Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 271657, 10 pages
doi:10.1155/2012/271657
Research Article
Characterization of Eigenvalues in Spectral Gap for
Singular Differential Operators
Zhaowen Zheng and Wenju Zhang
School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China
Correspondence should be addressed to Zhaowen Zheng,
Received 5 June 2012; Revised 14 August 2012; Accepted 26 August 2012
Academic Editor: Michiel Bertsch
Copyright q 2012 Z. Zheng and W. Zhang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The spectral properties for n order differential operators are considered. When given a spectral gap
a, b of the minimal operator T0 with deficiency index r, arbitrary m points βi i 1, 2, . . . , m in
a, b, and a positive integer function p such that m
i1 pβi ≤ r, T0 has a self-adjoint extension T
such that each βi i 1, 2, . . . , m is an eigenvalue of T with multiplicity at least pβi .
1. Introduction
In this paper, we consider the following nth-order formal symmetric differential expression:
⎧
⎫
n−1/2
⎨n/2
j
j1
j ⎬
τy w−1
− qj∗ yj1
,
−1j pj yj
−1j qj yj
⎩ j0
⎭
j0
1.1
on α, β, where −∞ ≤ α < β ≤ ∞, y yt is a complex-valued m-vector function,
pj t, qj t and wt are measurable and locally integrable m × m matrices, pj t, wt are
Hermitian, and wt > 0. a.e. t ∈ α, β.
The spectral properties of nth-order differential expression, particularly the distribution of eigenvalues, have been widely researched in these years see 1–8 and references
cited therein.
Let us recall some known results due to Neumann 1, Stone 2, Friedrichs 3 and
Krein 4. For convenience of the reader in addition to these original sources we will also
give text book references 7.
2
Abstract and Applied Analysis
An open interval a, b with −∞ < a < b < ∞ is called a spectral gap of a symmetric
operator A if
A − a b f ≥ b − a f
w
2
2
w
∀f ∈ DA.
1.2
· w and ·, · w are defined in Section 2. It is easy to find that a b/2 is a real regular
point of A. If Af, f w ≥ bf2w , we shall also say that −∞, b is a spectral gap of A the
latter definition is not generally used but convenient for our purpose; and it is justified since
each interval a, b with a < b is a spectral gap of A. Let a, b be a spectral gap of A, then
of A for instance the famous Friedrichs extension
there exists a self-adjoint extension A
which will be defined in
such that a, b ⊂ ΓA here ΓA is the regular-form domain of A
Definition 2.4, this is the reason why we call a, b a spectral gap of A.
Suppose in addition that the deficiency index of a symmetric operator A is equal
be a self-adjoint extension of A. The sum of the multiplicities of the
to n n ∈ N. Let A
within the interval a, b is at most n, and no point of the continuous
eigenvalues of A
lies in the interval a, b cf. 5, Theorem 8.19 and Corollary 2 in Section 8.3.
spectrum of A
Conversely, given any finite subset E of a, b and positive integers pλ, λ ∈ E, such that
λ∈E pλ ≤ n there exists a self-adjoint extension A of A such that σA ∩ a, b E and
are equal pλ for each λ ∈ E cf. 4. If one
the multiplicities of λ as an eigenvalue of A
only requires that a, b is some interval within the set of real regular points of A, then the
such that
corresponding statement is false. For instance, one can give a symmetric operator A,
the deficiency index of A
is equal one and each self-adjoint
each λ ∈ R is a regular point of A,
cf. 4.
extension A of A has a periodic point spectrum with period p independent of A
This paper consists of three sections including the introduction. In Section 2, we
present some preliminary materials that include definitions and theorems needed for the rest
of the paper. In Section 3, we give three main results in the study of self-adjoint extensions
of a minimal operator generated by differential expression τ. Firstly, we present a partial
self-adjointness of the minimal operator T0 . Secondly, if a, b is a spectral gap of T0 , for all
β1 , β2 , . . . βm ∈ a, b m ≤ r, r is the deficiency index of T0 and positive integer function p
satisfying m
i1 pβi r, T0 has a self-adjoint extension T with the following properties:
i σp T ∩ a, b {β1 , β2 , . . . βm };
ii each βi i 1, 2, . . . m is an eigenvalue of T with multiplicity equal to pβi ;
iii T has pure point spectrum within a, b.
Finally, given a symmetric operator T0 with the deficiency index being equal to r, a, b ⊂ R,
β1 , β2 , . . . βm ∈ a, b m ≤ r being real regular points of T0 , the interval a, b need not be
a spectral gap of T0 and positive integer function p satisfying m
i1 pβi ≤ r, T0 has a self
adjoint extension T with the properties that each βi i 1, 2, . . . m is an eigenvalue of T with
multiplicity at least βi .
Abstract and Applied Analysis
3
2. Preliminaries
In this section, we introduce notations, definitions, and some theorems that are needed in this
paper.
First, we define the following space:
L2 α, β; w :
β
f:
∗
f twtft < ∞ ,
2.1
α
with the inner product
f, g w
β
g ∗ twtftdt,
2.2
α
where the weight function wt is the same as that in 1.1. Denote fw f, f w 1/2 for
f ∈ L2 α, β; w. If f ∈ L2 α, β; w, then f is called square integrable. Here, we note that if w is
singular, L2 α, β; w is a quotient space in the sense that y z if and only if y − zw 0. In
this case, L2 α, β; w is a Hilbert space.
Now we introduce the maximal operator T and minimal operator T0 generated by the
expression τ.
Definition 2.1. The maximal operator T generated by τ is defined by
DT f ∈ L2 α, β; w : f {0} , f {1} , . . . f {n−1} ∈ ACloc α, β , τf ∈ L2 α, β; w ,
T f τf
2.3
for f ∈ DT ,
where ACloc α, β denotes the collection of functions on α, β which are absolutely
continuous locally. Roughly speaking the jth quasi-derivative u{j} will be collection of terms
that, if differentiated n − j times, is “part” of the differential expression rτu see 7 for
details. We know that T is densely defined and closed.
Definition 2.2. The preminimal operator T00 generated by τ is defined by
DT00 f ∈ DT : f has compact support in α, β ,
T00 f τf T f
for f ∈ DT00 .
2.4
Obviously, T00 ⊂ T and T00 is Hermitian. It is easy to know that DT00 is dense, so T00 is
symmetric, and T00 is not closed 8. The closure of T00 is called the closed minimal operator
denoted by T0 .
Definition 2.3. Given a linear operator A with domain and range in a Hilbert space H, the
resolvent set of A is
ρA λ (...truncated)
