Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 41–66.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

SPECTRAL THEORY OF C-SYMMETRIC NON-SELFADJOINT

DIFFERENTIAL OPERATORS OF ORDER 2n

HORST BEHNCKE, DON HINTON

Abstract. We continue the spectral analysis of differential operators with

complex coefficients, extending some results for Sturm-Liouville operators to

higher order operators. We give conditions for the essential spectrum to be
empty, and for the operator to have compact resolvent. Conditions are given

on the coefficients for the resolvent to be Hilbert-Schmidt. These conditions

are new even for real coefficients, i.e., the selfadjoint case. Asymptotic analysis
is a central tool.

1. Introduction

Non-selfadjoint operators (NSA) arise in many areas of theoretical physics. Yet,
compared to thousands of papers on selfadjoint operators arising in differential
equations, there are fewer papers on their NSA counterparts. This is so mainly
because of the absence of the spectral theorem and the order properties of the
real numbers. Thus there are no Sturm theorems and no spectral representations.
Special tools like subordinacy and the m-matrix as the Borel transforms of the
spectral measure are likewise missing. This is not surprising and it can be already
seen for Sturm-Liouvillve operators. New phenomena arise, like empty spectrum,
all of C as spectrum, or higher order poles for the resolvent. Hence the analysis the
NSA differential operators so far lacks order and guiding principles.

Most studies rely on numerical range conditions [9, 10] which led among others
the analysis of dissipative operators. The Russian school, Naimark, Ljance, et al.
[30] in their study of NSA Sturm-Liouville operators use function theoretic methods
and Fourier transforms in their analyses of the eigenfunction expansion of Sturm-
Liouville NSA operators. Yet this approach came to a stop in the mid 1970’s.
Sims [38] was able to extend Weyl’s program by construction of the m-function for
−y′′ + q(x)y where q is complex valued with Im q(x) ≤ 0. This was extended by
Brown, McCormack, Evans, and Plum [9] to general Sturm-Liouville operators and
later to complex Hamiltonians [10, 29]. All these studies rely heavily on numerical
range conditions to localize the spectrum and construct the m-matrix by further
assumptions on the numerical range. However, unlike the real case, the meaning

2020 Mathematics Subject Classification. 34L05, 34B20, 34B27, 34B40, 34B60.
Key words and phrases. m-functions; singular operators; essential spectrum;

non-selfadjoint operators; C-symmetric operators; Green’s function; asymptotic solutions.
©2023 This work is licensed under a CC BY 4.0 license.

Published March 27, 2023.

41



42 H. BEHNCKE, D. HINTON EJDE/SI/02

of the m-matrix remains obscure, even for Sturm-Liouville constant coefficients
operators.

This article is devoted to the spectral analysis of differential operators of the
form

T [y] =
1

w

n∑
k=0

(−1)k(pky
(k))(k) on L2

w(I). (1.1)

Here I is an interval in R, and w is the weight function which defines the scalar
product. Mostly we will be dealing with the one singular endpoint case I = [a,∞).
The coefficients pk are assumed to fulfill the usual requirements, i.e., 1/pn, pk for
k = 0, . . . , n− 1, are locally integrable and complex valued. The maximal operator
Tmax has domain D(Tmax) consisting of all functions y in L2

w(I) to which T can
be applied and have T [y] ∈ L2

w(I). D(Tmin), the domain of the minimal operator,
is the closure of the set of all functions in D(Tmax) with compact support in the
interior of I. Both Tmin and Tmax are closed, densely defined operators. For the
systems formulation of (1.1) below in (2.7), we define the quasi-derivatives y[k] by
[2]

y[k] = y(k), 0 ≤ k ≤ n− 1, y[n] = pny
(n),

y[n+k] = −y[n+k−1]
′
+ pn−ky

[n−k],
(1.2)

for 1 ≤ k ≤ n− 1, in which case

T [y] =
1

w

[
y[2n−1]

′
+ p0y

]
.

For y ∈ D(Tmin), I = [a,∞), it is known that y[k](a) = 0, k = 0, . . . , 2n− 1, and
from this it follows that Tmin has no eigenvalues as the existence-uniqueness theory
for (1.1) implies y ≡ 0 if y ∈ D(Tmin) and Tmin[y] = zy.

The formal adjoint T+ of T is then given by

T+[y] =
1

w

n∑
k=0

(p̄ky
(k))(k) on L2

w(I). (1.3)

We have the adjoint relations e.g., Goldberg [18, p. 130] or Kauffman, Read, and
Zettl [22, p. 14],

T ∗min = T+
max, Tmax = T+∗

min, Tmin = T+∗
max, T ∗max = T+

min,

where * is the Hilbert space adjoint. Further we have Tmin is C−symmetric, i.e.,
Tmin ⊂ CT+

maxC where C is complex conjugation. In the case of C-symmetric opera-
tors like Tmin, we are interested in the spectral theory of the C-selfadjoint extensions
Ts of Tmin, i.e., operators Ts which satisfy the relations

Tmin ⊂ Ts ⊂ Tmax, T ∗s = CTsC. (1.4)

In general a C−symmetric map C on a Hilbert space is one that is conjugate lin-
ear, involutive, and isometric. Our C-symmetry is usually called J -symmetry for
complex conjugation.

These operators are obtained by imposing appropriate boundary conditions on
the elements of D(Tmax), see Knowles [25]. In [8] the spectral analysis of such
maximal C-symmetric operators was based on asymptotic integration, because a
good knowledge of the eigenfunctions allows to deduce spectral properties of T .
Earlier asymptotic integration for differential equations with complex coefficients
was mainly used for computing the deficiency index. References for this approach



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 43

may be found in the book of Eastham [12] which is entirely devoted to asymptotic
integration.

In fact we will follow the presentations of [6, 7, 8] closely so that we may cut the
first three sections rather short. The spectrum, resolvent, the domain, null space,
and the range of an operator T will be denoted by σ(T )ρ(T ), D(T ), N(T ), and
R(T ) respectively. The numerical range of T is defined by

N (T ) = {〈Tx, x〉 : x ∈ D(T ), ‖x‖ = 1}. (1.5)

The set N (T ) is convex, but need not be closed. N (T ) is an important tool in
finding the spectrum - see section 2. Clearly, eigenvalues are contained in N (T ).
Let

Nn(T ) = {〈Tx, x〉 : x ∈ D(T ), ‖x‖ = 1, supportx ⊆ [n.∞)}.
Since many properties of T depend only on the asymptotics of the eigenfunctions,

we define the essential numerical range of T by

N∞(T ) = ∩Nn(T ). (1.6)

Note that the numerical range for Tmin can be computed from

〈Tminy, y〉 =

∫ ∞
a

n∑
k=0

pk(x)|y(k)k (x)|2 w(x)dx. (1.7)

Thus the numerical range of Tmin will lie in a sector of C of angle ≤ π if all values
of the coefficients lie in this sector.

Lp0 = Lp0([a,∞)), 1 ≤ p < ∞, will denote the set of all p-integrable functions
vanishing at infinity. For functions f and g we write f � g if |f | = o(|g|) and f ≈ g
if for some K > 0, K−1|f | ≤ g ≤ K|f |.

For the general theory of linear differential operators we refer to the books by
Glazman [17], Naimark [30], and Weidmann [39]. In a sense this article may be
considered an extension of [3].

This article is organized as follows: Introduction, spectral theory and asymp-
totic integration, the resolvent, conditions for σess(Tmax) 6= C, eigenvalues of equal
magnitude, and other higher order equations.

2. Spectral theory and asymptotic integration

2.1. Spectral theory. For a closed, densely defined operator S in a Hilbert space
H, the regularity field is defined by

Π(S) = {z ∈ C : ‖(S − z)(x)‖ ≥ kz‖x‖, x ∈ D(S), for some kz > 0}.
The resolvent set ρ(S) of S is the set of all z in Π(S) such that the range of

S − z is H. The spectrum σ(S) of S is the complement of ρ(S). The set σ(S)
is the union three sets: the eigenvalues of S, σp(S), the residual spectrum σr(S)
which is the set of values of z /∈ σp(S) for which the range of S − z is closed but
different from H and finally, the essential spectrum of S, σess(S) which is the set
of z such that the range of S − z is not closed. Glazman [17, p. 9] proves that this
is equivalent (when there are no eigenvalues of infinite geometric multiplicity) to
there being a singular sequence for z, i.e., a bounded noncompact sequence {fn}
such that (S − z)(fn)→ 0 as n→∞ or equivalently there is a sequence {fn} with
‖fn‖ = 1 such that (S − z)(fn)→ 0 as n→∞ and fn → 0 weakly.

If S is a C-selfadjoint operator, then σr(S) = ∅ since z ∈ σr(S) implies N(S∗ −
z̄) 6= ∅. (see (2.1) below) If S∗φ = z̄φ, then Sφ̄ = CS∗Cφ̄ = CS∗φ = C(z̄φ) = zφ̄



44 H. BEHNCKE, D. HINTON EJDE/SI/02

which is contrary to z ∈ σr(S). Thus a C-selfadjoint operator has no residual spec-
trum. This parallels the selfadjoint case as selfadjoint operators have no residual
spectrum. One difference however, is that a C-symmetric operator always has a C-
selfadjoint extension, see Knowles [24], while a symmetric operator may not have a
selfadjoint extension. This occurs for symmetric operators with finite and unequal
deficiency indices.

In general, σ(S) = σp(S) ∪ σr(S) ∪ σess(S) and σ(S) = σp(S) ∪ σess(S) if S is
a C-selfadjoint operator. While for both selfadjoint and C-selfadjoint operators the
spectrum is the union of the point and essential spectrum, there is an important
difference. If S is selfadjoint, then both σ(S) 6= ∅ ( by the spectral theorem) and
ρ(S) 6= ∅ (as z ∈ ρ(S) if z 6= 0). If S is a C-selfadjoint operator, then it is possible
for σ(S) = ∅ (see Example 2.2 below) or for ρ(S) = ∅ (see Example 2.1 below).

From the fact that D(Tmax) is a finite dimensional extension of D(Tmin), it can
be proved , see [22, p. 16], that when one of Tmin − z, Tmax − z, T+

min − z̄, T+
max − z̄

has a closed range, then all do. From this it follows that

σess(Tmin) = σess(Tmax).

For S = Tmin, I = [a,∞), as noted before, we have σ(Tmin) = σr(Tmin) ∪
σess(Tmin). We have the well known relations, Kato [21, p. 267],

N(T+
max − z̄) = (R(Tmin − z))⊥, N(Tmax − z) = (R(T+

min − z̄))
⊥. (2.1)

Note that the conjugation map y → ȳ shows that dimN(T+
max− z̄) = dimN(Tmax−

z). In the general case studied here the numerical range of Tmin may be C.

For z /∈ N(Tmin), we have from Kato [21, p. 268] that Tmin − z has a closed
range, nullity Tmin−z = 0, and the defect of Tmin−z is constant on each connected

component of N(Tmin)
C

. Thus one has σess(Tmin) ⊆ N(Tmin).
If z /∈ σess(S) and z /∈ σp(S), then by the closed graph theorem, z ∈ Π(S); the

converse also holds so that

Π(S) = σess(S)C ∩ σp(S)C and

C = Π(S) ∪Π(S)C = Π(S) ∪ σess(S) ∪ σp(S).
(2.2)

We define
s = dim(D(Tmax)/D(Tmin)). (2.3)

In the one singular endpoint case I = [a,∞), s ≥ 2n since one can construct 2n
compactly supported independent functions in D(Tmax)/D(Tmin) [30]. Further, it
follows in the one singular endpoint case from Kauffman, Read, and Zettl [22, p.
16], that when Tmin − z has a closed range,

s = nul(Tmax − z) + nul(T+
max − z̄) = 2 nul(Tmax − z). (2.4)

From these in this case and for all z /∈ σess(Tmin), we obtain

def(Tmin − z) := dim(R(Tmin − z))⊥ ≥ n, and def(T+
min − z̄) ≥ n. (2.5)

For a C-symmetric operator these defect numbers are independent of z /∈ σess(Tmin)
[25], and we refer to them as def Tmin and def T+

min.

Example 2.1. McLeod [28] gave the example of the equation

τ [y] := −y′′ − 2i(exp(2(1 + i)x))y = zy, 0 ≤ x <∞,
whose solutions can be expressed in terms of Bessel functions, and no nontrivial
solution is in L2([0,∞)) for any z. If z /∈ σess(Tmin) = σess(Tmax) for some z ∈ C,



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 45

then (2.4)-(2.5) yields that nul(Tmax − z) ≥ 1 which is a contradiction. Thus

σess(Tmin) = C which also implies N(Tmin) = C as σess(Tmin) ⊆ N(Tmin).
This result holds for powers of τ as well. For simplicity, consider τ2. First we

show τ2 has no eigenvalues. Suppose

τ2[y] = z2y, y 6= 0, y ∈ L2([0,∞)).

Then
τ2[y]− z2y = (τ − z)(τ + z)[y] = 0.

This implies g := (τ + z)[y] 6= 0 as τ + z has no nontrivial solutions in L2([0,∞))
which in turn implies (τ−z)[g] 6= 0 as τ−z has no nontrivial solutions in L2([0,∞))

which is a contradiction. Since dimD(T̂max/D(T̂min) ≥ 4, where T̂ refers to τ2,
we reach a contradiction as before Hence there are examples of even powers of
differential operators with essential spectrum C.

Example 2.2. In [8], the eigenvalue problem on 0 ≤ x ≤ 1,

τ [y] = −y′′,
with boundary conditions

A

[
y(0)
y′(0)

]
+B

[
y(1)
y′(1)

]
=

[
0
0

]
, rank[A,B] = 2,

with complex matrices A,B was proved to be C- symmetric if an only if the bound-
ary conditions are of the form

y(0) = −cy(1), y′(0) = cy′(1), c = ±i.
Further, it was shown than with c = ±i there are no eigenvalues. Hence we have an
example of a C-symmetric operator with empty spectrum as the minimal operator
for a compact interval has empty essential spectrum.

Example 2.3. First we recall how singular sequences are used to find points in the
essential spectrum. To show the minimal operator Tmin for τ [y] = (−1)ny(2n), 0 ≤
x < ∞, has σess(Tmin) = [0,∞), one can construct a singular sequence. For ex-
ample, let {In}, In = [an, bn], be a sequence of disjoint intervals in [0,∞) so that
3 ≤ |In| = bn−an and |In| → ∞ as n→∞. Let φn be a C∞ function with support
In so that φn(x) = 1 on [an + 1, bn − 1], n = 1, 2, . . . .

Let yn(x)) = φn(x) sin(λ1/2nx) for λ > 0. Then for φn(x) ≡ 1,

(Tmin − λ)[yn] = (−1)ny(2n)n − λyn = λyn − λyn = 0.

and it is clear that

||(Tmin − λ)[yn]||/||yn|| → 0 as x→∞,
so that {yn} is a singular sequence for Tmin establishing (0,∞) ⊆ σess(Tmin). Since
σess(Tmin) is closed and N (Tmin) ⊆ [0,∞), this gives σess(Tmin) = [0,∞). If c ∈ C,
and f(x) = c on the supports of the φn and zero elsewhere, then the same singular
sequence as above shows that σess((T + f)min) = R(c) where R(c) is the ray

R(c) = {µ : µ = λ+ c, λ ≥ 0}.
That σess((T + f)min) = R(c) follows from (4.3) below, but the construction above
shows how singular sequences generate points in the essential spectrum.

Now suppose K is a convex set in C and let {cn} be a countable sequence whose
closure is K. Suppose [0,∞) is decomposed into disjoint intervals Iij such that



46 H. BEHNCKE, D. HINTON EJDE/SI/02

|Iij | → ∞ as i → ∞ and |Iij | → ∞ for j → ∞. Define the potential V by
V (x) = ci if x ∈ Iij .

By the above argument we see that for all n, R(cn) ⊆ σess((T + V )min). Since
the essential spectrum is closed,

∪(n)R(cn) ⊆ σess((T + V )min).

Note the left side of this equation is a closed convex set which is also contained in
the closure of the numerical range of (T + V )min which is itself a convex set. This
example illustrates the variety of sets that can be essential spectrum for Sturm-
Liouville operators with complex coefficients. In particular, if K is the left half
plane, then σess((T + V )min) = C.

2.2. Asymptotic integration. Asymptotic integration has been a major tool to
derive properties of the eigenfunctions of the maximal C− symmetric extensions
of Tmin. In the beginning it was mainly used to compute the deficiency index of
differential operators. The application to spectral theory began with [3]. Edmunds
and Evans [13] gave five Fredholm type definitions of essential spectrum. For C−
symmetric operators, the first four define the same object. The eigenvalues define
the point spectrum σp(T ). It is obvious that σess(T ) will only depend on the
asymptotics of the coefficients of T . In particular it will generally be independent
of the boundary conditions at the left endpoint.

In asymptotic integration one first writes

Ty = zy

in systems form. With the quasi-derivatives (1.2) and

u =
[
y[0], y[1], . . . , y[n−1], y[2n−1], . . . , y[n]

]t
, (2.6)

where t is transpose, the equation Ty = zy can be written as as a system

J u′ = [zA + B]u (2.7)

where

J =

[
0 −In
In 0

]
, A = diag[w, 0, . . . , 0], B =

[
−C At

A B

]
,

and the nonzero elements of A,B,C are [2] (Note that B = Bt, C = Ct)

Ai,i+1 = 1, Bnn = 1/pn, Cii = pi−1.

We will write the system (2.7) for short as

u′ = Cu with C =

[
A B

C̃ −A∗
]

(2.8)

where C̃ is C modified by replacing C11 = p0 with p0−zw. This system formulation
is similar, but different from that of [12, p. 105]. In fact, if G is a constant
nonsingular matrix, then multiplying (2.8) by G gives an equivalent system with C
replaced by GCG−1.

For asymptotic integration of (2.7) or (2.8) this system has to be brought into
Levinson form [12],

v′ = [Λ +R]v withΛ = diag[λi(x.z)], Rij = L1, i, j = 1, . . . , 2n, (2.9)



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 47

because solutions of (2.9) will almost look like the solutions of the unperturbed
system v′ = Λv if Λ satisfies the dichotomy conditions. To transform (2.8) into
(2.9) the matrix C has to be diagonalized. Assume

S−1CS = Λ.

Then the system for Sv = u becomes

v′ =
(
S−1CS − S−1S′

)
v =

(
Λ− S−1S′

)
v. (2.10)

Thus such a transformation makes sense if S, respectively C, is differentiable so
that S−1S′,i.e., C′ is small. This seems to restrict systems (2.8) to those with
differentiable coefficients. However, if C can be written as C = C1 + C2 with C1

differentiable and C2 integrable it suffices to diagonalize the smooth part C1 of C.
Such diagonalizations may be repeated. Thus, if one can write C = C0 + C1 +

C2 + · · · + Cm, with Ckm − k times differentiable, C0 a constant, C
(l)
k = o(1), and

C
(n)
n ∈ L0, this will essentially lead to Levinson form. In addition note that it

suffices to apply the diagonalization only to the off-diagonal parts of the systems
matrix arising. For simplicity we shall restrict the analysis to m = 3 and require
for the coefficients of C a decomposition of the form

f = f1 + f2 + f3 with f1 twice differentiable, f2 once differentiable and

f ′′1 /(1 + |f1|), f ′2/(1 + |f1|), f + f ′21 /(1 + |f1|), f3 ∈ L1
0.

(2.11)

In this case we write

C = C1 + C2 + C3 (2.12)

and we restrict the diagonalization to C1 + C2. This is done in three steps.
I: Determine the eigenvalues of C1 + C2. II: Determine the eigenvectors.
III: Compute the diagonlaization matrix S.

As regards (I) we also have to require that all eigenvalues of C1 +C2 are distinct.
Otherwise we may get a Jordan normal form type expressions which we will not
study here. The eigenvalues of C1+C2 are the roots of the characteristic polynomial
of C1 + C2

P (λ, x, z) =

n∑
k=1

(−1)kpk(x)λ2k + (p0(x)− zw)

= pn det
(
C1(x) + C2(x)− λ

) (2.13)

Here pk are just the smooth parts of the coefficients. It is advantageous to replace
λ by by iλ, Then one obtains the characteristic Fourier polynomial

PF (λ, x, z) = P (iλ, x, z) =

n∑
k=1

pk(x)λ2k + (p0(x)− zw). (2.14)

For the remainder we will work with PF only. Note that PF is a function of λ2,
so that −λ is an eigenvalue if λ is. Note that the Fourier polynomial of (2.7) is
given by

PF (λ, x, z) = −pn det
(
zA + B− iλJ

)
. (2.15)



48 H. BEHNCKE, D. HINTON EJDE/SI/02

II: If the eigenvalues λk = λk(x, z), k = 1, . . . , 2n, of (2.13) are distinct, the
eigenvectors ρk(x, z) of Cρ = λρ are given by [3]

(ρk)s(x, z) = λk(x, z)s−1, 1 ≤ s ≤ n,

(ρk)n+s(x, z) =

n∑
ν=s

(−1)s+νpν(x)λk(x, z)2ν−s, 1 ≤ s ≤ n.
(2.16)

III: It is advantageous to base the diagonalization on the eigenvetors ν with

νk(x, z) = M
−1/2
k (x, z)

[
ρ1, ρ2, . . . , ρn, ρ2n, . . . , ρn+1

]t
(x, z) (2.17)

with Mk = ∂pF
∂λ

∣∣
λ=λk

because the formulas for the selfadjoint case extends directly

to this situation. Thus

S(x, z) =
[
ν1(x, z), ν2(x, z), . . . , ν2n(x, z)

]
(2.18)

will be used as the diagonalizing matrix.
In the situation envisaged above, S diagonalizes C1 + C2 and Sv = u leads to

v′ =
[
Λ− S−1S′ + S−1C3S

]
v, Λ = diag

(
λ(x, z)

)
. (2.19)

The matrix S−1S′ in (2.19) is given below; similar formulas may be found in East-
ham [12],

(S−1S′)jk =
(
λj − λk

)−1
M
−1/2
j M

−1/2
k

n∑
l=0

plλ
l
kλ

l
j , j 6= k,

(S−1S′)kk = 0.

(2.20)

This latter relation is a consequence of the normalization of eigenvectors with

M
−1/2
k . Note that these formulas also extend to the case where T has odd compo-

nents.
The transformed system (2.19) is

Sv = u, v′ =
(
Λ +Q+R

)
v (2.21)

where Q is the smooth off diagonal part of −S−1S′ and where R is the remainder.
If the terms of Q are sufficiently small, Qij = o(1), a further matrix of the form
(I+B) may be applied. For this one needs that the eigenvectors in Λ are sufficiently
distinct, With the usual perturbation Ansatz,(

I + εB
)−1(

Λ + εQ
)(
I + εB! + ε2B2 + . . .

)
= Λ + εΛ1 + ε2Λ2 + . . . ,

one gets in this case for B1,

Bii = 0, Bij =
(
λj − λk

)
Rij , Λ1 = 0. (2.22)

Higher order corrections are of the order O(λj − λk
)−2

Qij . . . Qlk. This requires,
for example,

|λi − λj | ≥ ε > 0, Qij ∈ L1
0. (2.23)

The correction terms are then (I +B)−1B′. These are integrable if B′ij ∈ L1.
If the remainder terms B,B′, BB′ are integrable, the system is in Levinson form.

If not, further transformations as above may be necessary. While the application
of asymptotic integration to problems of the deficiency index are rather straight-
forward. The use for spectral theory requires that the transformations above can
be performed uniformly in the spectral parameters at least for z in a small neigh-
borhood of a given z0.



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 49

2.2.1. Transformations. To reduce the multitude of cases somewhat and to simplify
the rising expressions, one should transform the variables. The best transformation
we know is the Kummer-Liouville transformation which is based on [1, 2].

y(x) = µ(t)z(t), dt/dx = γ(x). (2.24)

For differential equations of order six or higher the transformed coefficients are prac-
tically impossible to compute. For this reason we had introduced a transformation
adapted to asymptotic integration [2], i.e., modulo Levinson terms the transformed
system has the same form as the original system. For this one requires

µ = µ1 + µ2, γ = γ1 + γ2, µ1, γ1 twice differentiable, µ2, γ2 once differentiable ,

µ′1/µγ, γ
′
1/γ = o(1), γ′′/γ2, µ′′1/µγ ∈ L(I).

(2.25)
Then we let

bk = kµ′1γ
k−1 +

1

2
k(k − 1)µγ′1γ

k−2. (2.26)

For the transformed coefficients one gets

p̃n = µ2γ2n−1pn, p̃k = µ2γ2k−1pk − µγk−1
(
bk+1pk+1

)′
+ bkbk+1pk+1, (2.27)

for k = 0, . . . , n− 1.
If pn > 0, the expressions for µ and γ can be obtained from p̃n = 1 and w̃ =

µ2w/γ. In this case the transformation is even unitary. If pn is not positive, base
the transformation on a smooth approximation of |pn|.

2.2.2. Dichotomy condition. Levinson’s Theorem states that a system on [a,∞),

v′ =
(
Λ +R

)
v, Λ = diag

[
λi
]
, R ∈ L1([a,∞), (2.28)

has solutions which almost look like the solutions of the unperturbed system u′ =
Λu if R is a small and if the λk satisfy a dichotomy condition, i.e.,

vk(x) =
(
ek + rk(x)

)
exp
(∫ x

λk(t) dt
)
, rk(x) = o(1) as x→∞, (2.29)

where ek is the unit vector with 1 as the kth component. The dichotomy condition
requires the exponential terms to grow at sufficiently different rates. Details may
be found in the book of Eastham [12], which is entirely devoted to Levinson’s
theorem. The dichotomy condition requires that for any pair of distinct indices
i, j ∈ {1, . . . , n} and for all a ≤ x ≤ t <∞, and for some constants K1,K2,

exp
(∫ x

t

Re(λi(s)− λj(s)) ds
)
≤ K1 or

exp
(∫ x

t

Re(λi(s)− λj(s)) ds
)
≥ K2.

(2.30)

Note that we are using the Fourier polynomial so that λ will have to be replaced
with iλ so that Re(λi(s)− λj(s)) = Im(iλi(s)− iλj(s)).

If system (2.8) has been transformed into Leviinson’s form and if the dichotomy
holds, then the solutions of (2.7) are given by

uk(x, z) = S(x, z)
(
I +B(x, z)

)[
ek + rk(x, λk, z)

]
exp
(∫ x

a

λk(t, z) dt
)
, (2.31)

with rk = o(1).



50 H. BEHNCKE, D. HINTON EJDE/SI/02

If further diagonalizations have been used, (I + B) will have to be replaced by
their product. If B = o(1) formula (2.31) can be refined to

uk(x, z) = M
−1/2
k (x, z)S(x, z)

(
ek + rk(x, λk, z)

)
exp
(∫ x

a

λk(t, z)dt
)
,

Mk = ∂pF /∂λ
∣∣
λ=λk

.

(2.32)

If one follows the proof of the asymptotic integration [12], one finds that the
corrections terms (S−1S′)jk in (2.20) are z- uniformly bounded and (S−1S′)jk → 0
as x→∞ and as |z| → ∞. This also holds for the correction terms arising in further
diagonalizations. Following the proof of Levinson’s Theorem [12] then shows that
this extends to the corrections term rk(x, λk, z) as well. Thus

rk(x, λk, z)→ 0 as x→∞, uniformly in z and

rk(x, λk, z)→ 0 as |z| → ∞.
(2.33)

These results clearly show the importance of the first diagonalization. Asymptotic
integration will be successful if the eigenvalues dominate the remainder terms. In
the following we will call the factors M−1/2 form factors, a term borrowed from
nuclear physics. Note that this whole procedure requires a combination of decay
and smoothness for the coefficients as well as the dichotomy condition for the eigen-
values.

At this point one should realize that asymptotic integration is stable with respect
perturbations of the coefficients pn by terms qn ∈ L1

0. In fact these terms define a
relatively compact perturbations so that the essential spectrum remains invariant.

3. Resolvent

The aim of this article is to study the spectrum of operators (1.1) via their re-
solvent. However, it is still an open question when a resolvent exists for general
C-symmetric differential operators. Examples for C-symmetric Sturm-Liouville op-
erators operators without a resolvent or with empty spectrum are known as noted
earlier. So far the only general information is based on the numerical range. In
the case of compact operators the existence of the resolvent can also be inferred
from the structure of the domain. In the situation we are considering here, we will
quite often construct the resolvent explicitly, see (3.1) below. The development
below has been presently worked out for C-symmetric Hamiltonian systems with
almost constant coefficients. We will be mainly deal with operators on the half-line.
Problems in R can be handled by the decomposition method.

For the remainder of this section we make the following hypothesis.

(H1) There is a z0 in C so that z0 /∈ σess(Tmin), and dimN(Tmax − z0) = n.

This implies that Tmin−z0 is a Fredholm operator and hence Tmin−z is a Fredholm
operator in a neighborhood K0 of z0 [27]. Since the Fredholm index of Tmin − z is
constant in K0 and dimN(Tmax− z) = dimN(T+

max− z̄), it follows that (H1) holds
in K0. Property (H1) holds for problems with almost constant coefficients [7], and
for operators with compact resolvent.

Note that if the coefficients in (1.1) are real and (1.1) is in the limit point
condition at infinity, then dimN(Tmax − z0) = n. for any non real z0.

The operator Tmin associated to (1.1) is C-symmetric. As in the symmetric
situation, we are, however, looking at the spectral theory of maximal C−symmetric
extensions H. Such extensions can be obtained from Tmax by imposing boundary



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 51

conditions. In general these are given at a and possibly at infinity, but under(H1)
we only impose them at a to define a C-selfadjoint operator [25]. This does not
work for a selfadjoint limit circle operator.

Since the operator is regular at a and its deficiency index is n, the boundary
conditions at a for a C-selfadjoint operator can be defined by the boundary matrix
[4],

Yα(a, z) =

(
α1 −α2

α2 α1

)
(3.1)

with α∗1α1 + α∗2α2 = In and αt1α2 = αt2α1. With this the boundary conditions for
the point a become, with u as in (2.6),

(α∗1, α
∗
2)u(a) = 0, (3.2)

and we define the operator Tα as T restricted to the domain

D(Tα) = {y ∈ D(T )|(α∗1, α∗2)u(a) = 0}. (3.3)

By Knowles [25], Tα is a C-selfadjoint operator. If z ∈ K0 and z is not an eigenvalue
of Tα, then z ∈ ρ(Tα).

The fundamental matrix for (2.7) with initial conditions (3.1) will be denoted
by Yα. When we write the fundamental matrix Yα of (1.1) satisfying (3.1) as

Yα(x, z) = (Θα(x, z),Φα(x, z)), (3.4)

then Φα satisfies the boundary condition at a.
Similar to the fundamental matrix Yα for Tα,we have a fundamental matrix for

the adjoint system. Throughout the remainder the adjoint system will be marked
with a tilde. Thus, see [8],

Ỹα = (Θ̃α, Φ̃α) and Ỹα(a, z) =

(
α1 −α2

α2 α1

)
(3.5)

will stand for the solution of the adjoint system. By checking initial conditions of
the fundamental matrices it follows from the symmetry conditions B = Bt, C = Ct

of (2.7) (see [7]) that

Ỹα(x, z) = Yα(x, z) and hence Θ̃α = Θα, Φ̃α = Φα. (3.6)

The proof in [7, Proposition 3.1] gives the following for z ∈ K when (H1) holds.

Lemma 3.1. (a) Let I = [a,∞). For z ∈ K0 and z not an eigenvalue of

Tα there exists a unique n by n matrix Mα(z) so that Yα(x, z)
(

In
Mα(z)

)
=

Θα(x, z) + Φα(x, z)Mα(z) ∈ L2
A(I). Here Yα is the fundamental matrix of

(2.9) satisfying (2.18). Mα is analytic for z ∈ C \ σ(Tα).

(b) The M -matrix of the adjoint problem M̃ satisfies M̃α(z) = M∗α(z) and
Mα(z) = M t

α(z).

With Mα determined this way, most of the properties derived in [7, sect. 4] can
be shown. To do so we fix again Yα as the fundamental matrix of (2.9) with initial

conditions (2.18) Similarly let Ỹα be the adjoint system for T+ with the adjoint
initial conditions (3.5). Then

Ỹ ∗αJ Yα ≡ J , Y ∗αJ Ỹα ≡ J , (3.7)

can be shown as in [7]. Write Yα = (Θα,Φα) and Ỹα = (Θ̃α, Φ̃)α then

χ = Θ + ΦMα ∈ L2
A, χ̃ = Θ̃ + Φ̃M∗α ∈ L2

A, (3.8)



52 H. BEHNCKE, D. HINTON EJDE/SI/02

where we have deleted the initial value index α. From (3.7) one gets J YαJ Ỹ ∗α =
−In and as in [7, sect. 4] one can deduce for z ∈ K0 and z ∈ ρ(Tα) that

G(z, x, t) =

{
Φ(x, z)χ̃∗(t, z), a ≤ x ≤ t,
χ(x, z)φ̃∗(t, z), a ≤ t < x,

(3.9)

and

G̃(z, x, t) =

{
Φ̃(x, z)χ∗(t, z), a ≤ x ≤ t,
χ(x, z)Φ∗(t, z), a ≤ t < x,

(3.10)

are the integral kernels or Green’s functions of the resolvents Rz = (Tα − z)−1,

respectively R̃z = (T+
α − z), i.e.,

(Rzf)(x) =

∫ ∞
a

G(z, x, t)A(t)F (t)dt.

where F is as in (4.5) below.
For Hamiltonian systems the integration is based on the weight matrix A. For

scalar equations, which we are considering here, the matrix A is diag[w, 0, . . . 0], so
that only the first component counts and the integration uses the weight function
w.

4. Conditions for σess(Tmin) 6= C

We saw in section 3 that in order to have a meaningful spectral theory, it was
important for Π(Tmin) 6= ∅, or equivalently, (when σp(Tmin) = ∅), σess(Tmin) 6= C.

As we noted in section 2, this occurs if N (Tmin) 6= C. A simple criterion is that the
values of the coefficients pk are all bounded below or more generally lie in a convex
cone.

Another case of σess(Tmin) 6= C which we will not use however is when nul(T −
z) = 2n. Race [33] proved this case for the limit circle one singular endpoint case
for the Sturm-Liouville equation to obtain Π(Tmin) = C. This result has been
extended by Niessen [31] to operators of order 2n.

4.1. Almost constant coefficient case. We say a function f on I = [a,∞) is
almost constant if it can be written as

f = f0 + f1 + f2, f0 = constant, f1 → 0 as x→∞, f2 ∈ L(I).

Such decompositions are considered only for functions that are locally integrable.
These conditions which are weaker that those needed for asymptotic integration,
but strong enough for an exponential dichotomy, see [20]. For (1.1) we then assume
that p0, . . . , pn, 1/pn, w are almost constant coefficient with w = w0 + w1 only.

First we define the polynomial PF0 in (2.13) as in (2.14),

PF0(λ, z) =

n∑
k=1

pk0λ
2k + (p00 − zw0). (4.1)

Note that the Fourier polynomial of (2.7) is also given by

PF (λ, x, z) = −pn0 det
(
zA0 + B0 − iλJ

)
. (4.2)

In this section we work with PF0 only and assume pn0 = 1 without lost of
generality. In the constant coefficient case PF0 is of course independent of x, and



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 53

for a given z the λ-roots of PF0(λ, z) = 0 give rise to solutions of the form exp(iλx)ξ
of Ty = zy. Also PF0 has the form

PF0(λ, z) = λ2n + a2n−1(z)λ2n−1 + · · ·+ a0(z),

where the coefficients aj(z) are polynomials of z. A root λ(z) of PF0(λ, z) = 0 is a
holomorphic function in any simply connected region where there are no multiple
roots. Further the set of multiple roots of PF0(λ, z) = 0 is either a finite set or C.
For further discussion of PF0(λ, z) = 0 see Lemma 3.3 of [3] which is extracted from
the discussion of algebraic functions in Knopp [23]. The set of multiple roots of PF0

is finite if the discriminant of PF0 is not identically zero which holds in particular
if PF0 is irreducible. In the constant coefficient case it is sufficient to assume the
discriminant is not identically zero.

Solutions with Imλ > 0 for I = [a,∞) may lead to bound states if the boundary
conditions fit. On the other hand solutions with λ ∈ R lead to bounded generalized
eigenstates. These may be converted into approximate eigenfunctions by smooth
cutoffs. These functions are approximate eigenfunctions independent of the partic-
ular boundary conditions at 0. Thus in the case of constant coefficients it is proved
in [7] that the essential spectrum σess(Tmax) = Σ is given by

Σ = {z : PF0(λ, z) = 0 for some λ ∈ R}. (4.3)

Then an additional assumption is made:

The discriminant of PF0 is not identically zero .

Then it is shown in [7] that for almost constant coefficients,

σess(Tmax) ⊆ Σ ∪ E ,
where E is the finite set of z where PF has multiple roots. This is proved by showing
that under these hypotheses there is an exponential dichotomy for (2.7), i.e., there
is fundamental matrix W of (2.7) and a projection matrix Q of rank n, and positive
constants K1,K2, α1, α2 such that for t, s ∈ [a,∞),

‖W (t)QW−1(s)‖ ≤ K1 exp
(
− α1(t− s)

)
for t ≥ s,

‖W (t)(I −Q)W−1(s)‖ ≤ K2 exp
(
− α2(s− t)

)
for s ≥ t.

(4.4)

The first equation in (4.4) shows that the columns of W (t)Q form an n dimen-
sional subspace of N(Tmax−z) so dimN(Tmax−z) ≥ n. To prove dimN(Tmax−z) =
n we must prove that the first elements of the columns of W (t)(I −Q) form a sub-
space that contains only the zero element of N(Tmax − z). Let the 2n × n matrix
Γ be a basis for the subspace formed by the columns of W (t)(I −Q). Suppose for
some vector c 6= 0 that the first element of Γc ∈ N(Tmax−z). Now Γ = W (I−Q)C
for some 2n× n matrix C of rank n. By (4.4), for s ≥ t,

‖W (t)(I −Q)W−1(s)Γ(s)c‖ ≤ K2 exp
(
− α2(s− t)

)
‖Γ(s)c‖.

Since

W (t)(I −Q)W−1(s)Γ(s)c = W (t)(I −Q)Cc = Γ(t)c,

we have

‖Γ(t)‖ ≤ K2 exp
(
− α2(s− t)

)
‖Γ(s)c‖ for s ≥ t.

But Γc ∈ N(Tmax−z) implies there is a sequence {sn} with sn →∞ and Γ(sn)c→ 0
as n→∞. This implies ‖Γ(t)c‖ = 0 for all t and thus c = 0 which is a contradiction.
Summarizing, we have the following theorem.



54 H. BEHNCKE, D. HINTON EJDE/SI/02

Theorem 4.1. If the coefficients of T are almost constant and PF0 is irreducible,
then Hypothsis (H1) holds for all z /∈ Σ ∪ E.

4.2. General case. We now develop a criterion based on the asymptotic solutions
of (1.1) which may apply when N (Tmin) = C. The nonhomegeous version of (1.1),
Ty = zy + f has the form

J u′ = [Az + B]u+ AF, (4.5)

where F =
[
f, 0, . . . , 0

]t
. We now order the eigenvalues of the characteristic poly-

nomial (2.13) as

Im iλ1 ≤ Im iλ2 ≤ · · · ≤ Im iλn < 0, λn+k = −λn+1−k, k = 1, . . . , n. (4.6)

Assuming the conditions for asymptotic integration, there are solutions uk of (2.7),

uk(x, z) = M
−1/2
k (x, z)S(x, z)

(
ek + rk(x, λk, z)

)
exp
(∫ x

a

λk(t, z)dt
)
,

Mk = ∂pF /∂λ
∣∣
λ=λK

.

(4.7)

The component (uk)1 is a solution of Ty = zy. We now prove that (uk)1 ∈
L2
w[a,∞) under the addition of some further hypotheses. We will see in the next

section that these conditions hold for a large class of operators where the eigenvalues
are of equal magnitude. The bounds assumed below will be found in terms of the
coefficients of (1.1) in section 5.

For k = 1, . . . , n, assume that for some δ > 0,

− π

2
+ δ ≤ φk(x, z) ≤ π

2
− δ, (4.8)

where λk(x, z) = −γk(x, z)eiφk(x,z), γk(x, z) = |λk(x, z)| and

1

|Mk(x, z)|1/2
≤ Lk(x, z)γ

1/2
k (x, z)[1 + o(1)] (4.9)

with w(x)L2
k(x, z) bounded on [a,∞). Then from (4.7) and (4.9),

|(uk)1(x, z)| =
∣∣∣M−1/2k (x, z)

∣∣∣ (1 + o(1)
)

exp
(
−
∫ x

a

γk(t, z) cos(φk(z, t)dt)
)

≤ Lk(x, z)
(
1 + o(1)

)
γ
1/2
k (x, z) exp

(
−
∫ x

a

γk(t, z) sin(δ)dt
)
.

(4.10)

where we have used λk = −γk[cosφk + i sinφk] so that∣∣ exp
(∫ x

a

−iγk(t, z) sinφk(t, z) dt
)∣∣ = 1. (4.11)

Note that cosφk(t, z) ≥ cos(π/2− δ) = sin δ. Thus for some constant C, indepen-
dent of x, ∫ ∞

a

w(x)|(uk)1(x, z)|2 dx

≤ C
∫ ∞
a

γk(x, z) exp
(
−2

∫ x

a

γk(t, z) sin(δ)dt
)
dx

=
−C

2 sin δ
exp
(
−2

∫ x

a

γk(t, z) sin(δ)dt
)∣∣∞
a

=
C

2 sin δ
<∞;

(4.12)



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 55

hence (uk)1 ∈ L2
w[a,∞) for k = 1, . . . , n.

We define now the fundamental matrix for (2.7),

U(z, x) =
[
u1(x, z), . . . , u2n(x, z)

]
(4.13)

Consider the variation of constants formula for (4.5),

u(x) =

∫ x

a

U(x, z)P1U(t, z)−1J−1A(t)F (t) dt

−
∫ ∞
x

U(x, z)P2U(t, z)−1J−1A(t)F (t) dt,

(4.14)

where

P1

[
I 0
0 0

]
, P2

[
0 0
0 I

]
.

Then u is a pointwise solution of (4.5), and the first component y of u belongs to
L2
A[a,∞) if f ∈ L2

w[a,∞) at least for f of compact (4.14) support. By considering
only y, we see that y is of the form

y(x) =

∫ ∞
a

K(x, t)w(t)f(t) dt =: T (f), (4.15)

where K is the kernel function defined implicitly by (4.14).
We will prove below that the operator T defined by (4.15) is a bounded operator

from L2
w[a,∞) to L2

w[a,∞). This implies y ∈ L2
w[a,∞) if f ∈ L2

w[a,∞). Since by
the variation of constants formula, (Tmax − z)y = f , this proves T is one-to-one,
D(T −1) = R(T ) ⊆ D(Tmax), and Tmax − z is onto L2

w[a,∞). Thus Tmax − z has a
closed range and therefore z /∈ σess(Tmax). It also proves that T −1 is a restriction
of Tmax − z. To prove T is bounded we use a theorem of Okikiolu [32, p.190].

Theorem 4.2. Let the measures on X,Y ⊆ [a,∞) be defined by mX(x) = w(x)dx,
mY (y) = w(y)dy, and let K0(x, y) be a measurable function on [a,∞)× [a,∞) such
that∫

X

|K0(x, y)|dmX(x) ≤M2
1 , a.e., y;

∫
Y

|K0(x, y)|dmY (y) ≤M2
2 , a.e., x

for some constants M1,M2. Let T0 be the integral operator defined on L2
w(X) by

T0(f)(y) =

∫
X

K0(x, y)f(x) dmX(x). (4.16)

Then T0 is a bounded operator from L2
w(X) to L2

w(Y ) with ‖T0‖ ≤M1M2.

To apply Okikiiolu’s Theorem, we must first compute K in (4.15). We first
see that for t ≤ x, K(x, t) is the (1, n + 1) entry of U(x, z)P1U(t, z)−1, and for
x ≤ t, K(x, t) is the (1, n+ 1) entry of U(x, z)P2U(t, z)−1. By (2.32),

U(x, z) = S(x, z)M(x, z)
(
I + o(1)

)
× diag

[
exp
(∫ x

a

λ1(t, z)dt, . . . , exp
(∫ x

a

λ2n(t, z) dt
))]

,
(4.17)

where

M = diag
[
M
−1/2
1 , . . . ,M

−1/2
2n

]
.



56 H. BEHNCKE, D. HINTON EJDE/SI/02

The first row of S is all one’s, and the last column of S−1 is given by [12, p. 106],[
1/M1, . . . , 1/M2n

]t
. From (4.17) we have

U(t, z)−1 = diag
[
exp
(
−
∫ t

a

λ1(t, z)dt, . . . , exp
(
−
∫ x

a

λ2n(t, z) dt
))]

×
(
1 + o(1)

)
M(x, z)−1S(x, z)−1.

(4.18)

A calculation shows that for t ≤ x, the (1, n+ 1) entry of U(x, z)P1U(t, z)−1 is

K(x, t) = K1,+(x, t) + · · ·+Kn,+(x, t), (4.19)

where

Kk,+(x, t) =
[1 + o(1)] exp

(∫ x
t
λk(s, z) ds[

Mk(x, z)Mk(t, z)
]1/2 (4.20)

A similar calculation shows that for x ≤ t, the (1, n+1) entry of U(x, z)P2U(t, z)−1.
is

K(x, t) = K1,−(x, t) + · · ·+Kn,−(x, t), (4.21)

where, as x→∞,

Kk,−(x, t) =
[1 + o(1)] exp

(∫ x
t
λn+k(s, z) ds[

Mk(x, z)Mk(t, z)
]1/2 . (4.22)

Now make the assumption, for t ≥ a, x ≥ a, where Mk = ∂λPF (λk),

1

|Mk(x, z)Mk(t, z)|1/2
≤ Nk(x, z)γk(t, z)

w(t)
exp

(∫ x

t

|o(γk(s, z))|ds
)

(4.23)

with Nk(x, z) bounded on [a,∞). Hence, we have for some constant C, independent
of x and t, such that∫ ∞

x

∣∣Kk,+(x, t)
∣∣w(t) dt

≤ C
∫ ∞
x

w(t)

∣∣ exp
(∫ x
t
λk(s, z)

∣∣ ds∣∣Mk(x, z)Mk(t, z)
∣∣1/2 dt

≤ C
∫ ∞
x

Nk(x, z)γk(t, z) exp
(
−
∫ t

x

[1 + o(1)]γ(s, z) sin(δ)ds
)
dt

≤ −CNk(x, z)

sin δ
exp
(
−
∫ x

a

γk(t, z) sin(δ)dt
)∣∣∞
a

=
CNk(x, z)

sin δ
<∞.

(4.24)

In a similar way, we find that∫ x

a

∣∣Kk,−(x, t)
∣∣w(t) dt ≤ CNk(x, z)

sin δ
<∞. (4.25)

and thus ∫ ∞
a

∣∣Kk(x, t)
∣∣w(t) dt ≤ CNk(x, z)

sin δ
<∞. (4.26)

where

K(x, t) = K1(x, t) + · · ·+Kn(x, t).



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 57

Hence, since (4.26) holds for every k, and with N = N1 + · · ·+Nn,∫ ∞
a

∣∣K(x, t)
∣∣w(t) dt ≤ CN(x, z)

sin δ
≤ sup
x≥a

CN(x, z)

sin δ
<∞. (4.27)

This establishes the second integral of Theorem 4.2.
To establish the first integral of Theorem4.2, we make an assumption parallel to

(4.23), i.e.,

1

|Mk(x, z)Mk(t, z)|1/2
≤ Ñk(t, z)γk(t, z)

w(x)
exp

(∫ x

t

|o(γk(s, z)|ds
)

(4.28)

with Ñk(x, z) bounded on [a,∞). Repeating the above argument yields∫ ∞
a

∣∣K(x, t)
∣∣w(x) dx ≤ CÑ(t, z)

sin δ
≤ sup

t≥a

CÑ(t, z)

sin δ
<∞. (4.29)

Hence Okikiolu’s Theorem applies to the operator T defined in (4.15). This proves
the following theorem.

Theorem 4.3. Under assumptions (4.8), (4.9), (4.23), and (4.28), it follows that
for such z the maximal operator Tmax and the minimal operator Tmin satisfy

z /∈ σess(Tmax) = σess(Tmin).

In addition to having T −1 a restriction of Tmax− z, we will now prove that T −1
is an extension of Tmin − z.

Theorem 4.4. Under the assumptions of Theorem 4.3 and def (Tmin− z) = n, the
operator T −1 is an extension of Tmin − z.

Proof. Let ỹ ∈ D(Tmin), f = Tmin(ỹ), and y = T (f). Then for ŷ := ỹ − y, we have
(Tmax − z)(ŷ) = f − f = 0. Now ỹ[i](a) = 0, i = 0, . . . , 2n − 1, and if we prove
y[i](a) = 0, i = 0, . . . , 2n−1 then ŷ ≡ 0 by uniqueness of initial value problems and
ỹ = y ∈ D(T −1).

We use the form of the Lagrange identity used by Knowles [25, p. 207] for
functions y1, y2 ∈ D(Tmax), i.e.,

[y1, y2] =

n∑
k=1

(
y
[k−1]
1 y

[2n−k]
2 − y[2n−k]1 y

[k−1]
2

)
. (4.30)

Let f1 = (Tmax − z)(y1), f2 = (Tmax − z)(y2), and let u, v be the vectors corre-
sponding to y1, y2 as in (2.6). Then a computation shows that

utJv = −[y1, y2], (utJv)′ = y1wTmax(y2)− y2wTmax(y2). (4.31)

With U as in (4.13) and using the fact that U tJU = C is a constant matric as
shown in [7]. we write with n× n blocks,

U tJU = C =

[
C11 C12

C21 C22

]
. (4.32)

The elements of C11 are the Lagrange forms [ui, uj ], i.j = 1, . . . , n. Knowles [25,
Lemma 4.8] gives that these forms are all zero under the condition def(Tmin−z) = n.
Hence C11 = 0. Now U is non singular since the first components of the vectors
u1, . . . , u2n form a basis for the solutions of (Tmax − z)(y) = 0. Hence C12, C21 are



58 H. BEHNCKE, D. HINTON EJDE/SI/02

non-singular and U tJU = C gives U−1 = C−1U tJ . Using n × n blocks for U , a
computation yields

P2U
−1 = P2C

−1U tJ

=

[
0 0
0 I

]
C−1

[
U t11 U t21
U t12 U t22

]
J

=

[
0 0

C−112 U
t
21 −C−112 U

t
11

]
.

(4.33)

We define the n vectors On, f̃ , by

On = [0, . . . , 0]t, f̃ = [wf, 0, . . . 0]t.

Then from (4.14),

u(a) = −U(a, z)

∫ ∞
a

[
0 0

C−112 U
t
21 −C−112 U

t
11

] [
On
f̃

]
. (4.34)

Let y1, . . . , yn be the first components of the vectors u1, . . . , un. Then∫ ∞
a

U t11f̃ = [< y1, f̄ >, . . . , < yn, f̄ >]t. (4.35)

Since (Tmax−z)(yi) = 0 by taking conjugates we have (T+
max− z̄) ¯(yi) = 0. Hence

by (2.1),

ȳi ∈ N(T+
max − z̄) = R(Tmin − z)⊥

This gives

〈yi, f̄〉 = 〈ȳi, f〉 = 0,

and so u(a) = 0 and thus y[i](a) = 0, for i = 0, . . . , 2n− 1. �

From (3.8) we have that dimN(Tmax − z) ≥ n. Since a nontrivial combi-
nation of functions not in L2

w[a,∞) may be in L2
w[a,∞), we have not proved

that dimN(Tmax − z) = n even though we have n independent non L2
w[a,∞)

functions. Under the assumptions (4.8), (4.9), and (4.23), it is not clear that
dimN(Tmax − z) = n.

For later purposes (Theorem 4.6 for the bound (4.45)) we may repeat the above
proof, under the assumption

1

|Mk(x, z)Mk(t, z)|1/2
≤ Vr,k(x, z)γ(t, z)1/r

w(t)1/r
exp

(∫ x

t

|o(γk(s, z)|ds
)

(4.36)

with Vr,k(x, z) bounded on [a,∞), to show that for r ≥ 1,∫ ∞
x

∣∣Kk,+(x, t)
∣∣rw(t) dt

≤ C
∫ ∞
x

V rr,k(x, z)γ(t, z) exp
(
−r
∫ t

x

γ(s, z) sin(δ)ds
)
dt

≤
−CV rr,k(x, z)

r sin δ
exp
(
−r
∫ ∞
a

γk(t, z) sin(δ)dt
)∣∣∞
a

=
CV rr,k(x, z)

r sin δ
<∞,

(4.37)



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 59

and similarly, ∫ x

a

∣∣Kk,−(x, t)
∣∣rw(t) dt ≤

CV rr,k(x, z)

r sin δ
<∞, (4.38)

and hence ∫ ∞
a

∣∣Kk(x, t)
∣∣rw(t) dt ≤

CV rr,k(x, z)

r sin δ
<∞. (4.39)

We can take Vr,k = Nkγ
1−1/r/w1−1/r, but we use Vr,k to have a simpler notation.

It then follows that with Vr = Vr,1 + · · ·+ Vr,n,∫ ∞
a

∣∣K(x, t)
∣∣rw(t) dt ≤ Cnr+1V rr (x, z)

r sin δ
<∞. (4.40)

Similar arguments also show that∫ ∞
a

∣∣K(x, t)
∣∣rw(t)r/2 dt ≤ Cnr+1w(x)−1+r/2V rr (x, z)

r sin δ
<∞. (4.41)

Theorem 4.5. Assume that (4.8), (4.9), and (4.23) hold, and that in (4.23)
N(x, z0) → 0 as x → ∞, then the operator T defined by (4.15) is a compact
operator from L2

w([a,∞)) to L2
w([a,∞)).

Proof. To show that T is compact, let the integral operators Tb and T̃b be defined
by the kernels χ[a,b](x)K(x, s) and χ[b,∞)(x)K(x, s), i.e.,

(Tbf)(x) =

∫ ∞
a

χ[a,b](x)K(x, s)w(s)f(s) ds, (4.42)

and similarly for T̃b. Applying Theorem 4.2 to T̃b and using (4.29) Y = [b,∞), and

X = [a,∞), it follows that T̃b has an arbitrary small norm if b is sufficiently large.
Thus T is compact if Tb is compact as it is the limit in operator norm of compact
operators. To see that Tb is compact we employ a similar decomposition writing
Tb = Tb1 + Tb2 where

(Tb1f)(x) =

∫ b′

a

χ[a,b](x)K(x, s)w(s)f(s) ds,

(Tb2f)(x) =

∫ ∞
b′

χ[a,b](x)K(x, s)w(s)f(s) ds.

Now Tb1 is compact since its kernel is continuous on the compact set [a, b]×[a, b′].
Repeating the argument above shows that Tb2 has arbitrary small norm if b′ is
sufficiently large; thus Tb is compact. �

Recall that a compact kernel operator is a Hilbert-Schmidt operator if the kernel
K satisfies ∫ ∞

a

∫ ∞
a

|K(x, s)|2w(s)w)x) ds dx <∞. (4.43)

A compact operator T defined on a Hilbert space belongs to the Schatten class Cs
for 1 ≤ s <∞ provided that

∑∞
1 µ(T )s <∞ where the µ(T ) are the s-numbers of

T , i.e., eigenvalues of the compact operator (T T )∗)1/2. C∞ is the class of compact
operators, and C1 is the class of trace class operators. If k < s, then Ck ⊂ Cs, and
the inclusion is proper. Thus for s > 2, a Schatten class Cs operator may fail to be
Hilbert-Schmidt.

The theorem below gives an upper bound for the norm, s ≥ 2, of the Schatten
class operator Cs generated by (1.1).



60 H. BEHNCKE, D. HINTON EJDE/SI/02

Theorem 4.6. Under the conditions (4.8), (4.9), (4.23), and s ≥ 2, the operator
T defined by (4.15) is a Schatten class Cs operator if (4.30) below holds for s = 2.∫ ∞

a

Vs(x, z)
sw(x) dx <∞, (4.44)

If (4.30) holds for some s > 2, then there is a constant C, independent of s, such
that the Schatten norm ‖T ‖s of T satisfies

‖T ‖ss ≤ C
∫ ∞
a

Vs(x, z)
sw(x) dx <∞. (4.45)

Proof. From (4.40) for s = 2, we have∫ ∞
a

∫ ∞
a

∣∣K(x, t)
∣∣2w(x)w(t) dt dx ≤

Cnr+1
∫∞
a
V 2
2 (x, z)w(x)dx

r sin δ
<∞ (4.46)

which proves T is Hilbert-Schmidt if (4.30) holds.
To establish (4.45) for s > 2, we use a theorem of Russo [37]. In his theorem we

use the fact that K(x, t) = K(t, x). Define the kernel by

k(x, t) = w(x)1/2K(x, t)w(t)1/2,

and the operator T̃ : L2[a,∞)→ L2[a,∞) by

(T̃ g)(x) =

∫ ∞
a

k(x, t)g(t)dt.

Note that (4.46) implies that k ∈ L2[a,∞) × L2[a,∞) so that Russo’s theorem
applies.

Russo’s Theorem states that the Schatten class Cs norm of T̃ satisfies, using also
k(x, t) = k(t, x)

‖T̃ ‖s ≤ ‖k‖ν,s =
(∫ ∞

a

(∫ ∞
a

|k(x, t)|νdt
)s/ν

dx
)ν/s

,
1

s
+

1

ν
= 1. (4.47)

From (4.41) with r = ν, we have for some constant C1, independent of ν,(∫ ∞
a

∣∣K(x, t)
∣∣νw(t)ν/2 dt

)1/ν
≤ C1

(
w(x)−1+ν/2V νν (x, z)

)1/ν
= C1w(x)(ν−2)/2νV νν (x, z)

(4.48)

Thus ∫ ∞
a

(∫ ∞
a

|k(x, t)|νdt
)s/ν

dx

=

∫ ∞
a

(∫ ∞
a

w(x)ν/2
∣∣K(x, t)

∣∣νw(t)ν/2 dt
)s/ν

dx

≤ C1

∫ ∞
a

w(x)V νν (x, z)dx

(4.49)

which will yield (4.45) by (4.47) after we verify that ‖T ‖t = ‖T̃ ‖t. To see this
let M : L2

w([a,∞)) → L2([a,∞)) be defined by M(y) = w1/2y. Then M−1(g) =
g/w1/2 , and ‖M(y)‖L2([a,∞)) = ‖y‖L2

w([a,∞)). ThusM is isomorphic from L2
w([a,∞))

onto L2([a,∞)). Since T̃ = MTM−1, it follows that T̃ and T are unitarily equiv-

alent and ‖T ‖t = ‖T̃ ‖t. �



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 61

Theorem 4.7. Assume the hypotheses of Theorems 4.4 and 4.5. Let Tα be as in
(3.3) and assume z is not an eigenvalue of Tα and (1.1) has def Tmin = n. Then
Tα−z has a compact resolvent. Further Tα has a Hilbert-Schimdt resolvent if (4.30)
holds for s = 2.

Proof. Note that (H1) holds for z as in Theorems 4.4 and 4.5, and by section 2,
0 belongs to the resolvents of T −1 and Tα − z. By [21, Corollary 6.34 p. 188],
T = (T −1)−1 is compact if and only if (Tα − z)−1 is. Further. by [21, Lemma 6.38
p. 188], the difference (Tα − z)−1 − T is a finite rank operator. Since a finite rank
operator is both compact and Hilbert-Schmidt, the result follows. �

5. Eigenvalues of equal magnitude

When the eigenvalues are of equal magnitude, the bounds (3.4), (4.23), and (4.36)
can be found explicitly. To illustrate this, we apply a theorem of Eastham [12, p.108]
which uses less general asymptotic hypotheses than those of Section 2, but for which
the explicit expressions we need have already been made. Eastham’s theorem allows
us to estimate the factors Mk above. Note the ordering of the coefficients in [12] are
reverse of ours, i.e., his p0, . . . , pn−1, pn is our pn, . . . , p1, p0 − zw. Also the matrix
S of [12] and S of section 2 both have the properties of the first row is all ones and
the last column of S−1 have the same formula so the computation of the kernel K
of (3.10) is the same.

We show how Eastham’s Theorem can be applied to give the bounds (4.9), (4.23),
(4.28), and (4.36). First we define

P =
(p0 − zw

pn

)1/2n
.

We formulate below Eastham’s theorem in our notation.

Theorem 5.1 ([12, p.108]). Let w, pr(0 ≤ r ≤ n) have locally absolutely continuous
first derivatives in [a,∞) , and let pn and p0 − zw be nowhere zero in [a,∞). Also
let for r = 1, . . . , n− 1

(i) pn−r/[pn(x)P 2r)(x)]→ cr as x→∞, where cr is a finite limit;
(ii) the polynomial (c0 = c1 = 1 in Eastham)

g(ξ) = ξ2n + c1ξ
2n−2 + · · ·+ cn−1ξ

2 + 1 (5.1)

have 2n distinct roots ξk (1 ≤ k ≤ 2n);

(iii)
p′n−r
pn

= o(P 2r+1) as x→∞, r = 0, . . . , n− 1,
p′0−zw

′

p0−zw = o(P );

(iv)
p′′n−r

pnP 2r+1 ∈ L[a,∞), r = 0, . . . , n− 1,
p′′0−zw

′′

(p0−zw)P ∈ L[a,∞);

(v)
p′2r

p2nP
4r+1 ∈ L[a,∞), r = 0, . . . , n.

Finally, let Re[λj(x, z)−λk(x, z) have only one sign in [a,∞) for each unequal pair
j, k in [1, 2n], where the λk are the solutions of

pnλ
2n + pn−1c1λ

2n−1 + · · ·+ cn−1p1λ
2 + p0 − zw = 0

Then T [y] = λy has solutions yk(x, z), (1 ≤ k ≤ 2n), such that, as x→∞,

yk(x, z) =
(
pn(p0 − zw)2n−1

)−1/4n
[1 + o(1)] exp

(∫ x

a

λk(t, z)dt
)
. (5.2)



62 H. BEHNCKE, D. HINTON EJDE/SI/02

Eastham’s conditions arise from the leading terms in the Kummer-Liouville
transformation. So the conditions in Theorem 5.1 just arise from operators with
almost constant coefficients.

Lemma 5.2. Let g, h be functions on [a,∞) such that g(x) 6= 0, h(x) > 0, g is
absolutely continuous, g′, h ∈ Lloc[a,∞), and |g′(x)/g(x)| = o(h(x) as x → ∞.
Then

|g(x)

g(t)
| ≤ exp

∣∣ ∫ x

t

|o(h(s))| ds
∣∣ as x, t→∞.

Proof. We have

|g(x)

g(t)
| =

∣∣ exp

∫ x

t

g′(s)

g(s)
ds
∣∣ ≤ exp

∣∣ ∫ x

t

|g
′(s)

g(s)
| ds
∣∣ ≤ exp

∣∣ ∫ x

t

|o(h(s))| ds
∣∣,

which completes the proof. �

First we note from [12] that

λk = Pξk[1 + o(1)], P =
[
p0 − zw)/pn

]1/2n
, (5.3)

and for some constant ck,

Mk = ckλ
−1
k pnP

2n[1 + o(1)] = ckλ
−1
k (p0 − zw)[1 + o(1)]. (5.4)

Hence

1

|Mk(x, z)|1/2
=
|λk(x, z)|1/2[1 + o(1)]

ck|p0(x)− zw(x)|1/2
=
γk(x, z)1/2[1 + o(1)]

ck|p0(x)− zw(x)|1/2
. (5.5)

so we have

Lk(x, z) :=
1

ck|p0(x)− zw(x)|1/2
,

Hence condition (4.9) becomes

w(x)

ck|p0(x)− zw(x)|1/2
is bounded on [a,∞). (5.6)

Hence, yk ∈ L2
w[a,∞) for k = 1, . . . , n, when (5.6) holds.

From (iii) above we have that |p′n/pn| = o(γ) and |(p′0−zw′)/(p0−zw)| = o(γ) so
we can apply Lemma 5.2 with g = pn, h = γ, and g = p0− zw, h = γ, respectively.

Further for some constant dk,

1

|Mk(x, z)Mk(t, z)|1/2

= dk
[γk(x, z)γk(t, z)]1/2[1 + o(1)]

|p0(x)− zw(x)|1/2|p0(t)− zw(t)|1/2

= dk
w(x)

|p0(x)− zw(x)|
w(t)

w(x)

[γk(x, z)

γk(t, z)

]1/2∣∣p0(x)− zw(x)

p0(t)− zw(t)

∣∣1/2 γk(t, z)

w(t)

= dk
w(x)

|p0(x)− zw(x)|
exp

∣∣ ∫ x

t

|o(γk(s, z)|ds
∣∣γk(t, z)

w(t)

(5.7)

where we have applied Lemma 5.2. Thus one can define

Nk(x, z) := dk
w(x)

|p0(x)− zw(x)|
. (5.8)



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 63

Hence condition (4.23) is

w(x)

|p0(x)− zw(x)|
is bounded on [a,∞). (5.9)

Note that Nk and γk are independent of k except for a constant.

To compute Vr for r ≥ 1, we will use Vr,k = Nkγ
1−1/r
k /w1−1/r mentioned earlier.

Also because of the independence of k in Nk, γk, we use just Vr. Hence from (5.3)
and (5.8),

Vr(x, z) = (const.)
Nk(x)γk(t, z)1−1/r

w(x)1−1/r

= (const.)
Nk(x)γk(x, z)1−1/r

w(x)1−1/r
γk(t, z)1−1/r

γk(x, z)1−1/r

= (const.)
w(x)1/r

|p0(x)− zw(x)|
∣∣p0(x)− zw(x)

pn(x)

∣∣ r−1
2rn

= (const.)
w(x)1/r

|pn(x)| r−1
2rn |p0(x)− zw(x)| 2rn+1−r

2rn

.

(5.10)

where we have applied Lemma 5.2 as in (5.7). Thus we can define

Vr(x, z) :=
w(x)1/r

|pn(x)| r−1
2rn |p0(x)− zw(x)| 2rn+1−r

2rn

. (5.11)

Hence condition (4.36) is

w(x)1/r

|pn(x)| r−1
2rn |p0(x)− zw(x)| 2rn+1−r

2rn

is bounded on [a,∞). (5.12)

Thus (4.30) is∫ ∞
a

Vs(x, z)
sw(x) dx =

∫ ∞
a

w(x)2

|pn(x)| s−1
2n |p0(x)− zw(x)| 2sn+1−s

2n

dx <∞. (5.13)

Finally, we come to the issue of (4.8). Here the roots of the polynomial (5.1)
are needed. This is particularly simple for the two term equation, i.e., pr = 0,
r = 1, . . . , n − 1. In this case, (5.1) is simply ξ2n + 1 = 0. For n = 1: ξ = ±i For

n = 2: ξ = ±(1± i)/
√

2, etc.
We examine the case n = 1 more in detail when w/p0 → 0 as x→∞. Suppose

a root of (p0/p1)1/2 satisfies

δ ≤ arg(p0/p1 ≤ 2π − δ (5.14)

for some δ > 0. From (5.3),

arg λk(x, z) =
(

arg ξk +
1

2
arg

p0
p1

)(
1 + o(1)

)
(5.15)

Choosing ξk = −i, gives

−π
2

+
δ

2
≤ arg ξk +

1

2
arg

p0
p1
≤ π

2
− δ

2
(5.16)

Thus for some δ′ > 0 and sufficiently large x,

−π
2

+ δ′ ≤ arg λk(x, z) ≤ π

2
− δ′; (5.17)



64 H. BEHNCKE, D. HINTON EJDE/SI/02

hence (4.8) holds. This agrees with the results of [8] when one takes into account
that the leading coefficient there preceded by a minus sign.

Finally, we give a fourth order example to illustrate that Theorem 4.6 gives new
results even for selfadjoint operators.

Example 5.3.
τ [y] = yiv + xαy, 0 < a ≤ x <∞. (5.18)

This equation is known to be limit point at infinity, i.e., def(Tmin−z) = 2, Im z 6= 0,
and to have spectrum that is discrete and bounded below. Let Tα be a selfadjoint
operator generated by (5.18). As the coefficients are real, for all z non real, hypoth-
esis (H1) holds. To apply Theorem 4.7, choose n = s = 2, z = i. The criteria for
a Hilbert-Schmidt kernel for the resolvent (Tα − z)−1 with z non real by Theorem
4.7 is then ∫ ∞

a

dx

|xα − i|7/4
<∞ (5.19)

which is equivalent to α > 7/4.

6. Other equations of higher order

The spectral analysis of higher order differential operators faces several diffi-
culties. First of all the characteristic polynomial has to be factored. Then the
dichotomy condition for the roots has to be shown. Finally, the eigenfunctions and
resolvents have to be analyzed. The fourth order differential operators are some-
how the gateway to higher order operators in as much new phenomena can first be
observed for this class of operators. However, determining the roots of the charac-
teristic polynomial is no problem at all, so that one can concentrate on the other
critical phenomena. To avoid any technical difficulties, we consider only operators
of the form (6.1) below. Here we only discuss the approach and refer to the litera-
ture for precise results in the case of real coefficients, and only indicate here how a
similar approach may be carried out for operators of the form,

τ [y] = (y′′)′′ + (p1y
′)′ + p0y. (6.1)

The characteristic polynomial

PF (x, λ) = λ4 + p1λ
2 + p0 − z (6.2)

has the roots, [12, p. 126],

λ1 = −λ2 =
1√
2

√
−p1 + ∆, λ3 = −λ4 =

1√
2

√
−p1 −∆,

∆ =
√
p21 − 4(p0 − z)

(6.3)

Of course we will also assume the usual properties of smoothness and decay
(2.11) for the coefficients p1 and p0. Even though we have an explicit factorization
of the Fourier polynomial, we will still have to demand the dichotomy condition,
even though it is mostly easy to check in this case. The form factors

Mj = 4λ3j + p1λj

are unbounded if p0 is. In this case the operator has a compact resolvent. If p0 and
p1 are bounded, then continuous spectrum may arise. The most interesting case is,
when p1 is dominant, i. e.,

(p0 − z) = o(p21). (6.4)



EJDE-2023/SI/02 C-SYMMETRIC NON-SELFADJOINT DIFFERENTIAL OPERATORS 65

In this case,

∆ = p1 − 2(p0 − z)/p1 +O((p0 − z)2/p31)

so that

p
−1/2
1 λ1 =

1√
2

(
− 2

p0 − z
p1

+O
( (po − z)2

p21

))1/2
,

λ2 =
1√
2

(
− 2p1 +O

(p0 − z
p1

))1/2 (6.5)

In this case the dichotomy condition holds if it holds for λ1, λ2 and for λ3, λ4. The

eigenvalues in the (1,2) block are proportional to p
−1/2
1 , while the off block elements

are proportional to p′1p
−3/2
1 . This means that a further diagonalization will turn

these integrable expressions so that the problem is essentially that of blocks (1,2)
and (3,4) Sturm-Liouville operators. One of these, the (2,4) block gives discrete
spectrum.

This phenomenon can be observed for higher order operators with a dominant
middle term. Even this can be generalized to operators with several classes of
eigenvalues of different magnitude. For real coefficients this has bee carried out by
Behncke and Nyamwala [5].

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66 H. BEHNCKE, D. HINTON EJDE/SI/02

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Horst Behncke
Fachbereich Mathematik/Informatik, Universitat Osnabruck, 49069 Osnabruck, Ger-
many

Email address: sabine.schroeder@uni-osnabrueck.de

Don Hinton

Mathematics Department, University of Tennessee, Knoxville, TN 37996, USA

Email address: dhinton1@tennessee.edu


	1. Introduction
	2. Spectral theory and asymptotic integration
	2.1. Spectral theory
	2.2. Asymptotic integration

	3. Resolvent
	4. Conditions for ess(Tmin)=C
	4.1. Almost constant coefficient case
	4.2. General case

	5. Eigenvalues of equal magnitude
	6. Other equations of higher order
	References