map ␳␣Hsb共M兲 → ␳␥Hrb共M兲 is continuous if ␣ ⱖ ␥ and s ⱖ r. It is
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Hunsicker, Nistor, and Sofo
compact if ␣ ⬎ ␥ and s ⬎ r.
The Fredholm and mapping properties for a b-elliptic, b-differential operator P are studied by
constructing various sorts of parametrices for P acting between weighted spaces. The first “small
parametrix” Q that is constructed corresponds essentially to the properly supported parametrix
coming from the standard theory of pseudodifferential operators such as in Refs. 38 and 39.
Without entering further into technical details, let us just note that Q is a pseudodifferential
operator whose symbol is the inverse of that of P.
Theorem II.6: If P is an order m b -elliptic, b -differential operator on a manifold M that is
¯ , then there exists an operator Q on M with the
the interior of a manifold with boundary M
following properties.
共1兲
共2兲
The map Q : ␳␥Hsb共M兲 → ␳␥Hs+m
b 共M兲 is bounded for all s , ␥ 苸 R.
The remainder maps, R1 = 1 − QP and R2 = 1 − PQ , are bounded and smoothing, i.e., for all a
and ␥ the maps Ri : ␳␥Hab共M兲 → ␳␥Hsb共M兲 are well defined and continuous for all a , s , ␥ 苸 R.
Fredholm and more refined regularity results for a b-elliptic operator P as in Eq. 共10兲 are
determined by a subset Specb共P兲 傺 C ⫻ Z+ defined by
Specb共P兲 = 兵共z,k兲 苸 C ⫻ Z+, Pˆ共␶兲−1 has a pole of order k + 1 at z其.
共19兲
For the Laplace operator we have the following.
Lemma II.7: Taking D = −␳2⌬ as before, we obtain Specb共D兲 = Z ⫻ 兵0其.
Proof: The eigenvalues of −⌬S2 are of the form l共l + 1兲, l 苸 Z+. Let ␾l be an eigenfunction with
ˆ 共z兲 = 共−z2 − z − ⌬ 2兲. We have that D
ˆ 共z兲 is not invertible if
eigenvalue −l共l + 1兲. Then we know that D
S
and only if z共z + 1兲 = l共l + 1兲 for some l 苸 Z+, which implies either z = l or z = −l − 1. Moreover, the
roots are simple. This gives the desired result.
䊐
Now again considering a general elliptic b-differential operator P, define
specb共P兲 = 兵z 苸 C,共z,k兲 苸 Specb共P兲 for some k 苸 Z+其.
共20兲
specb共␳2⌬兲 = Z.
共21兲
In particular,
In terms of this set, the b-calculus tells us the following.
␥ k
Theorem II.8: An elliptic b -differential operator P : ␳␥Hk+m
b 共M兲 → ␳ Hb共M兲 is Fredholm if
and only if ␥ + ıc 苸 specb共P兲 for all c 苸 R. That is, in this case, there exists a continuous operator
Q␥ : ␳␥Hkb共M兲 → ␳␥Hk+m
b 共M兲 such that 1 − PQ␥ and 1 − Q␥ P are compact operators on their respective spaces.
Note that the operator Q␥ as above is a true parametrix, as opposed to the “small parametrix”
Q from Theorem II.6, which inverts P on these spaces only up to bounded smoothing operators,
which are not compact on a noncompact manifold such as T \ S. The operators Q␥ all can be taken
to play the role of Q in Theorem II.6, but they cannot be chosen to be independent of ␥. The
theory of the b-calculus, in fact, provides a much stronger regularity result than implied by
Theorem II.6. It is stated in terms of two more definitions.
Definition II.9: An index set is a subset E 傺 C ⫻ Z+ with the following three properties.
共1兲
共2兲
共3兲
E 艚 兵␣ ⱕ Re共z兲其 ⫻ Z+ is finite for any ␣ 苸 R.
If 共z , n兲 苸 E then so is 共z , k兲 for all 0 ⱕ k ⱕ n.
If 共z , n兲 苸 E then so is 共z + k , n兲 for all k 苸 Z+.
¯ with boundary, we say that a function
Then for a manifold M that is the interior of a manifold M
¯ if for each 共z , n兲 苸 E there exist a
f : M → C is polyhomogeneous with index set E near ⳵ M
z,n
⬁ ¯
苸 C 共⳵ M 兲 such that for all N,
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Periodic Schrödinger operators
gN ª f −
兺
共z,n兲苸E
¯ 兲,
␳z共log ␳兲naz,n 苸 ␳NCN共M
共22兲
Re共z兲 ⱕ N
¯ and vanishes at ⳵ M
¯ with all derivatives up
that is, gN is N times continuously differentiable on M
¯ is denoted
to order N. The space of all polyhomogeneous functions on M with index set E on ⳵ M
by AE共M兲.
Each of the functions ␳z共log ␳兲naz,n is called a singular function of u and Eq. 共22兲 is called the
singular function expansion of u. The singular function corresponding to Re共z兲 minimal is called
the first singular function of u.
Now we can state the refined regularity result for b-elliptic operators.
Theorem II.10: Let P be a b -elliptic operator on functions over M , and suppose that u
␣
苸 ␳␣H−⬁
b 共M兲 satisfies Pu = 0. Then u is polyhomogeneous with index set E , the smallest index set
containing all 共z , n兲 苸 Specb共P兲 with Re共z兲 ⱖ ␣.
An alternative approach to Theorem II.10, called the theory of “singular functions,” was
developed by Kondratiev40 in the framework of boundary value problems 共see also Refs. 35 and
41兲. Singular functions have a long history of applications in physics and engineering.
Corollary II. 11: The only possible index sets for D = −␳2⌬ are the sets of the form Es
ª 兵共n , 0兲 , n 苸 Z , n ⱖ s其.
Proof: This follows from Lemma II.7 and from the definition of index sets.
䊐
III. MAPPING AND FREDHOLM PROPERTIES: PROOFS OF THEOREMS I.1 AND
I.3
We now want to apply the tools from Sec. II to the Schrödinger operator
3
Hk ª − 兺 共⳵ j + ık j兲2 + V
共23兲
j=1
defined in Sec. I by Eq. 共2兲.
¯ = 共T \ S兲 艛 共S ⫻ S2兲 was introduced in Eq.
Recall that the compact manifold with boundary M
S
共11兲. Its interior identifies with T \ S and its boundary are a disjoint union of 2-spheres. The
assumptions on the potential V for the two main regularity theorems stated in Sec. I are as follows.
¯ 兲.
Assumption 1. We have ␳V 苸 C⬁共M
S
¯
That is, the function ␳V extends to a smooth function on the manifold with boundary M
S
introduced in Eq. 共11兲.
Assumption 2. We have ␳V 苸 W⬁,⬁共T \ S兲.
If V satisfies Assumption 1, then it also satisfies Assumption 2. This is simply because
⬁ ¯
C 共M S兲 傺 W⬁,⬁共T \ S兲. In most applications, V has a Coulomb-type singularity, in which case we
¯ 兲 and hence V satisfies Assumption 1 共the stronger assumption兲. As
have ␳V 苸 C⬁共T兲 傺 C⬁共M
S
mentioned in Sec. I, we need the weaker Assumption 2 for an application to the Hartree–Fock
method.
At a first reading, one may assume in this section that V satisfies Assumption 1, which is the
one needed for Theorem I.1. However, with an eye on the proof of Theorem I.2 in Sec. IV, we
show how some of our results generalize to the weaker case when V satisfies only Assumption 2.
The proofs in either case 共Assumption 1 or Assumption 2兲 come from the fact that our
operators Hk are perturbations of the Laplacian. In Sec. II, we saw how to obtain regularity results
using the b-calculus by considering the operator D = −␳2⌬. Let us therefore introduce the perturbation operators Bk,V,␭ by
冉兺
3
␳Bk,V,␭ ª ␳ 共Hk − ␭兲 − D = ␳
2
2
j=1
冊
共− 2ık j⳵ j + k2j 兲 + V − ␭ .
共24兲
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Hunsicker, Nistor, and Sofo
In the proofs of both Theorems I.1 and I.2 we will use the fact that Hk − ␭ is a “small”
perturbation of the Laplacian, in which it is a relatively compact perturbation. Since the Laplacian
⌬ is self-adjoint on L2共T兲 and has domain H2共T兲, this will imply 共see Theorem I.3兲 that Hk is also
self-adjoint on L2共T兲 with domain H2共T兲. This in turn implies that under either assumption on V,
all eigenfunctions of Hk will lie in H2共T兲.
In the proof of Theorem I.1, we will also use that the perturbation ␳Bk,V,␭ of D is small in the
b-calculus sense, that is, that it has order less than the order of D and adding it to D results in a
new b-differential operator with the same indicial family 关see 共iii兲 of Lemma III.1兴. We will thus
be able to get all the same regularity properties for Hk − ␭ as for ⌬ when V satisfies Assumption 1.
To prove Theorem I.2, we will no longer be able to use the b-calculus because in this
situation, Hk − ␭ is no longer a b-differential operator. We will still be able to use Theorem I.3, but
this is not sufficient to give the regularity needed for the Hartree–Fock method. Thus in addition
we will use a refined regularity result.
A. Proof of Theorem I.1
In this subsection, we will give the proof of Theorem I.1 modulo the proof of Theorem I.3.
Then we will prove a series of lemmas that culminate in the proof of this theorem as a corollary.
When V satisfies Assumption 1, it is not difficult to see that Hk − ␭ is a b-differential operator.
Lemma III.1: Assume that the potential V satisfies Assumption 1. Then
共1兲
共2兲
共3兲
the operator Bk,V,␭ of Eq. 共24兲 is a b -differential operator of order 1,
␳2共Hk − ␭兲 is an elliptic b -differential operator of order 2, and,
The indicial family of ␳2共Hk − ␭兲 is the same as that of D = −␳2⌬.
¯
Proof: Since V satisfies Assumption 1, 共i兲 follows from the fact that a smooth function on M
S
¯
¯
is a b-differential operator of order zero on M S and ␳⳵ j is a b-differential of order one on M S. 共ii兲
follows from 共i兲 using also Eq. 共24兲 and the fact that ␳2⌬ is an elliptic b-differential operator of
¯ . 共It has the same principal symbol as the Laplacian of a smooth metric on M
¯ .兲 共iii兲
order 2 on M
S
S
follows from the fact that the indicial family of ␳Bk,V,␭ is zero and from Eq. 共24兲 defining Bk,V,␭.䊐
Equipped with this, we can now give the proof of Theorem I.1. We first prove an apparently
weaker form of the theorem, under the additional assumption that u 苸 H2共T兲, using the concepts
introduced in Sec. II. Then in Theorem I.3, which we give at the end of this section, we show the
additional assumption that u 苸 H2共T兲 is automatically satisfied.
Theorem III.2: Assume that V satisfies Assumption 1 and fix k 苸 R3 and m 苸 Z+. Let Hku
= ␭u, u 苸 H2共T兲, be an eigenfunction of Hk. Then, in the neighborhood of each singular point p
苸 S, u is polyhomogeneous with index set E = Z+ ⫻ 兵0其 and its first singular function (the one
m
corresponding to ␳0 ) is constant. In particular u 苸 AE共T \ S兲 艚 H5/2−⑀共T兲 艚 K3/2−
⑀共T \ S兲 , for any
⑀ ⬎ 0.
Proof: By elliptic regularity away from the singular set S, we know that any eigenfunction u
of Hk is smooth on T \ S. Now we use the regularity theory from the b-calculus. We have from
Lemma III.1 that ␳2Hk is a b-differential operator whose indicial set is the same as the index set
of D, namely, Es = 兵共n , 0兲 , n ⱖ s其, by Corollary II.11. Theorem II.10 then gives that, for each
element p 苸 S, there exist smooth functions ␾k共␪兲 苸 C⬁共S2兲 with the following property. In a small
neighborhood V p of p, u has an asymptotic expansion of the form
⬁
u⬃
兺 ␳k␾k共␪兲.
k=−1
Recall that this means that for all N,
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Periodic Schrödinger operators
N
h N共 ␳ , ␪ 兲 ª u −
兺 ␳k␾k共␪兲 苸 C˙N共M¯ S 艚 Vp兲,
k=−1
¯ 艚 V 兲 is the space of all functions on M
¯ 艚 V , which are N times differentiable and
˙ N共M
where C
S
p
S
p
¯ . Thus we can also say that
which vanish together with all N derivatives up to order N at ⳵ M
S
N
˙ 共V 兲, that is, each such h is N-times differentiable near p 苸 S and vanishes with all of
h共␳ , ␪兲 苸 C
p
its derivatives up to order N at p.
Now we also know by Theorem I.3 that u 苸 H2共T兲. Thus we see that, in fact, the −1 term of
this series must vanish or ⳵␳u would not be in L2共T兲. Further, we know that 共Hk − ␭兲u = 0 since u
is an eigenfunction of Hk. Write
u = ␾ 0共 ␪ 兲 + ␳ ␾ 1共 ␪ 兲 + ␳ 2␾ 2共 ␪ 兲 + h 2共 ␳ , ␪ 兲
and apply 共Hk − ␭兲 to this. Then the leading order term in ␳ must in particular vanish 共since the
whole thing vanishes兲. This term is −␳−2⌬S2␾0, so we find further that ␾0 must be a constant
function.
It is standard that in R3, a function of the form ␳k␾k共␪兲 is in Hk+3/2−⑀共B共0 ; 1兲兲 共proven by a
combination of direct calculation and interpolation兲. Thus consider
N
hN共␳, ␪兲 = u − 兺 ␳k␾k共␪兲 = ␳N+1␾N+1 + ␳N+2␾N+2 + ␳N+3␾N+3 + hN+3共␳, ␪兲.
k=0
¯ 艚 V 兲 傺 HN+5/2−⑀共V 兲, and each ␳N+k␾ 苸 HN+k+3/2−⑀共B共0 ; 1兲兲, we
˙ N+3共M
Since hN+3共␳ , ␪兲 苸 C
S
p
p
N+k
have the better regularity hN共␳ , ␪兲 苸 HN+5/2−⑀共V p兲. Finally, since ␾0共␪兲 is constant, it is, in fact, in
C⬁共V p兲. Thus overall we may conclude that u 苸 H5/2−⑀共V p兲 around each p 苸 S, and u 苸 C⬁共T \ S兲,
䊐
so u 苸 H5/2−⑀共T兲.
Now we need to prove Theorem I.3. We start with several lemmas.
Lemma III.3: For any f 苸 W⬁,⬁共T \ S兲 , the multiplication map
m
a−3/2 m
Hb 共T \ S兲
Km
a 共T \ S兲 苹 u → fu 苸 Ka 共T \ S兲 = ␳
共25兲
is continuous for all m 苸 Z+ and all a 苸 R.
a−3/2 m
Proof: Recall that Km
Hb 共T \ S兲. Let us write D␳␣ = 共␳⳵␳兲␣1⳵␣⬘. Then the well
a 共T \ S兲 = ␳
␣ ␤ ␣−␤
␣
⳵ f ⳵ u extends to D␳␣共fu兲 = 兺␤ⱕ␣ D␳␤ fD␳␣−␤u. The result then
known formula ⳵␣共fu兲 = 兺␤ⱕ␣
␤
␤
¯ 兲, whereas D␤u 苸 L2共T \ S兲, by definitions.
follows from the fact that D␳␤ f 苸 L⬁共M
䊐
S
␳
Lemma III.4. Assume that the potential V satisfies Assumption 2. Let m , a 苸 R. Then
共兲
共i兲
共ii兲
共iii兲
共兲
m+1
m−1
共T \ S兲 to Ka−1
共T \ S兲 continuously,
the operator Hk − ␭ maps Ka+1
m+1
−1
the operator ␳ Bk,V,␭ maps Ka+1 共T \ S兲 to Km
a 共T \ S兲 continuously, and
m+1
m−1
␳−1Bk,V,␭ : Ka+1
共T \ S兲 → Ka−1
共T \ S兲 is compact.
Proof: First consider the case that V satisfies Assumption 1. Then 共i兲 follows from Lemma
III.1 共i兲 and 共ii兲 using also Lemma II.4. Next, we have ␳2共Hk − ␭ + ⌬兲 = ␳Bk,V,␭. Thus 共ii兲 follows
from Lemma III.1 共i兲 and Lemma II.4. 共iii兲 follows from 共ii兲 and the compactness of the embedm−1
ding Km
a 共T \ S兲 → Ka−1 共T \ S兲 共see Lemma II.5兲.
To prove that properties 共i兲–共iii兲 remain valid if V satisfies only Assumption 2, let us write
␳2共Hk − ␭ + ⌬兲 = ␳Bk,0,␭ + ␳共␳V兲. Then we use the results already proven for V = 0 and Lemma III.3,
䊐
which states that multiplication by ␳V is bounded on all spaces Km
a 共T \ S兲.
This gives us the following corollary.
m−1
Corollary III.5. Let V satisfy Assumption 2; then the map Bk,V,␭ : Km
a 共T \ S兲 → Ka 共T \ S兲 is
bounded for all m and a.
Proof: Assume V = 0 first. Then the result follows since Bk,0,␭ is a b-differential operator of
order 1. In general, the result follows from the decomposition Bk,V,␭ = Bk,0,␭ + ␳V and Lemma III.3,
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Hunsicker, Nistor, and Sofo
J. Math. Phys. 49, 083501 共2008兲
which says that multiplication by ␳V 苸 W⬁,⬁共T \ S兲 is bounded on all spaces Km
䊐
a 共T \ S兲.
Next, using the Fredholm results in Theorem II.8, we get the following.
m+1
m−1
共T \ S兲 → Ka−1
共T \ S兲 is FredLemma III.6: Let V satisfy Assumption 2. The map Hk − ␭ : Ka+1
1
” Z.
holm if and only if a − 2 苸
Proof: Lemma III.4 共iii兲 states that ␳−1Bk,V,␭ = 共Hk − ␭兲 − ␳−2D is compact. Since the property of
being Fredholm is preserved by adding a compact operator, it suffices to prove this theorem for
k = 0, ␭ = 0, and V = 0, that is, for −⌬. Let D ª −␳2⌬, as before. Thus
specb共P␭兲 = specb共␳2⌬兲 = Z,
共26兲
as determined in Sec. II 关Eq. 共21兲兴.
Note that this is independent of which point in S we expand around. Thus this is the correct
set for all the boundary components of T \ S. So we have that D is Fredholm for all ␥ 苸 Z.
Multiplying by −␳−2 to get ⌬ and rewriting in terms of the spaces introduced in Sec. I using Eq.
共18兲, we thus obtain the desired result for ⌬, and the full result follows when we add a compact
operator to get Hk − ␭. Note that the shift of 21 in weights is due to the factor ␳a−3/2 in Eq. 共18兲. 䊐
We shall need the following standard result, which we state for further use.
2
共T \ S兲 and that v
Lemma III.7: Let a 苸 R be arbitrary and assume that u 苸 K1+a
2
苸 K1−a共T \ S兲. Then 共⌬u , v兲 + 共ⵜu , ⵜv兲 = 0.
Proof: Let C共u , v兲 ª 共⌬u , v兲 + 共ⵜu , ⵜv兲. Then it is known that C = 0 if u , v 苸 C⬁c 共T \ S兲, simply
by integration by parts 共no boundary appears in our lemma兲. The result follows from the continuity
䊐
of C on the indicated spaces and from the density of C⬁c 共T \ S兲 in K2b共T \ S兲, b 苸 R.
This gives the following proposition, which is of independent interest.
m+1
m−1
共T \ S兲 → Ka−1
共T \ S兲 is an isomorphism for all
Proposition III.8: The operator 1 − ⌬ : Ka+1
兩a兩 ⬍ 1 / 2.
1
−1
共T \ S兲 → Ka−1
共T \ S兲, that is to
Proof: By regularity, we can assume m = 0. Let Da = 1 − ⌬ : Ka+1
ⴱ
say, 1 − ⌬ with fixed domain and range. Then Da = D−a.
By Lemma III.6, Da is Fredholm for 兩a兩 ⬍ 1 / 2. Since D0 is self-adjoint it has index zero.
Further, for such a, the family ␳aDa␳−a is a continuous family of Fredholm operators between the
same pair of spaces. Since the index is constant over such families, we have that ind共Da兲 = 0 for all
兩a兩 ⬍ 21 .
1
Let 21 ⬎ a ⱖ 0. The inclusion Ka+1
共T \ S兲 傺 K11共T \ S兲 allows us to compute 共Dau , u兲 = 共ⵜu , ⵜu兲
1
+ 共u , u兲 for u 苸 K1+a共T \ S兲, by Lemma III.7. Assume Dau = 0, then 共Dau , u兲 = 0, and hence u = 0.
m+1
m−1
共T \ S兲 → Ka−1
共T \ S兲 is injective for 0 ⱕ a ⬍ 21 . Since it is
This implies that the operator 1 − ⌬ : Ka+1
Fredholm of index zero, it is also an isomorphism. This proves our result for 0 ⱕ a ⬍ 21 .
For − 21 ⬍ a ⱕ 0, we take adjoints and use Da = 共D−a兲ⴱ. The proof is now complete.
䊐
Although we do not use this to prove Theorem I.3, it is useful to note here the following
result, which will be needed for the proof of Theorem I.2.
2
Corollary III.9: We have H2共T兲 傺 K3/2−
⑀共T \ S兲 for all ⑀ ⬎ 0.
Proof: If the result is true for ⑀, it will be true for all ⑀⬘ ⬎ ⑀. We can therefore assume ⑀
苸 共0 , 21 兲. We have L2共T兲 = K00共T \ S兲 and hence
0
0
H2共T兲 = 共1 − ⌬兲−1L2共T兲 傺 共1 − ⌬兲−1K−1/2−
⑀共T \ S兲 傺 K3/2−⑀共T \ S兲,
共27兲
where for the last result we have used Proposition III.8 for a = 21 − ⑀ 苸 共0 , 1 / 2兲.
䊐
Now we have the tools needed to prove that V is a relatively compact perturbation of ⌬.
Lemma III.10: Using the notation of Eq. (24), we have that the operator
␳−1Bk,V,0共1 − ⌬兲−1:L2共T兲 → L2共T兲
is well defined and is compact for all V satisfying Assumption 2.
0
Proof: We have L2共T兲 = K00共T \ S兲 傺 K−1/2−
⑀共T \ S兲. Next we use Corollary III.5 and Proposition
III.8 for ⑀ 苸 共0 , 1 / 2兲 to conclude that the maps
0
2
共1 − ⌬兲−1:K−1/2−
⑀共T \ S兲 → K3/2−⑀共T \ S兲,
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J. Math. Phys. 49, 083501 共2008兲
Periodic Schrödinger operators
2
1
␳−1Bk,V,0:K3/2−
⑀共T \ S兲 → K1/2−⑀共T \ S兲
1
are well defined and bounded. The result then follows from the compactness of K1/2−
⑀共T \ S兲
0
2
→ K0共T \ S兲 = L 共T兲 共Lemma II.5兲.
䊐
Finally, using standard results as in Refs. 42 and 43 we can complete the proof of Theorem
I.3.
Proof: 共Proof of Theorem I.3兲 We have that 1 + Hk = 共1 − ⌬兲 + ␳−1Bk,V,0. The proof is then
obtained from Lemma III.10 because ⌬ is self-adjoint with domain H2共T兲 and a relatively compact
perturbation preserves self-adjointness and the domain 共see Ref. 43, pp. 162, 163, and 340; in that
reference, the term “⌬-compact” is used兲. The self-adjointness of Hk shows that 共ı
+ Hk兲−1 : L2共T兲 → H2共T兲 is an isomorphism. The compactness of 共ı + Hk兲−1 : L2共T兲 → L2共T兲 then
䊐
follows from the fact that the inclusion H2共T兲 → L2共T兲 is compact.
Note that, in particular, the issue of the domain of our self-adjoint extension does not appear,
unlike the related case of conical manifolds.44
We are ready now to conclude the proof of Theorem I.1.
Proof: 共Proof of Theorem I.1兲 The only difference between the statements of Theorems I.1
and III.2 is that in the latter, we make the apparently stronger assumption that u 苸 H2共T兲. Let us
show that this stronger assumption is not really necessary. Indeed, if u 苸 L2共T兲 is such that Hku
䊐
= ␭u, then 共ı + Hk兲u = 共ı + ␭兲u 苸 L2共T兲. Therefore u = 共ı + ␭兲共ı + Hk兲−1u 苸 H2共T兲.
IV. PROOF OF THEOREM I.2
Although in the case of Theorem I.2, that is, where V satisfies only the weaker Assumption 2,
the operator Hk is no longer a multiple of a b-differential operator, we can nevertheless use the
weaker regularity theorem from the b-calculus, Theorem II.6, to get the following regularity result,
which applies to the case that V satisfies only Assumption 2.
0
共T \ S兲 is such that 共Hk − ␭兲u
Theorem IV.1: Assume ␳2V 苸 W⬁,⬁共T \ S兲. Suppose u 苸 Ka+1
m−1
m+1
苸 Ka−1 共T \ S兲 , then u 苸 Ka+1 共T \ S兲. Moreover, there exists a constant C ⬎ 0 such that
储u储Km+1 ⱕ C共储共Hk − ␭兲u储Km−1 + 储u储K0 兲.
a+1
a−1
共28兲
a+1
Proof: Let us take first V = 0 and P␭ ª ␳2共Hk − ␭兲, which is a b-differential operator, by Lemma
m−1
共T \ S兲. Now apply the small parametrix Q from
III.1. Our assumption means that v ª P␭u 苸 Ka+1
Theorem II.6 to both sides to get
m+1
共T \ S兲.
共1 − R1兲u = QP␭u = Qv 苸 Ka+1
共29兲
m+1
⬁
Rearranging Eq. 共29兲, we get u = Qv + R1u. Since R1u 苸 Ka+1
共T \ S兲, we have u 苸 Ka+1
共T \ S兲. A
more careful look at this rearrangement gives
储u储Km+1 ⱕ 储Qv储Km+1 + 储R1u储Km+1 ⱕ C共储P␭u储Km−1 + 储u储K0 兲,
a+1
a+1
a+1
a+1
a+1
共30兲
which is the desired inequality 共28兲.
Let us now take V with ␳2V 苸 W⬁,⬁共T \ S兲. Then ␳2共Hk − ␭兲 = P␭ + ␳2V. The rest of the proof is
based on the general fact that if P␭ satisfies a regularity property, then P␭ + ␳2V will also satisfy
that regularity property since ␳2V is of lower order, as we will explain next.
Let us recall this standard argument for the benefit of the reader. We proceed by induction on
m. For m = 0 this claim is automatically satisfied by the second part of the assumption. For m
m
共T \ S兲. Then by Lemma III.3, we get
ⱖ 1 we use the induction hypothesis to get that u 苸 Ka+1
m−1
m
m
2
␳Vu 苸 Ka+1共T \ S兲 and thus ␳ Vu 苸 Ka+2共T \ S兲. By assumption, 共Hk − ␭兲u 苸 Ka−1
共T \ S兲. Thus
m−1
m−1
m
2
2
P␭u = ␳ 共Hk − ␭兲u − ␳ Vu 苸 Ka+1 共T \ S兲 + Ka+2共T \ S兲 傺 Ka+1 共T \ S兲. We can then use the result for
m+1
共T \ S兲. The desired inequality follows
P␭ from the first part of the proof to conclude that u 苸 Ka+1
from Eq. 共30兲 as follows:
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Hunsicker, Nistor, and Sofo
储u储Km+1 ⱕ C共储P␭u储Km−1 + 储u储K0 兲 ⱕ C共储共P␭ + ␳2V兲u储Km−1 + 储u储Km 兲 ⱕ C共储共P␭ + ␳2V兲u储Km−1
a+1
a+1
a+1
a+1
a+1
+ 储u储K0 兲 = C共储共Hk − ␭兲u储Km−1 + 储u储K0 兲,
a+1
a−1
a+1
a+1
共31兲
where the last inequality is from the induction hypothesis and the last equality is by definition. The
proof is now complete.
䊐
Throughout the rest of this section, we shall assume that V satisfies Assumption 2, namely,
that ␳V 苸 W⬁,⬁共T \ S兲. For such a V, the stronger conditions of the previous result are obviously
satisfied.
The following lemma gives us our first piece of Theorem I.2.
m
Lemma IV.2: Let u 苸 L2共T兲 be such that Hku = ␭u. Then u 苸 H2共T兲 and u 苸 K3/2−
⑀共T \ S兲 for all
m 苸 Z+ and ⑀ ⬎ 0.
Proof: We have already seen in the proof of Theorem I.1 共at the conclusion of the previous
2
section兲 that u 苸 H2共T兲. Corollary III.9 gives then u 苸 K3/2−
⑀共T \ S兲. Since 共Hk − ␭兲u = 0
m
苸 K3/2−⑀共T \ S兲 for any m 苸 Z+, we can use the regularity theorem, Theorem IV.1, to conclude that
m
䊐
u 苸 K3/2−
⑀共T \ S兲.
In view of Lemma IV.2, to complete the proof of Theorem I.2, we need only to show that if
␳V 苸 W⬁,⬁共T \ S兲 and u is an eigenfunction of Hk, then u 苸 W⬁,⬁共T \ S兲 and ⌬−1兩u兩2 苸 W⬁,⬁共T \ S兲.
To even make sense of this last statement, we first have to define ⌬−1. We precompose with
the projection of u onto the orthogonal complement of 共1兲. The fact that the resulting map is well
defined comes from the following lemma, which is similar to Proposition III. 8.
Lemma
IV.3:
Let
兵1其⬜ ª 兵u 苸 Km
,
where
b ⬎ − 23 .
Then
b 共T \ S兲 , 兰Tudx = 0其
m+1
m−1
⬜
⬜
⌬ : Ka+1 共T \ S兲 艚 兵1其 → Ka−1 共T \ S兲 艚 兵1其 , m 苸 Z+, is an isomorphism for all 兩a兩 ⬍ 21 .
m+1
m−1
共T \ S兲 艚 兵1其⬜ → Ka−1
共T \ S兲 艚 兵1其⬜ the induced map. Lemma
Proof: Let us denote by ⌬a : Ka+1
m+1
共T \ S兲 艚 兵1其⬜
II.5 gives that the identity map defines a compact operator Ka+1
m−1
⬜
→ Ka−1 共T \ S兲 艚 兵1其 . Therefore ⌬a and −Da = ⌬ − 1 are compact perturbations of each other on the
indicated spaces. Therefore they have the same index 共when they are Fredholm兲. Proposition III.8
then shows that ⌬a has index zero for 兩a兩 ⬍ 21 .
Next we proceed as in the proof of Proposition III.8. Let us assume that 0 ⱕ a ⬍ 21 . Lemma
m+1
III.7 shows that 共⌬au , u兲 = −共ⵜu , ⵜu兲 for 0 ⱕ a ⬍ 21 , and hence ⌬a is injective on Ka+1
共T \ S兲 艚 兵1其⬜.
1
This shows that ⌬a is an isomorphism for 0 ⱕ a ⬍ 2 .
Finally, the relation Dⴱa = D−a shows that ⌬a is an isomorphism also for − 21 ⬍ a ⱕ 0. This
completes the proof.
䊐
We shall need the following improvements on Lemma IV.2. As above, let ␹ p be a smooth
function equal to 1 in the neighborhood of p 苸 S. We can assume that the supports of ␹ p are
disjoint. Let Vs be the linear span of the functions ␹ p. We fix ⑀ 苸 共0 , 21 兲 arbitrary.
1
m−1
m+1
共 1兲
Lemma IV.4: If v 苸 K3/2−
⑀共T \ S兲 and ⌬v 苸 K1/2−⑀共T \ S兲, ⑀ 苸 0 , 2 , then v 苸 Vs + K5/2−⑀共T \ S兲.
Proof: We shall use the notation and the results of Lemma IV.3 above. Let us notice that
m+1
共⌬v , 1兲 = 0. Hence v = ⌬−1共⌬v兲 + c, where c is a constant. Since c 苸 Vs + K5/2−
⑀共T \ S兲, it is enough to
m+1
m+1
m−1
show that ⌬−1共⌬v兲 苸 Vs + K5/2−⑀共T \ S兲. The map ⌬ : K5/2−⑀共T \ S兲 → K1/2−⑀共T \ S兲 is Fredholm, by
Lemma III.6. By the results of Ref. 29, for instance, the index of this map is given by the sum over
all boundary components of T \ S of minus the order of the pole of the indicial family of ⌬ at 23 .
This shows that each point in S contributes −1 to this index. Hence the index of ⌬ on these spaces
is equal to −dim Vs = − # S 共the number of elements of S兲. 共This also follows from the results of
m+1
Ref. 29, 35, and 41, or 45.兲 Our index calculation shows that the map ⌬ : Vs + K5/2−
⑀共T \ S兲
m−1
2
→ K1/2−⑀共T \ S兲 is still Fredholm, this time of index zero. Since K2共T \ S兲 傺 H2 the energy estimate
m+1
is still satisfied to show that ⌬ 关now acting on the space Vs + K5/2−
⑀共T \ S兲兴 still has kernel consisting of the multiples of the function 共1兲. We then obtain that
m+1
m−1
⬜
⬜
⌬:共Vs + K5/2−
⑀共T \ S兲兲 艚 兵1其 → K1/2−⑀共T \ S兲 艚 兵1其
m−1
m+1
is an isomorphism. Hence ⌬−1共K1/2−
⑀共T \ S兲兲 傺 Vs + K5/2−⑀共T \ S兲. As explained above, this is
enough to complete the result 共see Ref. 46 for a similar result兲.
䊐
We have the following embedding result.
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Periodic Schrödinger operators
J. Math. Phys. 49, 083501 共2008兲
⬁
⬁,⬁
Lemma IV.5: We have Vs + K5/2−
共T \ S兲, ⑀ 苸 共0 , 21 兲.
⑀共T \ S兲 傺 W
1
2
⬁
2
兩␣兩 ␣
Proof: For ⑀ 苸 共0 , 2 兲, we get K5/2−⑀ 傺 H . So if u 苸 K5/2−
⑀, then for any ␣, ␳ ⳵ 共u兲
2
2
⬁,⬁
苸 K5/2−⑀ 傺 H so all b-derivatives of u must be continuous, hence bounded. So u 苸 W 共T \ S兲. It
䊐
is also clear that Vs 傺 W⬁,⬁共T \ S兲. This completes the proof.
A more general result is proved in Ref. 34, showing that the right range of ⑀ for which the
above result holds true is ⑀ 苸 共0 , 1兲. The proof provided here is more elementary.
Theorem IV.6: Let V be such that ␳V 苸 W⬁,⬁共T \ S兲 (that is, V satisfies Assumption 2). Let
m
u 苸 L2共T兲 be such that Hku = ␭u. Then u 苸 K5/2−
⑀共T \ S兲 + Vs for all m 苸 Z+ and ⑀ ⬎ 0. In particular,
⬁,⬁
u 苸 W 共T \ S兲.
0
m−1
Proof: By hypothesis, u 苸 L2共T兲 = K3/2−
⑀共T \ S兲, and 共Hk − ␭兲u = 0 苸 K3/2−⑀共T \ S兲 for all m by
m
Theorem IV.1, so we have that u 苸 K3/2−⑀共T \ S兲 for all m. Then Lemma III.4 共ii兲 gives that
m−1
m
−1
␳−1Bk,V,␭u 苸 K1/2−
⑀共T \ S兲 for all m. We next apply Lemma IV.4 to ⌬u = ␳ Bk,V,␭u 苸 K1/2−⑀共T \ S兲 to
m
䊐
conclude that u 苸 K5/2−⑀共T \ S兲 for all m 苸 Z+.
The following shows that the potentials appearing in the iterations defined in the Hartree–
Fock and density-functional method iterations all satisfy Assumption 2. Hence the eigenfunctions
of the corresponding Schrödinger operators will all satisfy the regularity result of Theorem I.2.
Corollary IV.7: Let V and u be as in Theorem IV.6. Then ⌬−1兩u兩2 苸 W⬁,⬁共T \ S兲.
m
m
−1 2
Proof: We have 兩u兩2 = ¯uu 苸 W⬁,⬁共T \ S兲K3/2−
⑀共T \ S兲 傺 K3/2−⑀共T \ S兲 for all m. Hence ⌬ 兩u兩
⬁
⬁,⬁
苸 Vs + K5/2−⑀共T \ S兲 傺 W 共T \ S兲.
䊐
V. PROOFS OF APPROXIMATION RESULTS
We first address the proof of Theorem I.4. The proof of Theorem I.5 is shorter and is included
at the end. The first step in proving Theorem I.4 is to construct our sequence of tetrahedralizations.
These will be based on the tetrahedralizations constructed in Ref. 47; thus we refer the reader to
that paper for the details and here only give an outline and state the critical properties. The second
step is to prove a sequence of simple lemmas used in the estimates. The third step is to prove the
estimate separately on smaller regions. Our proof uses the scaling properties of the tetrahedralizations and the following important approximation result.48–51 To state this result, let us recall the
definition of the “degree m Lagrange interpolant.”
A. Lagrange interpolants and approximation
Let T = 兵T其 be a mesh on T, that is, a tetrahedralization of T with tetrahedra T. We can identify
this T with a mesh T ⬘ of the fundamental region of the lattice L 共that is, to the Brillouin zone of
L兲. We fix in what follows an integer m 苸 N that will play the role of the order of approximation.
We shall denote by S共T , m兲 the finite element space associated with the degree m Lagrange
tetrahedron. That is, S共T , m兲 consists of all continuous functions ␹ : T → R such that ␹ coincides
with a polynomial of degree ⱕm on each tetrahedron T 苸 T. We shall denote by uI = uI,T,m
苸 S共T , m兲 the Lagrange interpolant of u 苸 H2共T兲. Let us recall the definition of uI,T,m. First, given
a tetrahedron T, let 关t0 , t1 , t2 , t3兴 be the barycentric coordinates on T. The nodes of the degree m
Lagrange triangle T are the points of T whose barycentric coordinates 关t0 , t1 , t2 , t3兴 satisfy mt j
苸 Z. The degree m Lagrange interpolant uI,T,m of u is the unique function uI,T,m 苸 S共T , m兲 such that
u = uI,T,m at the nodes of each tetrahedron T 苸 T. The shorter notation uI will be used only when
only one mesh is understood in the discussion. The same definitions and concepts apply to any
polyhedral domain P 傺 T.
The proof of Theorem I.4 follows from the standard result of any of the following basic
references.48–51
Theorem V.1: Let T be a tetrahedralization of a polyhedral domain P 傺 T with the property
that all tetrahedra comprising T have angles ⱖ␣ and edges ⱕh. Then there exists an absolute
constant C共␣ , m兲 such that, for any u 苸 Hm+1共P兲,
储u − uI储H1共P兲 ⱕ C共␣,m兲hm储u储Hm+1共P兲 .
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B. Constructing the tetrahedralizations
We continue to keep the approximation degree m fixed throughout this section. Fix a parameter a 苸 共0 , 1 / 2兲 and let ␬ = 2−m/a. In our estimates, we will choose a such that 1 + a = 3 / 2 − ⑀. Let
l denote the smallest distance between points in S. Choose an edge length H ⬍ ␬l / 4 and an initial
tetrahedralization of T0 with tetrahedra with sides ⱕH such that all singular points of V 共i.e., all
points of S兲 are among the nodes of T0. Let ␣ be the minimum of all the angles of the tetrahedra
of T0.
For each point p 苸 S, we denote by V p the union of all tetrahedra of T0 that have p as a vertex.
Now subdivide these tetrahedra as follows. For any integer 1 ⱕ j ⱕ n, let V pj denote the union of
tetrahedra obtained by scaling the tetrahedra defining V p by a factor of ␬ j with center p.
An important step in our refinement is to set up a level k uniform refinement of an arbitrary
tetrahedron T. This refinement procedure will divide each edge of T into 2k equal segments. Let
关t0 , t1 , t2 , t3兴 be barycentric coordinates in T. Then our refinement is obtained by considering all
planes t j = l / 2k and ti + t j = l / 2k, where i and j are arbitrary indices in 兵0,1,2,3其 and 0 ⱕ l ⱕ 2k is an
arbitrary integer. By definition, the level k + 1 refinement of T is a refinement of the level k
refinement of T.
Let us apply now the level 1 of refinement to all tetrahedra comprising V p. Then we deform
the new nodes on the sides containing p to divide their edge in the ratio ␬. Then we deform all the
other edges and planes accordingly. This will give a new tetrahedralization of V p such that all
tetrahedra defining V p1 are contained in this new tetrahedralization. In particular, we have obtained
a tetrahedralization of R p1 ª V p \ V p1. Then we dilate this refinement to a similar refinement of
R pj ª V p共j−1兲 \ V pj.
We can now define the tetrahedralization 共or mesh兲 Tn of T. Define
P ª T \ 艛 p苸SV p .
共32兲
T = P艛 p苸S共艛nj=1R pj 艛 V pn兲.
共33兲
We first decompose T as the union
Then we apply to each tetrahedron T 苸 T0 contained in P the level n refinement. Then we apply to
each tetrahedron T 傺 R pj the level n − j refinement. The tetrahedra defining V pn are not refined. The
resulting tetrahedra 共including the ones defining V pn兲 define the tetrahedralization Tn. The fact that
Tn is a tetrahedralization follows from the way our refinement was performed. This and more
details of these constructions are given in Ref. 47.
Let us denote by uI,n = uI,Tn,m the degree m Lagrange interpolant of u associated with the mesh
Tn. By construction, the restriction of Tn to R pj scales to the restriction of Tn−j+1 to R p1. This gives
the following.
Lemma V.2: We have uI,n共x兲 = uI,n−j+1共␬−共j−1兲共x兲兲 , for all x 苸 R pj , where ␬−共j−1兲共x兲 ª p + 共x
− p兲 / ␬共j−1兲 is the dilation with ratio ␬−共j−1兲 and center p.
The size of each simplex of Tn contained in P is ⱖh2−n. Similarly, the size of each simplex of
Tn contained in R pj is ⱖC␬ j2−共n−j兲. All angles of the tetrahedra used in our meshes are bounded
uniformly from below by an angle ␣ ⬎ 0 共independent of the order of refinement n兲. This shows
that the volumes of the tetrahedra are ⱖCh2−3n for the tetrahedra in P. Similarly, the volumes of
the tetrahedra in R pj are ⱖC␬3j2−3共n−j兲. The constant C ⬎ 0 is independent of n or j. Since the
volume of R pj is ⱕC␬3j, we obtain that the total number of simplices of Tn is ⱕC23n. Hence
dim S共Tn,m兲 ⱕ C23n .
共34兲
C. Preliminary lemmas
None of the lemmas of this subsection is very difficult and analogous results have been proven
in various papers, but it is useful to have them collected here.
Lemma V.3: Let D be a small neighborhood of a point p 苸 S such that on D, ␳ is given by
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Periodic Schrödinger operators
distance to p. Let 0 ⬍ ␥ ⬍ 1 and denote by ␥D the region obtained by radially shrinking around p
by a factor of ␥. Then
储u储Km共D兲 = 共␥兲a−3/2储u储Km共␥D兲 .
a
a
Proof: Let x denote the coordinates on ␥D. For simplicity of notation, we will denote here by
⳵x a derivative with respect to any of these coordinates.
兺
2
储u储Km共␥D兲 =
a
储␳兩␣兩−a⳵x␣u储L2共␥D兲 =
2
兩␣兩ⱕm
兺
兩␣兩ⱕm
冕
␥D
␳2兩␣兩−2a共x兲兩⳵x␣u兩2dx.
Do the change of variables w = ␥−1x. Then ⳵x = ␥−1⳵w, ␥D becomes D, and dx becomes ␥3dw.
Further, since ␳共x兲 = distance to p, ␳共x兲 = ␥␳共w兲. Thus we get
兺
2
储u储Km共␥D兲 =
a
兩␣兩ⱕm
= 共␥
冕
共␥␳共w兲兲2兩␣兩−2a兩␥−兩␣兩⳵w␣ u兩2␥3dw = ␥3−2a
D
3/2−a
兺
兩␣兩ⱕm
冕
␳共w兲2兩␣兩−2a兩⳵w␣ u兩2dw
D
储u储Km共D兲兲 ,
共35兲
2
a
䊐
as required.
Lemma V.4: If m ⱖ m⬘ , a ⱖ a⬘ , and 0 ⬍ ␳ ⬍ ␦ on D , then
储v储Km⬘共D兲 ⱕ c共m兲␦a−a⬘储v储Km共D兲 .
a⬘
a
Proof: We have
2
储v储Km⬘共D兲 =
a⬘
兺
储␳兩␣兩−a⬘⳵␣v储L2共D兲 ⱕ
兩␣兩ⱕm⬘
冕
兺
2
兩␣兩ⱕm
兩␳a−a⬘␳兩␣兩−a⳵␣v兩2dx ⱕ
D
兺
兩␣兩ⱕm
冕
␦2共a−a⬘兲兩␳兩␣兩−a⳵␣v兩2dx
D
= ␦2共a−a⬘兲储v储Km共D兲 ,
2
共36兲
a
䊐
as stated.
Lemma V.5: For any m , if ␳ ⱕ b , then
储v储Km共D兲 ⱕ b␣储v储Hm共D兲 .
0
Proof: We compute
2
储v储Km共D兲 =
0
兺
兩␣兩ⱕm
冕
兩␳兩␣兩⳵␣v兩2dx ⱕ
D
兺
兩␣兩ⱕm
冕
D
b2兩␣兩兩⳵␣v兩2dx = b2␣储v储Hm共D兲 ,
2
which gives the result.
Lemma V.6: If 1 ⱖ ␳ ⬎ ␤ ⬎ 0 on D , and m ⱖ a , then
共37兲
䊐
储v储Hm共D兲 ⱕ ␦a−m储v储Km共D兲 .
a
Proof: We have
2
储v储Km共D兲 =
a
Since ␳ ⱕ 1 and 兩␣兩 ⱕ m, we get ␳
兩␣兩−a
ⱖ␳
兺
兩␣兩ⱕm
m−a
冕
兩␳兩␣兩−a⳵␣v兩2dx.
D
. Thus this is
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Hunsicker, Nistor, and Sofo
ⱖ
兺
兩␣兩ⱕm
冕
␳2共m−a兲兩⳵␣v兩2dx ⱖ
D
兺
兩␣兩ⱕm
冕
␦2共m−a兲兩⳵␣v兩2dx = ␦m−a储v储Hm共D兲 .
D
䊐
Lemma V.7: If 0 ⱕ ␳ ⱕ ␦ ⱕ 1 on D and a ⱖ m , then
储v储Hm共D兲 ⱕ ␦a−m储v储Km共D兲 .
a
Proof:
储v储Km共D兲 =
a
兺
兩␣兩ⱕm
冕
兩␳兩␣兩−a⳵␣v兩2dx.
D
Since a ⬎ m ⱖ 兩␣兩 and ␳ ⱕ ␦ ⱕ 1, we know ␳2兩␣兩−2a ⱖ ␦2兩␣兩−2a ⱖ ␦2m−2a; thus this is
ⱖ
兺
兩␣兩ⱕm
冕
D
␦2共m−a兲兩⳵␣v兩2dx = ␦2共m−a兲储v储H2 m共D兲 .
䊐
The proof is complete.
D. Breaking the estimates into regions and proof of Theorem I.4
Now we can prove Theorem I.4 stated in Sec. I. Recall that V p consists of the tetrahedra of the
initial mesh T0 that have p as a vertex and that all the regions V p are away from each other 共they
are closed and disjoint兲. We used this to define P ª T \ 艛 pV p. The region V pj is obtained by dilating
V p with the ratio ␬ j ⬍ 1 and center p. Finally, recall that R pj = V p共j−1兲 \ V pj. Let R be any of the
regions P, R pj, or V pn. Since the union of these regions is T, it is enough to prove that
储u − uI,Tn,m储K1共R\S兲 ⱕ C dim共Sn兲−m/3共储u储Km+1
3/2−⑀
1
共R\S兲
+ max 兩u共p兲兩兲.
p苸S艚R
共38兲
The result will follow by squaring all these inequalities and adding them up.
Since 2−nm ⱕ C dim共Sn兲−m/3, by Eq. 共34兲, it is enough to prove
储u − uI,Tn,m储K1共R\S兲 ⱕ C2−nm共储u储Km+1
3/2−⑀
1
共R\S兲
+ max 兩u共p兲兩兲.
p苸S艚R
共39兲
If R = P, the estimate of Eq. 共39兲 follows right away from Theorem V.1. For the other estimates, we need to choose 0 ⬍ ␬ ⱕ 2−m/a, where 1 + a = 3 / 2 − ⑀, with a 苸 共0 , 21 兲.
We next establish the desired interpolation estimate on the region R = R pj, for any fixed p
苸 S and j = 1 , 2 , . . . , n. Let w共x兲 = u共␬ j−1x兲. From Lemmas V.3 and V.2, we have
储u − uI,n储K1共Rpj兲 = 共␬ j−1兲1/2储w − wI,n−j+1储K1共Rp1兲 .
1
Now we can apply Theorem V.1 with h = 2
1
−共n−j+1兲
H to get that this is
ⱕC共␬ j−1兲1/22−m共n−j+1兲储w储Km+1共Rp1兲 .
1+a
Now applying Lemma V.3 to scale back again and using also ␬ = 2−m/a, we get that this last
quantity is equal to
=C共␬ j−1兲a2−m共n−j+1兲储u储Km+1共Rpj兲 ⱕ C2−mn储u储Km+1共Rpj兲 .
1+a
1+a
This proves the estimate of Eq. 共39兲 for R = R pj.
It remains to prove this estimate for R = V pn. Then on V pn, we have u = v + u共p兲, where v共p兲
= 0. For any function w on V pn, we let wn共x兲 = w共␬nx兲, a function on V p. For simplicity, below, we
shall denote all interpolants in the same way 关so uI = uI,n and un,I = 共un兲I,0兴. So
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Periodic Schrödinger operators
储u − uI储K1共Vpn兲 = 共␬n兲1/2储共u − uI兲n储K1共Vp兲
1
1
by scaling Lemma V.3 and
共␬n兲1/2储共u − uI兲n储K1共Vp兲 = 共␬n兲1/2储un − un,I储K1共Vp兲
1
1
by Lemma V.2 共which follows from the definition of the tetrahedralizations Tk and from the fact
that interpolation commutes with changes of variables兲. Then the decomposition u = v + u共p兲 gives
共␬n兲1/2储un − un,I储K1共Vp兲 = 共␬n兲1/2储共v + u共p兲兲n − 共v + u共p兲兲n,I储K1共Vp兲 = 共␬n兲1/2储vn + u共p兲 − vn,I
1
1
− u共p兲储
K11共V p兲
= 共␬ 兲 储vn − v 储
n 1/2
共40兲
n,I K11共V p兲
by properties of interpolation. Now let ␹ be a smooth cutoff function on V p which =0 in a
neighborhood of p and =1 at every other interpolation point of V p. Then
共␬n兲1/2储vn − vn,I储K1共Vp兲 ⱕ 共␬n兲1/2共储vn − ␹vn储K1共Vp兲 + 储␹vn − vn,I储K1共Vp兲兲
1
1
1
by the triangle inequality and
=共␬n兲1/2共储vn − ␹vn储K1共Vp兲 + 储␹vn − 共␹vn兲I储K1共Vp兲兲
1
1
by properties of interpolation, since ␹vn = vn at each interpolation point. This is
ⱕC共␬n兲1/2共储vn储K1共Vp兲 + 储␹vn − 共␹vn兲I储K1共Vp兲兲
1
1
by bounds on ␹ and its first derivatives. Now, since ␹vn is in H 共V p兲, we can apply the standard
result of Theorem V.1 and the fact that ␹vn is supported away from the singularity to get
m+1
ⱕC共␬n兲1/2共储vn储K1共Vp兲 + 储␹vn储Hm+1共Vp兲兲 ⱕ C共␬n兲1/2储vn储Km+1共Vp兲 = C共␬n兲a储v储Km+1共Vp兲
1
ⱕ C2
−mn
1+a
储v储Km+1共Vp兲 .
1+a
共41兲
1+a
m+1
Since constants are in K1+a
共R兲, we have that 储v储Km+1共Vp兲 ⱕ 储u储Km+1共Vp\S兲 + 兩u共p兲兩. This proves the
1+a
1+a
estimate of Eq. 共39兲 for R = V pn and completes the proof of Theorem I.4.
E. Proof of approximation Theorem I.5
We now address Theorem I.5. A much more detailed discussion of the following issues will be
included in the sequel to this paper, where also numerical tests will be included.
Proof: 共Proof of Theorem I.5兲 Let v 苸 Hs共T兲. Then the standard estimate of the growth of the
Fourier coefficients of v 关see Eq. 共3兲兴 gives that
储v − PNv储H1共T兲 ⱕ CN−共s−1兲储v储Hs共T兲 ,
共42兲
and no better estimate will hold for any v in Hs共T兲. Now consider u as in the hypotheses of the
theorem. By Theorem I.1, u 苸 H5/2−⑀共T兲. So putting the estimate above with the fact that
dim共SN兲 ⬃ N3 proves the first relation.
For the second relation, let us assume for simplicity that l = 1. We remove the order ␳ term
from the expansion for u in Theorem I.1 by choosing h p 苸 C⬁共S2兲 such that
vªu−
⬁
7/2−⑀
共T兲.
兺p ␳hp␹p 苸 K7/2−
⑀共T \ S兲 傺 H
Note that we did not have to subtract the first singular function ␾0 = 1 since it is already smooth
and that we may fix ␳ and ␹ p depending only on S 共if we choose differently we obtain an
equivalent norm p below兲.
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Hunsicker, Nistor, and Sofo
We then approximate v with elements in SN. By the same standard arguments as for the first
part of the theorem, this approximation is of order N5/2−⑀. Now we can approximate the h p with
linear combinations h pN of spherical harmonics such that 兺 p␳h pN␹ p 苸 Ws. This approximation is of
order k for any k since v is smooth in polar coordinates around any singular point. To be precise,
since each h p 苸 C⬁共S2兲, for s sufficiently large we have
储h p − h pN储H1共S2兲 ⱕ CN−a共s−1兲/2储h p储Hs共S2兲 ,
where a is the constant used to define the spaces SN,a,l
⬘ . Thus integrating, we get in a neighborhood
U p of each p 苸 S,
储␳␹ p共h p − h pN兲储H1共Up兲 ⱕ C储h p − h pN储H1共S2兲 ⱕ CN−a共s−1兲/2储h p储Hs共S2兲 .
Since for sufficiently large N the dimension of the enlarged space SN⬘ is dim共SN,3,1兲 ⱕ CN3
共from Weyl’s theorem on the asymptotics of eigenvalues as in Sec. I兲, all together we obtain
储u − PN⬘ u储H1共T兲 ⱕ 储v − PNv储H1共T兲 + 兺 储␳␹ p共h p − h pN兲储H1共T兲 ⱕ CN−共5/2−⑀兲储v储H7/2−⑀共T兲
p
⬘ 兲兲−共5−⑀兲/6 p共u兲,
+ CN−a共s−1兲 兺 储h p储Hs共S2兲 ⱕ C共dim共SN,3,1
共43兲
p
by choosing s ⬎ 5 / 2 − ⑀ and setting p共u兲 = 储v储H7/2−⑀共T兲 + 兺 p储h p储Hs共S2兲 for the decomposition given as
above.
䊐
The general case l ⱖ 1 is completely similar. In addition, we note that for potentials satisfying
Assumption 1, better 共i.e., smaller兲 approximation space than SN,a,l
⬘ could be used to approximate
the eigenfunctions u of Hk. This could be done by looking in more detail at the expansion of
Theorem I.1. Using the eigenfunction equation, supplemental information may be derived about
the functions ␾k共x⬘兲 in this expansion. Using this information would allow us to restrict to a
smaller space of supplementary orbital functions. This is similar to the technique used in the case
of the usual Schrödinger operator for the hydrogen atom.24 Again, this will be discussed in more
detail in a later paper.
ACKNOWLEDGMENTS
We would like to thank Hengguang Li and Anna Mazzucato for useful discussions. V.N. and
E.H. would like to thank the Max Planck Institute for Mathematics in Bonn, where part of this
work was completed. We also thank the referee for carefully reading our paper. V.N. was supported in part by NSF Grant Nos. DMS 0555831, DMS 0713743, and OCI 0749202. We are
grateful to Hundertmark for suggesting Refs. 8 and 26–28.
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