PERTURBATIONS OF SELFADJOINT OPERATORS WITH DISCRETE SPECTRUM
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of
Philosophy in the Graduate School of the Ohio State University
By
James Adduci, BS
Graduate Program in Mathematics
The Ohio State University
2011
Dissertation Committee:
Professor Boris Mityagin, Advisor
Professor Gregory Baker
Professor Paul Nevai c Copyright by
James Adduci
2011 ABSTRACT
Consider a selfadjoint operator A whose spectrum is a set of eigenvalues {λ0 < λ1 < . . .}
∞ with corresponding eigenvectors {φk}k=0. Now introduce a perturbation B and set
α−1 1−α L = A + B. We prove that if λn+1 − λn ≥ cn ∀n and kBφnk n → 0 for some fixed α > 1/2 then the spectrum of L = A + B is discrete, eventually simple and the set of eigenvectors of L = A + B plus at most finitely many associated vectors form an unconditional basis. As an application we consider Schr¨odingeroperators of
β the form Ly = −y00 + |x| y + b (x) y on L2 (R) where b is a possibly complex-valued function and β > 1.
ii To Helen and Anthony
iii ACKNOWLEDGMENTS
I would like to express gratitude to my advisor Professor Boris Mityagin for his generous instruction, support and guidance over the past six years. I would also like to thank Professor Greg Baker and Professor Paul Nevai for participating in my dissertation committee.
iv VITA
1983 ...... Born in Geneva, Illinois
2004 ...... B.Sc. in Applied Mathematics,
2005-Present ...... Graduate Teaching Associate, The Ohio State University
PUBLICATIONS
J. Adduci, P. Djakov, B. Mityagin, Convergence radii for the eigenvalues of tri- diagonal matrices. Lett Math. Phys. 91, 45–60 (2010).
FIELDS OF STUDY
Major Field: Mathematics
Specialization: Analysis
v TABLE OF CONTENTS
Abstract ...... ii
Dedication ...... ii
Acknowledgments ...... iv
Vita...... v
CHAPTER PAGE
1 Introduction ...... 1
2 Statement of Main Results ...... 6
2.1 Notational conventions ...... 6 2.2 Localization of the spectrum ...... 8 2.3 Discrete Hilbert transform ...... 10 2.4 Main results ...... 12
3 Localization of the spectrum ...... 14
3.1 A technical lemma ...... 14 3.2 Proof of Proposition 2.2.1 ...... 16 3.3 Proof of Proposition 2.2.2 ...... 22
4 The discrete Hilbert transform ...... 23
4.1 Technical preliminaries ...... 23 4.2 Proof of Proposition 2.3.1 ...... 26
5 Proof of Main Results ...... 33
5.1 Proof of Theorem 2.4.1 ...... 33
6 Application to Schr¨odinger operators ...... 36
6.1 Eigenvalues ...... 36 6.2 Eigenfunctions ...... 37
vi 7 Eigenfunction asymptotics ...... 41
7.1 Pointwise bounds ...... 41 7.2 Pseudodifferential operators ...... 44
Bibliography ...... 46
vii CHAPTER 1
INTRODUCTION
Consider a selfadjoint operator A defined on a separable Hilbert space H. Suppose that the resolvent
R0 (z) = (z − A)−1 is compact for some z∈ / SpA and therefore all z∈ / SpA. Then the spectrum of A
∞ consists of a denumerable set {λn}n=0 with
lim |λn| = ∞. n→∞
Denote by φn the eigenvector corresponding to λn: Aφn = λnφn. Expansions in the basis of eigenfunctions of A have several desirable properties. For any f ∈ H we have
∞ X f = fnφn and n=0
∞ X the action of A on f is then determined by Af = λnfnφn. The eigenfunctions are n=0 0, m 6= n orthogonal hφm, φni = and we have the Parseval identity: 1, m = n
∞ 2 X 2 kfk = |fn| . (1.0.1) n=0 The introduction of a non-selfadjoint perturbation to A, A + B with domA ⊆
1 domB, can spoil these properties. Consider for example the 2 × 2 perturbed matrix introduced in [2]: n 0 0 ρ Mn (ρ) = + . 0 n + 2 −ρ 0
For 0 < ρ < 1 an elementary calculation shows that q q 1 + p1 − ρ2 1 − p1 − ρ2 √ √ 2 2 φ1 = , φ2 = ρ ρ √ q √ q 2 1 + p1 − ρ2 2 1 − p1 − ρ2
p 2 are eigenvectors corresponding to the eigenvalues λ1 = n + 1 + 1 − ρ , λ2 =
p 2 ⊥ n + 1 − 1 − ρ respectively. Now, denoting by φ1 a vector perpendicular to φ1 ρ √ q p 2 1 + 1 − ρ2 ⊥ φ = , 1 q 1 + p1 − ρ2 − √ 2
⊥ p 2 we have hφ1, φ2i = ρ and φ1 , φ2 = − (1/2) 1 − ρ . From this we derive the decomposition:
⊥ 2ρ 2 φ = φ1 − φ2. 1 p1 − ρ2 p1 − ρ2
So we see that Parseval’s inequality fails and as ρ ↑ 1 the coefficients of φ1, φ2 in the
⊥ expansion of φ1 approach ∞ and −∞ respectively.
2 We now take this example one step further. Consider now the infinite matrices: 1 0 0 0 0 ... 0 3 0 0 0 ... 0 0 5 0 0 ... M = , (1.0.2) 0 0 0 7 0 ... 0 0 0 0 9 ... ......
M1 (0) 0 0 0 0 ... 0 M (2/3) 0 0 0 ... 5 0 0 M (4/5) 0 0 ... 0 7 M = (1.0.3) 0 0 0 M9 (6/7) 0 ... 0 0 0 0 M (8/9) ... 11 ......
2 2 acting on ` (N) defined in terms of 2 × 2 blocks. If we decompose the space ` (N) into the blocks H0 = span (e0, e1) ,H2 = span (e2, e3) ,..., then
2 ` (R) = H0 × H2 × H4 × ...
0 and each H2n is an invariant subspace for M . However, if we denote the eigenfunc-
0 tions of M in H2n by φ2n and φ2n+1 then by the formula above we have
2n − 1 4n2 φ⊥ = 2n φ − φ . 2n 4n − 1 2n 4n − 1 2n+1
0 So the eigenfunctions {φn} of the operator M do not satisfy the Parseval identity (1.0.1) or even the following weakened variant of the Parseval identity:
∞ X 2 X 2 f = fkφk implies kfk ≤ C |fn| . (1.0.4) n=0 3 In this work we will give conditions under which the eigenbasis of a perturbation of a selfadjoint operator is similar to the (orthogonal) eigenbasis of the unperturbed operator. The following 1967 result of Kato will play an important role in our analysis.
0 ∞ Theorem 1.0.1. [12] Let Qk k=0 be a complete family of orthogonal projections ∞ in a Hilbert space H and let {Qk}k=0 be a family of (not necessarily orthogonal) projections such that QjQk = δj,kQj. Assume that