1 Finite dimensional perturbation theory

Theorem 1.1. Let T (β) be a matrix-valued analytic function near β_{0} and suppose λ_{0} be a simple eigenvalue of T (β_{0}). Then

1. For β near β_{0}, T (β) has exactly one eigenvalue λ_{0}(β) near λ_{0}.

2. λ_{0}(β) is a simple eigenvalue if β is near β_{0}.

3. λ_{0}(β) is analytic near β = β_{0}.

Theorem 1.2. If T (β) is a matrix-valued analytic function near be_{0} and if λ_{0} is an eigenvalue of T (β_{0}) of algebraic multiplicity m, then for β near β_{0} T (β) has exactly m eigenvalues (counting multiplicity) near λ_{0}. These eigenvalues are all the branches of one or more multivalued functions analytic near β_{0} with at worst algebraic singularities at β_{0}.

Theorem 1.3 (Rellich’s theorem). Suppose that T (β) is a matrix-valued analytic function in a region R containing a section of the real axis, and that T (β) is self-adjoint for β on the real axis. Let λ_{0} be an eigenvalue of T (β_{0}) of multiplicity m. If β_{0} is real, there are p < m distinct functions λ_{1}(β), ⋯, λ_{p}(β), single-valued and analytic in a neighborhood of β_{0}, which ara all the eigenvalues.

The main problem we want to consider is of the form H(β) = H_{0} + βV. Suppose that E_{0} is a nondegenerate eigenvalue of H_{0}. From theorem 1.1 we know that for small β, H(β) has a unique eigenvalue E(β) near E_{0} and that E(β) is analytic near β = 0. The coefficient of its Taylor series are called Rayleigh-Schrödinger coefficients and the Taylor series is called the Rayleigh-Schrödinger series. For the sake of simplicity, we only consider the case of the self-adjoint operator H_{0}. Then E(β) is the only eigenvalue of T (β) in {E | ∥E −E_{0}∥ ≤ ϵ} for small ϵ. We can have the projection onto the eigenvector associated with the eigenvalue E(β)

P (β) = −\frac{1}{2πi} ∮_{|E−E_{0}|=ϵ}(H_{0} + βV − E)^{−1}dE.

It can be shown that (H_{0} + βV − E)^{−1} is analytic in β near β = 0. Thus P (β) is analytic in β near β = 0.

Lemma 1.4. Let Ω_{0} be an eigenvector associated with the eigenvalue E_{0}, that it, H_{0}Ω_{0} = E_{0}Ω_{0}. Then

E(β) = \frac{(Ω_{0}, H(β)P (β)Ω_{0})}{(Ω_{0}, P (β)Ω_{0})} = E_{0} + \frac{(Ω_{0}, V P (β)Ω_{0})}{(Ω_{0}, P (β)Ω_{0})}

Proof. First, we have to make sure the denominator is nonzero. Since P (β)Ω_{0} → Ω_{0}, P (β)Ω_{0} ≠ 0 for small β. As P (β)Ω_{0} is an eigenvector of H(β)

E(β) = \frac{(Ω_{0}, H(β)P (β)Ω_{0})}{(Ω_{0}, P (β)Ω_{0})}.

The last equation follows from the fact that H_{0} is self-adjoint and E_{0} is an eigenvalue of H_{0}.

As a final result in finite dimensional perturbation theory:

Theorem 1.5. Let Ω_{0} be a nondegenerate eigenvalue for T_{0} with an eigenvalue E_{0}, and let T (β) be a matrix-valued analytic function with T (0) = T_{0}. Then for small β there is vector-valued analytic function Ω(β) that satisfies T (β)Ω(be) = E(β)Ω(β), where E(β) is the eigenvalue of T (β) near E_{0}. Moreove if T (β) is self-adjoint for β real, Ω(β) can be chosen so that ∥Ω(β)∥ = 1 for β real.

Proof. By theorem 1.1 there exist ϵ such that if |E − E_{0}| ≤ ϵ, then for some neighorbood of β_{0} T (β) has only one eigenvalue in {E | |E − E_{0}| ≤ ϵ}. So one can define

ψ(β) = −\frac{1}{2πi} ∮_{|E−E_{0}|=ϵ}(T (β) − E)^{−1}Ω_{0}dE ≡ P (β)Ω_{0}.

2 Regular perturbation theory

This section is divided into four parts:

1. A brief discussion of the discrete spectra of not necessarily self-adjoint operators.

2. A proof of the analyticity of discrete eigenvalues in the nondegenerate case for analytic families of operators.

3. Two simple criteria (type (A) and type (B)) for H_{0} + βV to be an analytic family.

4. A brief discussion of degenerate perturbation theory.

The main goal is to prove that under very general circumstances the Rayleigh-Schrödinger series has a nonzero radius of convergence for perturbations of unbounded operators in infinite dimensional Hilbert spaces.

Definition 2.1. Let X be a Hilbert space and T be in L(X).

1. The resolvent set ρ(T) is the set of complex numbers λ such that T − λI is a bijective and has a continuous inverse.

2. The mapping λ ↦ (T − λI)^{−1} from the resolvent set ρ(T) to the set of the isomorphisms of X, B(X) is called the resolvent.

3. The spectrum σ(T) is the complement of the resolvent set. The spectrum is classfied into three classes:

(a) The point spectrum σ_{p}(T) is the set of the complex numbers λ such that T − λI is not injective, that is, λ is an eigenvalue of T.

(b) The residual spectrum σ_{r}(T) is the set of the complex numbers λ such that R(T − λI) is not dense in X.

(c) The continuum σ_{c}(T) is the set of the complex numbers λ such that R(T − λI) is dense in X but not X.

Remark 2.2. 1. ρ(T) is open and therefore σ(T) is closed.

2. If T is self-adjoint, then σ(T) is a subset of the real numbers.

3. Thus σ(T) = σ_{p}(T) ∪ σ_{r}(T) ∪ σ_{c}(T).

Here is another classification of the spectrum.

Definition 2.3. Assume the condion of the definiton 2.1.

1. The discrete spectrum σ_{d}(T) is the set of isolated eigenvalues of finite multiplicity.

2. The essential spectrum σ_{ess}(T) is the complement of σ_{d}(T) in σ(T). σ_{ess}(T) contains all

eigenvalues of infinite multiplicity and all accumulation points of σ(T). A point λ ∈ σ(T) is called discrete if λ is isolated and P_{λ} is finite dimensional meaning that R(P_{λ}) is finite dimensional; if P_{λ} is one dimensional, λ is said to be a nondegenerate eigenvalue. Theorem 2.4. Suppose A is a closed operator and let λ be an isolated point of σ(A). Explicitly, suppose that {μ||μ − λ| < ϵ} ∩ σ(A) = {λ}.

1. For any r with 0 < r < ϵ,

P_{λ} = −\frac{1}{2πi} ∮_{|μ−λ|=r}(A − μI)^{−1}dμ

exists and is independent of r.

2. P_{λ}^{2} = P_{λ}. Thus P_{λ} is a (not necessarily orthogonal) projection.

3. If G_{λ} = R(P_{λ}) and F_{λ} = ker P_{λ}, then G_{λ} and F_{λ} are complementary (not necessarily orthogonal) closed subspaces, that is, G_{λ} + F_{λ} = X and G_{λ} ∩ F_{λ} = 0. Moreover, A leaves G_{λ} and F_{λ} invariant in the following precise sense: G_{λ} ⊂ D(A), AG_{λ} ⊂ G_{λ}, F_{λ} ∩ D(A) is dense in F_{λ} and A(F_{λ} ∩ D(A)) ⊂ F_{λ}.

4. If ψ ∈ G_{λ} and G_{λ} is finite dimensional, then (A − λI)^{n}ψ = 0 for some n.

If B := A|_{F_{λ}}, then λ ∉ σ(B). The next is a converse to Theorem 2.4.

Theorem 2.5. Let A be an operator with {μ||μ − λ| = r} ⊂ ρ(A).

1. Then P = (−2πi)^{−1} ∮_{|μ−λ|=r}(A − μI)^{−1}dμ is a projection.

2. If P has dimension n < ∞, then A has at most n points of its spectrum in {μ||μ − λ| < r} and each is discrete.

3. If n = 1, there is exactly one spectral point in {μ||μ − λ| < r} and it is nondegenerate. Definition 2.6. A (possibly unbounded) operator-valued function T (β) on a complex domain R is called an analytic family or an analytic family in the sense of Kato if and only if

1. For each β ∈ R, T (β) is closed and has a nonempty resolvent set.

2. For every β_{0} ∈ R, there is a λ_{0} ∈ ρ(T (β_{0})) so that λ_{0} ∈ ρ(T (β)) for β near β_{0} and (T (β) − λI)^{−1} is an analytic operator-valued function of β near β_{0}.