Schrödinger operators on the Wiener space^{∗}

Ichiro Shigekawa^{†} (Kyoto University)

We consider a Schrödinger operator A = L − V on an abstract Wiener space (B, H, μ). Here L is the Ornstein-Uhlenbeck operator and V is a scalar potential. Our goal is to determine the domain of L − V. To be precise, we will show that Dom(A) = Dom(L) ∩ Dom(V) under a suitable condition. We also show the spectral of gap of this operator.

Essential self-adjointness

We define FC_{0}^{∞} to be the set of all functions f: B → ℝ such that there exist n ∈ ℕ, F ∈ C_{0}^{∞}(ℝ^{n}) and φ_{1}, …, φ_{n} ∈ B^{∗} with

f(x) = F (⟨x, φ_{1}⟩, …, ⟨x, φ_{n}⟩). (1)

A Schrödinger operator L − V is defined on L^{2}(μ). Essentially self-adjointness of L − V is a fundamental problem and we have the following theorem:

Theorem 1. For a Schödinger operator L − V on an abstract wiener space (B, H, μ), we suppose that V_{+}, e^{V_{−}} ∈ L^{2+} where V_{+} := max{V, 0}, V_{−} := max{−V, 0}. Then L − V is essentially self-adjoint on FC_{0}^{∞}.

To show this theorem, we need the followig result for the logarithmic Sobolev inequalities. In general setting, the (defective) logarithmic Sobolev inequality for a Dirichlet form E is written as

∫_{B} |f|^{2} log(|f|/∥f∥_{2}) dμ ≤ αE(f, f) + β∥f∥_{2}^{2}. (2)

Here (B, μ) is a general measure space. We also denote the associated generator by L (not specify to the Ornstein-Uhlenbeck operator). For the Dirichlet form E, we assume the local property and the existence of square field opetator, i.e.,

E(f, g) = ∫_{B} Γ(f, g) dμ (3)

and Γ has the derivation property. In the case of an abstract Wiener space, Γ(f, f) = |∇f|^{2}, ∇ being a gradient operator. We have the following theorem.

Theorem 2. Asume that the logarithmic Sobolev (2) holds. Then, for any ε > 0, there exist positive constants K_{1}, K_{2} such that

∫_{B} f^{2} log_{+}^{2} f dμ ≤ α^{2}(1 + ε)∥Lf∥_{2}^{2} + K_{1} + K_{2}∥f∥_{2}^{6}. (4)

∗Stochastic analysis and related topics, Osaka University, January 7–9, 2005

†E-mail: ichiro@math.kyoto-u.ac.jp URL: http://www.math.kyoto-u.ac.jp/~ichiro

The domain of a Schrödinger operator

We consider an issue of the domain of a Schrödinger operator of the form A = L − V + W. Here we decompose the potential as follows:

(A.1) V ≥ 1 and V ∈ L^{2+}.

(A.2) W is non-positive and there exists a constant 0 < α < 1 such that e^{W} ∈ L^{2/α}.

Tough there are many ways to give sufficient conditions, we restrict ourselves to typical ones. One of them is

e^{W +|b|^{2}} ∈ L^{2/α}. (5)

The other is that there exists a constant C > 0 such that

|b|^{2} ≤ αV + C. (6)

Then we have the following

Theorem 3. We assume the same assumptions as before. Then we have that Dom(A) = Dom(L) ∩ Dom(V). Moreover, for sufficiently large λ, there exist positive constants K_{1}, K_{2} such that

K_{1}∥(A − λ)f∥_{2} ≤ ∥Lf∥_{2} + ∥V f∥_{2} ≤ K_{2}∥(A − λ)f∥_{2}. (7)

Spectral gap of a Schrödinger oerator

We denote the set of spectrum of A = L − V + W by σ(A) and set l = sup σ(A). Then, using Theorem 3, we have

Theorem 4. Assume that V, W sasisfy the conditions (A.1), (A.2). We also assume that either (5) or (6) is fulfilled. Then the spectrum of A = L − V + W is discrete on (l − 1, l], i.e., it consists of point spectrums of finite multiplicity.

References

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