This contradicts the assumption, so m(a2 ) = 2r is even. Then by the multiplicity

lemma (50.7) we have ai (t) = ai,ir (t)tir (i = 2, . . . , n) for real analytic ai,ir , and we

may consider the following real analytic curve of polynomials

Pr (t)(x) = xn + a2,2r (t)xn’2 ’ a3,3r (t)xn’3 · · · + (’1)n an,nr (t)

with all roots real. If Pr (t) is real analytically solvable and xk (t) are its real analytic

roots then xk (t)tr are the roots of P (t), and the original curve P is real analytically

solvable too. Now a2,2r (0) = 0 and we are done by the case above.

Claim. Let x = (x1 , . . . , xn ) : I ’ Rn be a real analytic curve of roots of P on an

open interval I ‚ R. Then any real analytic curve of roots of P on I is of the form

± —¦ x for some permutation ±.

Let y : I ’ Rn be another real analytic curve of roots of P . Let tk ’ t0 be a con-

vergent sequence of distinct points in I. Then y(tk ) = ±k (x(tk )) = (x±k 1 , . . . , x±k n )

for permutations ±k . By choosing a subsequence, we may assume that all ±k are

the same permutation ±. But then the real analytic curves y and ± —¦ x coincide on

a converging sequence, so they coincide on I and the claim follows.

Now from the local smooth solvability above and the uniqueness of smooth solutions

up to permutations we can glue a global smooth solution on the whole of R.

50.13. Now we consider the following situation: Let A(t) = (Aij (t)) be a smooth

(real analytic, holomorphic) curve of real (complex) (n — n)-matrices or operators,

50.13

546 Chapter X. Further Applications 50.14

depending on a real (complex) parameter t near 0. What can we say about the

eigenvalues and eigenfunctions of A(t)?

In the following theorem (50.14) the condition that A(t) is Hermitian cannot be

omitted. Consider the following example of real semisimple (not normal) matrices

2t + t3 t

A(t) := ,

’t 0

t2 t2

t

±t 1+

1+

t2

»± (t) = t + ± t2 1+ 4, x± (t) = ,

2 4

2 ’1

where at t = 0 we do not get a base of eigenvectors.

50.14. Theorem. Let A(t) = (Aij (t)) be a smooth curve of complex Hermitian

(n — n)-matrices, depending on a real parameter t ∈ R, acting on a Hermitian space

V = Cn , such that no two of the continuous eigenvalues meet of in¬nite order at

any t ∈ R if they are not equal for all t.

Then the eigenvalues and the eigenvectors can be chosen smoothly in t, on the whole

parameter domain R.

Let A(t) = (Aij (t)) be a real analytic curve of complex Hermitian (n — n)-matrices,

depending on a real parameter t ∈ R, acting on a Hermitian space V = Cn . Then

the eigenvalues and the eigenvectors can be chosen real analytically in t on the whole

parameter domain R.

The condition on meeting of eigenvalues permits that some eigenvalues agree for

all t ” we speak of higher ˜generic multiplicity™ in this situation.

The real analytic version of this theorem is due to [Rellich, 1940]. Our proof is

di¬erent.

Proof. We prove the smooth case and indicate the changes for the real analytic

case. The proof will use an algorithm.

Note ¬rst that by (50.10) (by (50.12) in the real analytic case) the characteristic

polynomial

P (A(t))(») = det(A(t) ’ »I)

(1)

= »n ’ a1 (t)»n’1 + a2 (t)»n’2 ’ · · · + (’1)n an (t)

n

tr(Λi A(t))»n’i

=

i=0

is smoothly solvable (real analytically solvable), with smooth (real analytic) roots

»1 (t), . . . , »n (t) on the whole parameter interval.

Case 1: distinct eigenvalues. If A(0) has some eigenvalues distinct, then one

can reorder them in such a way that for i0 = 0 < 1 ¤ i1 < i2 < · · · < ik < n = ik+1

we have

»1 (0) = · · · = »i1 (0) < »i1 +1 (0) = · · · = »i2 (0) < · · · < »ik +1 (0) = · · · = »n (0).

50.14

50.14 50. Applications to perturbation theory of operators 547

For t near 0 we still have

»1 (t), . . . , »i1 (t) < »i1 +1 (t), . . . , »i2 (t) < · · · < »ik +1 (t), . . . , »n (t).

For j = 1, . . . , k + 1 we consider the subspaces

ij

(j)

{v ∈ V : (A(t) ’ »i (t))v = 0}.

Vt =

i=ij’1 +1

(j)

Then each Vt runs through a smooth (real analytic) vector subbundle of the

trivial bundle (’µ, µ) — V ’ (’µ, µ), which admits a smooth (real analytic) framing

k+1 (j)

eij’1 +1 (t), . . . , eij (t). We have V = j=1 Vt for each t.

In order to prove this statement, note that

(j)

= ker (A(t) ’ »ij’1 +1 (t)) —¦ . . . —¦ (A(t) ’ »ij (t)) ,

Vt

(j)

so Vt is the kernel of a smooth (real analytic) vector bundle homomorphism B(t)

of constant rank (even of constant dimension of the kernel), and thus is a smooth

(real analytic) vector subbundle. This together with a smooth (real analytic) frame

¬eld can be shown as follows: Choose a basis of V , constant in t, such that A(0)

is diagonal. Then by the elimination procedure one can construct a basis for the

kernel of B(0). For t near 0, the elimination procedure (with the same choices)

gives then a basis of the kernel of B(t); the elements of this basis are then smooth

(real analytic) in t for t near 0.

From the last result it follows that it su¬ces to ¬nd smooth (real analytic) eigen-

vectors in each subbundle V (j) separately, expanded in the smooth (real analytic)

frame ¬eld. But in this frame ¬eld the vector subbundle looks again like a constant

vector space. So feed each of these parts (A restricted to V (j) , as matrix with

respect to the frame ¬eld) into case 2 below.

Case 2: All eigenvalues at 0 are equal. So suppose that A(t) : V ’ V is

Hermitian with all eigenvalues at t = 0 equal to a1n , see (1).

(0)

Eigenvectors of A(t) are also eigenvectors of A(t)’ a1n I, so we may replace A(t) by

(t)

A(t) ’ a1n I and assume that for the characteristic polynomial (1) we have a1 = 0,

(t)

or assume without loss that »i (0) = 0 for all i, and so A(0) = 0.

If A(t) = 0 for all t we choose the eigenvectors constant.

(1)

Otherwise, let Aij (t) = tAij (t). From (1) we see that the characteristic polynomial

of the Hermitian matrix A(1) (t) is P1 (t) in the notation of (50.8), thus m(ai ) ≥ i

for 2 ¤ i ¤ n, which also follows from (50.5).

The eigenvalues of A(1) (t) are the roots of P1 (t), which may be chosen in a smooth

way, since they again satisfy the condition of theorem (50.10). In the real analytic

case we just have to invoke (50.12). Note that eigenvectors of A(1) are also eigen-

vectors of A. If the eigenvalues are still all equal, we apply the same procedure

50.14

548 Chapter X. Further Applications 50.16

again, until they are not all equal: we arrive at this situation by the assumption of

the theorem in the smooth case, and automatically in the real analytic case. Then

we apply case 1.

This algorithm shows that one may choose the eigenvectors xi (t) of Ai (t) in a

smooth (real analytic) way, locally in t. It remains to extend this to the whole

parameter interval.

If some eigenvalues coincide locally then on the whole of R, by the assumption. The

corresponding eigenspaces then form a smooth (real analytic) vector bundle over

R, by case 1, since those eigenvalues, which meet in isolated points are di¬erent

after application of case 2.

(j) (j)

So we we get V = Wt where the Wt are real analytic sub vector bundles of

V —R, whose dimension is the generic multiplicity of the corresponding smooth (real

analytic) eigenvalue function. It su¬ces to ¬nd global orthonormal smooth (real

analytic) frames for each of these; this exists since the vector bundle is smoothly

(real analytically) trivial, by using parallel transport with respect to a smooth (real

analytic) Hermitian connection.

50.15. Example. (see [Rellich, 1937, section 2]) That the last result cannot be

improved is shown by the following example which rotates a lot:

cos 1 ’ sin 1 1

»± (t) = ±e’ t2 ,

t t

x+ (t) := , x’ (t) := ,

1 1

sin t cos t

»+ (t) 0

(x+ (t), x’ (t))’1

A(t) := (x+ (t), x’ (t))

0 »’ (t)

cos 2 sin 2

’ t1 t t

=e .

2

2

’ cos 2

sin t t

Here t ’ A(t) and t ’ »± (t) are smooth, whereas the eigenvectors cannot be

chosen continuously.

50.16. Theorem. Let t ’ A(t) be a smooth curve of unbounded self-adjoint oper-

ators in a Hilbert space with common domain of de¬nition and compact resolvent.

Then the eigenvalues of A(t) may be arranged increasingly ordered in such a way

that each eigenvalue is continuous, and they can be rearranged in such a way that

they become C 1 -functions.

Suppose, moreover, that no two of the continuous eigenvalues meet of in¬nite order

at any t ∈ R if they are not equal. Then the eigenvalues and the eigenvectors can

be chosen smoothly in t on the whole parameter domain.

If on the other hand t ’ A(t) is a real analytic curve of unbounded self-adjoint

operators in a Hilbert space with common domain of de¬nition and with compact

resolvent. Then the eigenvalues and the eigenvectors can be chosen smoothly in t,

on the whole parameter domain.

The real analytic version of this theorem is due to [Rellich, 1940], see also [Kato,

1976, VII, 3.9] the smooth version is due to [Alekseevsky, Kriegl, Losik, Michor,

1996]; the proof follows the lines of the latter paper.

50.16

50.16 50. Applications to perturbation theory of operators 549

That A(t) is a smooth curve of unbounded operators means the following: There is

a dense subspace V of the Hilbert space H such that V is the domain of de¬nition

of each A(t) and such that A(t)— = A(t) with the same domains V , where the

adjoint operator A(t)— is de¬ned by A(t)u, v = u, A(t)— v for all v for which the

left hand side is bounded as functional in u ∈ V ‚ H. Moreover, we require that

t ’ A(t)u, v is smooth for each u ∈ V and v ∈ H. This implies that t ’ A(t)u is

smooth R ’ H for each v ∈ V by (2.3). Similar for the real analytic case, by (7.4).

The ¬rst part of the proof will show that t ’ A(t) smooth implies that the resolvent

(A(t) ’ z)’1 is smooth in t and z jointly, and mainly this is used later in the proof.

It is well known and in the proof we will show that if for some (t, z) the resolvent

(A(t) ’ z)’1 is compact then for all t ∈ R and z in the resolvent set of A(t).

Proof. We shall prove the smooth case and indicate the changes for the real ana-

lytic case.

For each t consider the norm u 2 := u 2 + A(t)u 2 on V . Since A(t) = A(t)—

t

is closed, (V, t ) is also a Hilbert space with inner product u, v t := u, v +

s ) ’ (V,

A(t)u, A(t)v . All these norms are equivalent since (V, t+ t)

is continuous and bijective, so an isomorphism by the open mapping theorem. Then

t ’ u, v t is smooth for ¬xed u, v ∈ V , and by the multilinear uniform boundedness

principle (5.18), the mapping t ’ , t is smooth and into the space of bounded

bilinear forms; in the real analytic case we use (11.14) instead. By the exponential

law (3.12) the mapping (t, u) ’ u 2 is smooth from R — (V, s ) ’ R for each

t

¬xed s. In the real analytic case we use (11.18) instead. Thus, all Hilbert norms

t are equivalent, since { u t : |t| ¤ K, u s ¤ 1} is bounded by LK,s in R, so

u t ¤ LK,s u s for all |t| ¤ K. Moreover, each A(s) is a globally de¬ned operator

t ) ’ H with closed graph and is thus bounded, and by using again the

(V,

(multi)linear uniform boundedness principle (5.18) (or (11.14) in the real analytic

case) as above we see that s ’ A(s) is smooth (real analytic) R ’ L((V, t ), H).

If for some (t, z) ∈ R — C the bounded operator A(t) ’ z : V ’ H is invertible, then

this is true locally and (t, z) ’ (A(t) ’ z)’1 : H ’ V is smooth since inversion is

smooth on Banach spaces.

Since each A(t) is Hermitian the global resolvent set {(t, z) ∈ R — C : (A(t) ’ z) :

V ’ H is invertible} is open, contains R — (C \ R), and hence is connected.

Moreover (A(t) ’ z)’1 : H ’ H is a compact operator for some (equivalently any)

(t, z) if and only if the inclusion i : V ’ H is compact, since i = (A(t) ’ z)’1 —¦

(A(t) ’ z) : V ’ H ’ H.

Let us ¬x a parameter s. We choose a simple smooth curve γ in the resolvent set

of A(s) for ¬xed s.

(1) Claim. For t near s, there are C 1 -functions t ’ »i (t) : 1 ¤ i ¤ N which

parameterize all eigenvalues (repeated according to their multiplicity) of

A(t) in the interior of γ. If no two of the generically di¬erent eigenvalues

meet of in¬nite order they can be chosen smoothly.

50.16

550 Chapter X. Further Applications 50.16

By replacing A(s) by A(s)’z0 if necessary we may assume that 0 is not an eigenvalue

of A(s). Since the global resolvent set is open, no eigenvalue of A(t) lies on γ or

equals 0, for t near s. Since

1

(A(t) ’ z)’1 dz =: P (t, γ)

t’’

2πi γ

is a smooth curve of projections (on the direct sum of all eigenspaces corresponding

to eigenvalues in the interior of γ) with ¬nite dimensional ranges, the ranks (i.e.

dimension of the ranges) must be constant: it is easy to see that the (¬nite) rank

cannot fall locally, and it cannot increase, since the distance in L(H, H) of P (t) to

the subset of operators of rank ¤ N = rank(P (s)) is continuous in t and is either

0 or 1. So for t near s, there are equally many eigenvalues in the interior, and we

may call them µi (t) : 1 ¤ i ¤ N (repeated with multiplicity). Let us denote by

ei (t) : 1 ¤ i ¤ N a corresponding system of eigenvectors of A(t). Then by the

residue theorem we have

N

1

z p (A(t) ’ z)’1 dz,

µi (t)p ei (t) ei (t), =’

2πi γ

i=1

which is smooth in t near s, as a curve of operators in L(H, H) of rank N , since 0

is not an eigenvalue.

(2) Claim. Let t ’ T (t) ∈ L(H, H) be a smooth curve of operators of rank

N in Hilbert space such that T (0)T (0)(H) = T (0)(H). Then t ’ tr(T (t))

is smooth (real analytic) (note that this implies T smooth (real analytic)

into the space of operators of trace class by (2.3) or (2.14.4), (by (10.3) and

(9.4) in the real analytic case) since all bounded linear functionals are of

the form A ’ tr(AB) for bounded B, see (52.33), e.g.

Let F := T (0)(H). Then T (t) = (T1 (t), T2 (t)) : H ’ F • F ⊥ and the image of

T (t) is the space

T (t)(H) = {(T1 (t)(x), T2 (t)(x)) : x ∈ H}

= {(T1 (t)(x), T2 (t)(x)) : x ∈ F } for t near 0

= {(y, S(t)(y)) : y ∈ F }, where S(t) := T2 (t) —¦ (T1 (t)|F )’1 .

Note that S(t) : F ’ F ⊥ is smooth (real analytic) in t by ¬nite dimensional

inversion for T1 (t)|F : F ’ F . Now

T1 (t)|F ⊥

1 0 T1 (t)|F 1 0

tr(T (t)) = tr

T2 (t)|F ⊥

’S(t) 1 T2 (t)|F S(t) 1

T1 (t)|F ⊥

T1 (t)|F 1 0

= tr

’S(t)T1 (t)|F ⊥ + T2 (t)|F ⊥

0 S(t) 1

T1 (t)|F ⊥

T1 (t)|F 1 0

= tr , since rank = N

0 0 S(t) 1

T1 (t)|F + (T1 (t)|F ⊥ )S(t) T1 (t)|F ⊥

= tr

0 0

= tr T1 (t)|F + (T1 (t)|F ⊥ )S(t) : F ’ F ,

50.16

50.16 50. Applications to perturbation theory of operators 551

which visibly is smooth (real analytic) since F is ¬nite dimensional.

From the claim (2) we now may conclude that

m

1

z p (A(t) ’ z)’1 dz

»i (t)p = ’ tr

2πi γ

i=’n

is smooth (real analytic) for t near s.

Thus, the Newton polynomial mapping sN (»’n (t), . . . , »m (t)) is smooth (real an-

alytic), so also the elementary symmetric polynomial σ N (»’n (t), . . . , »m (t)) is

smooth, and thus {µi (t) : 1 ¤ i ¤ N } is the set of roots of a polynomial with

smooth (real analytic) coe¬cients. By theorem (50.11), there is an arrangement

of these roots such that they become di¬erentiable. If no two of the generically

di¬erent ones meet of in¬nite order, by theorem (50.10) there is even a smooth ar-

rangement. In the real analytic case, by theorem (50.12) the roots may be arranged

in a real analytic way.

To see that in the general smooth case they are even C 1 note that the images of

the projections P (t, γ) of constant rank for t near s describe the ¬bers of a smooth

vector bundle. The restriction of A(t) to this bundle, viewed in a smooth framing,

becomes a smooth curve of symmetric matrices, for which by Rellich™s result (50.17)

below the eigenvalues can be chosen C 1 . This ¬nishes the proof of claim (1).

(3) Claim. Let t ’ »i (t) be a di¬erentiable eigenvalue of A(t), de¬ned on

some interval. Then

|»i (t1 ) ’ »i (t2 )| ¤ (1 + |»i (t2 )|)(ea|t1 ’t2 | ’ 1)

holds for a continuous positive function a = a(t1 , t2 ) which is independent

of the choice of the eigenvalue.

For ¬xed t near s take all roots »j which meet »i at t, order them di¬erentiably near

t, and consider the projector P (t, γ) onto the joint eigenspaces for only those roots

(where γ is a simple smooth curve containing only »i (t) in its interior, of all the

eigenvalues at t). Then the image of u ’ P (u, γ), for u near t, describes a smooth

¬nite dimensional vector subbundle of R — H, since its rank is constant. For each u

choose an orthonormal system of eigenvectors vj (u) of A(u) corresponding to these

»j (u). They form a (not necessarily continuous) framing of this bundle. For any

sequence tk ’ t there is a subsequence such that each vj (tk ) ’ wj (t) where wj (t)

is again an orthonormal system of eigenvectors of A(t) for the eigenspace of »i (t).

Now consider

A(t) ’ »i (t) A(tk ) ’ A(t) »i (tk ) ’ »i (t)

vi (tk ) ’

vi (tk ) + vi (tk ) = 0,

tk ’ t tk ’ t tk ’ t

take the inner product of this with wi (t), note that then the ¬rst summand vanishes,

and let tk ’ t to obtain

»i (t) = A (t)wi (t), wi (t) for an eigenvector wi (t) of A(t) with eigenvalue »i (t).

50.16

552 Chapter X. Further Applications 50.17

This implies, where Vt = (V, t ),

|»i (t)| ¤ A (t) wi (t) wi (t)

Vt H

L(Vt ,H)

2 2

= A (t) wi (t) + A(t)wi (t)

L(Vt ,H) H H

1 + »i (t)2 ¤ a + a|»i (t)|,

= A (t) L(Vt ,H)

2

for a constant a which is valid for a compact interval of t™s since t ’ t is

smooth on V . By Gronwall™s lemma (see e.g. [Dieudonn´, 1960,] (10.5.1.3)) this

e

implies claim (3).

By the following arguments we can conclude that all eigenvalues may be numbered

as »i (t) for i in N or Z in such a way that they are C 1 , or C ∞ under the stronger

assumption, or real analytic in the real analytic case, in t ∈ R. Note ¬rst that by

claim (3) no eigenvalue can go o¬ to in¬nity in ¬nite time since it may increase at

most exponentially. Let us ¬rst number all eigenvalues of A(0) increasingly.

We claim that for one eigenvalue (say »0 (0)) there exists a C 1 (or C ∞ or real

analytic) extension to all of R; namely the set of all t ∈ R with a C 1 (or C ∞ or

real analytic) extension of »0 on the segment from 0 to t is open and closed. Open

follows from claim (1). If this interval does not reach in¬nity, from claim (3) it

follows that (t, »0 (t)) has an accumulation point (s, x) at the the end s. Clearly

x is an eigenvalue of A(s), and by claim (1) the eigenvalues passing through (s, x)

can be arranged C 1 (or C ∞ or real analytic), and thus »0 (t) converges to x and

can be extended C 1 (or C ∞ or real analytic) beyond s.

By the same argument we can extend iteratively all eigenvalues C 1 (or C ∞ or real

analytic) to all t ∈ R: if it meets an already chosen one, the proof of (50.11) shows

that we may pass through it coherently. In the smooth case look at (50.10) instead,