CHAPTER 1

BASIC CONCEPTS AND FORMULAS

The main tools in investigation of the Schrödinger equation are methods of functional analysis and theory of distributions: the Fourier transform, the Fredholm Theorem, and the Sobolev Embedding Theorems. In particular, the Fourier transform allows us to obtain an integral representation for the free Schrödinger propagator. The bounds for differential operators and the Sobolev Embedding Theorem extend to the weighted Sobolev spaces using the techniques of pseudodifferential operators.

1 DISTRIBUTIONS AND FOURIER TRANSFORM

A detailed theory of tempered distributions can be found in [24, 40, 41, 72, 78]. It is one of the main tools of the modern theory of partial differential equations.

Definition 1.1. The Schwartz space of test functions is the space ofall smooth complex-valued functions ψ(x) on with finite seminorms

for all N > 0 and all multi-indices α = (α1, …, αn) with αk = 0, 1,….

Thus, functions in with all their derivatives decay faster than the inverse of any polynomial. A sequence φk converges to 0 in if for all N, α

Definition 1.2. The Schwartz space of tempered distributions ′ = ′() is the space of all linear continuous functionals f : → . By definition,

(1.1)

For f(x) ∈ C() or f(x) ∈ 2 := 2(), the corresponding distribution is defined by

The Fourier representation for φ(x) ∈ () reads

(1.2)

(1.3)

The Fourier transform F is a linear bicontinuous bijection () → (). It can be extended by continuity to tempered distributions by the formula

so F : ′ → ′ is also a linear bicontinuous bijection. Let us note the following basic properties of the Fourier transform in ′:

(1.4)

where .

F2. F : 2 → 2 is a unitary operator, and the Plancherel identity holds

(1.5)

where ψ, ψ1 ψ2 ∈ 2 and || · || stands for the norm in 2.

F3. For ∈ 1 := 1(), formula (1.2) remains valid. Similarly, the second formula holds for ψ ∈ 1.

2 FUNCTIONAL SPACES

We will work with various versions of the Sobolev functional spaces. The introduction of the spaces and weak derivatives by Sobolev around 1930 resulted later in the Schwartz theory of distributions and turned the theory of partial differential equations into a chapter on modern functional analysis.

2.1 Sobolev spaces

We denote by s = s(3) the Sobolev space for s ∈ . For s = 0, 1, 2, … this is the Hilbert space of functions which belong to the 2 space as well as their distribution derivatives up to the order s. In particular, 0 = 2. For an arbitrary s ∈ the Sobolev space is defined in terms of the Fourier transform: s is the Hilbert space of tempered distributions ψ(x) with the finite norm

(2.1)

By the Plancherel identity, the Sobolev norm (2.1) can be written as

Obviously:

i) The embedding s1 ⊂ s2 is continuous for s1 ≥ s2.

ii) The scalar product (·, ·) in 2 extends to the duality between s and –s for every s ∈ :

where ψ1 ∈ s and ψ2 ∈ –s.

For any subset B ⊂ 3, denote

(2.2)

The following Sobolev Embedding Theorems play a crucial role every where below.

Theorem 2.1. ([40, Theorem 5.3]) For any s > 3/2 the embedding s(3) ⊂ Cb(3) is a bounded operator.

Theorem 2.2. ([40, Theorem 7.2]) For any bounded subset B ⊂ 3 and s1 > s2, the embedding

(2.3)

is a compact operator.

By definition (see [40, p. 19], [55, p. 233], and [96, p. 277]), embedding (2.3) is compact if

(2.4)

2.2 Agmon-Sobolev weighted spaces

For σ ∈ denote by the Hilbert space of functions with the finite norm

The weighted norms are a suitable tool for characterization of diverging waves with the conserved L2-norm. Namely, let us consider a function ψ(x, t) such that

(2.5)

where σ > 3/2. Then obviously

(2.6)

for any R > 0. The inverse is true if

(2.7)

Exercise 2.3. Check that (2.6) together with (2.7) implies (2.5).

Further, let us define weighted Agmon-Sobolev spaces. For s, σ ∈ we will denote by the Hilbert space of tempered distributions ψ(x) with the finite norm

(2.8)

In particular, and . By definition, the operator (1 – Δ)p : is continuous for any p, s, σ ∈ . The following lemma will play an important role below.

Lemma 2.4. For any s, σ ∈ :

i) The operator of multiplication by xj : is continuous.

ii) The operator of dijferentiation •j : is continuous.

Proof i) We should check that

In other words,

(2.9)

Let us denote f = 〈x〉σ〈〉sψ. Then ψ = 〈〉–s[〈x〉–σ f], and hence (2.9) reads

The product of the operator 〈x〉σ–1 〈〉sxj 〈〉–s〈x〉–σ is a continuous operator in 2 by theorems on composition and boundedness of pseudodifferential operators (PDOs). The theorems for the classes of PDOs, generated by the operators 〈x〉σ and 〈〉s with any s, σ ∈ , can be proved by Standard PDO technique [3, 40, 77].

ii) The continuity of the operator follows similarly.

The Sobolev Embedding Theorems 2.1 and 2.2 extend to the weighted Sobolev spaces:

Theorem 2.5. i) For s > 3/2 and any σ ∈ the embedding ⊂ C(3) is continuous.

ii) For s1 > s2 and σ > σ2 the embedding is a compact operator.

2.3 Operator-valued functions

Let H1 and H2 be two Hilbert spaces. Denote by (H1, H2) the space of linear continuous operators A : H1 → H2 with the norm

Let Ω be a subset in and A(ω) : H1 → H2 be an operator-valued function defined for ω ∈ Ω.

Definition 2.6. i) An operator-valued function A(ω) is uniformly continuous if

for any ω ∈ Ω.

ii) An operator-valued function A(ω) is strongly continuous if A(ω)ψ ∈ C(Ω, H2) for each ψ ∈ H1.

3 FREE PROPAGATOR

The free Schrödinger equation

(3.1)

corresponds to the zero potential V(x) = 0. Here all the derivatives are understood in the sense of distributions. The solution is defined uniquely by initial condition

(3.2)

3.1 Fourier transform

A formula for solutions to the initial problem (3.1), (3.2) can be calculated by the Fourier transform using the methods of analytic functions. Let us consider solutions ψ(· t) ∈ C(, 2) to (3.1), (3.2).

Proposition 3.1. For every initial data ψ0 ∈ 2 ∩ 1, the solution ψ(·, t) ∈ C(, 2) exists and is unique. For t ∈ \ 0 it is given by

(3.3)

Proof Step i) After the Fourier...