Schrödinger operators on the Wiener space

Ichiro Shigekawa ^{∗} Kyoto University

1. Introduction

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.

This kind of problem was considered in the context of scalar field of quantum field theory. Essential self-adjointness was discussed by Segal [5], B. Simon [10] and others. Issues of determining the domain and the spectral gap was considered by Glimm and Jaffe [1], Simon and Høegh-Krohn [11], etc. Their methods depend on the hypercontractivity but we use logarithcic Sobolev inequality , which is known to be equivalent with the hypercontractivity. To determine the operator domain we use the intertwining properties of operators.

The organization of the paper is as follows. In Section 2, we discuss the essential selfadjointness of a Schrödinger operator. We use the perturbation theory and the logaritmic Sobolev inequality. We also give a refinement of the logaritmic Sobolev inequality in terms of the generator itself. In Section 3, we determine the domain of the Schrödinger operator. The intertwining proerty of operaotrs plays an important role. Last, in Section 4, we discuss the spectral gap of the operator.

2. Essential self-adjointness

We first fix notations. Let (B, H, μ) be an abstract Wiener space, i.e., B is a Banach space, H is a Hilbert space inbedded in B, and μ is the Wiener measure with the characteristic function

∫_{B} e^{\sqrt{−1}⟨x,φ⟩}μ(dx) = exp\{−\frac{1}{2}|φ|_{H^{∗}}^{2}\}, φ ∈ B^{∗} ⊂ H^{∗}. (2.1)

Let ℱ C_{0}^{∞} 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}⟩). (2.2)

∗e-mail: ichiro@math.kyoto-u.ac.jp, URL: http://www.math.kyoto-u.ac.jp/~ichiro/

We suppose that a Schrödinger operator L − V is defined on L^{2}(μ), where is L is the Ornstein-Uhlenbeck operator, and V is a scalar potential. We first consider when L − V is essentially self-adjoin on ℱ C_{0}^{∞}. To d this, we use the theory of positive generalized function (see, e.g., [3]). We denote the L^{2}-norm by ∥ ∥_{2} and the operator norm by ∥ ∥_{op}. We devide V into the positive part V_{+} := max{V, 0} and the negative part V_{−} := max{−V, 0}. We have the following

Proposition 2.1. We assume that V_{+} ∈ L^{2+}. For the negative part, weassume that there exist 0 < a < 1, b > 0 such that

∥V_{−}f∥_{2} ≤ a∥Lf∥_{2} + b∥f∥_{2}. (2.3)

Then −L + V is essentially selfadjoint on ℱ C_{0}^{∞}.

Proof. From (2.3), we have

∥V_{−}(c − L)^{−1}f∥_{2} ≤ a∥L(c − L)^{−1}f∥_{2} + b∥(c − L)^{−1}f∥_{2}.

Further, by using the spectral decomposition, we have ∥L(c−L)^{−1}∥_{op} ≤ 1, ∥(c−L)^{−1}∥_{op} ≤ 1/c and thereby

∥V_{−}(c − L)^{−1}f∥_{2} ≤ (a + \frac{b}{c})∥f∥_{2}.

Take c to be large so that a+ \frac{b}{c} < 1. Since L is symmetric, we also have ∥(c−L)^{−1}V_{−}∥_{op} < 1. From the assumption V_{+} ∈ L^{2+}, we can teke p, p′ > 1 so that \frac{1}{p} + \frac{1}{p′} + \frac{1}{2} = 1 and V_{+} ∈ L^{p}.

c − L + V is clearly well-defined on ℱ C_{0}^{∞}. To prove the essential self-adjointness, it suffices to show that

Ker(c − L + V)^{∗} = {0}. (2.4)

Now we take any g ∈ Ker(c − L + V)^{∗}. Then we have, for all f ∈ ℱ C_{0}^{∞},

⟨(c − L + V)f, g⟩ = 0. (2.5)

Let q′ be a conjugate exponent of p′. Then V g ∈ L^{q′}. By noting that ℱ C_{0}^{∞} is dense in ℱ_{2,p′}, we can see that (2.5) holds for all f ∈ ℱ_{2,p′}. Let {T_{t}} be a semigroup generated by L − c. Since ℱ_{2,p′} is invariant under T_{t}, we have, for all f ∈ ℱ_{2,p′},

⟨T_{t}f, g⟩ − ⟨f, g⟩ = ∫_{0}^{t} \frac{d}{ds}⟨T_{s}f, g⟩ds = ∫_{0}^{t}⟨(L − c)T_{s}f, g⟩ds = ∫_{0}^{t}⟨T_{s}f, V g⟩ds.

Recalling that ℱ_{2,p′} is dense in L^{p′} and using the continuity, we have

⟨f, T_{t}g⟩ − ⟨f, g⟩ = ∫_{0}^{t}⟨T_{s}f, V g⟩ds, ∀f ∈ L^{p′}.

In particular, taking u ∈ (ℱ_{2,p′})_{+}, f = u sgn g ∈ L^{p′}, it follows that

⟨u sgn g, T_{t}g⟩ − ⟨u, |g|⟩ = ∫_{0}^{t}⟨T_{s}(u sgn g), V g⟩ds.

Since the semigroup {T_{t}} preserves the positivity, we have

⟨u, T_{t}|g|⟩ ≥ ⟨u, sgn g T_{t}g⟩

and therefore

⟨u, T_{t}|g|⟩ − ⟨u, |g|⟩ ≥ ∫_{0}^{t}⟨T_{s}(u sgn g), V g⟩ds.

Dividing the both hand by t and letting t → 0,

⟨u, (L − c)|g|⟩ ≥ ⟨u sgn g, V g⟩ = ⟨u, V |g|⟩ ≥ −⟨u, V_{−}|g|⟩.

Here (L−c)|g| is regarded a generalized function as an element of ℱ_{−2,q′}. Since the identy above holds for all u ∈ (ℱ_{2,p′})_{+}, it follows that (L − c)|g| + V_{−}|g| ∈ (ℱ_{−2,q′})_{+}.

On the other hand, for f ∈ ℱ_{2,2}, we have

|⟨V_{−}|g|, f⟩| = |⟨|g|, V_{−}f⟩| ≤ ∥g∥_{2}∥V_{−}f∥_{2} ≤ ∥g∥_{2}(a∥Lf∥_{2} + b∥f∥_{2}).

Then we can see that V_{−}|g| ∈ ℱ_{−2,2}. The same is true for (L − c)|g| and hence we have (L−c)|g|+V_{−}|g| ∈ ℱ_{−2,2}. Now, using [3, Proposition 3.5], we can get (L−c)|g|+V_{−}|g| ∈ (ℱ_{−2,2})_{+}, which means

(c − L)|g| ≤ V_{−}|g| in ℱ_{−2,2} .

Since (c − L)^{−1} preserves the positivity, we have

|g| ≤ (c − L)^{−1}V_{−}|g|

and further

∥g∥_{2} ≤ ∥(c − L)^{−1}V_{−}|g|∥_{2} ≤ ∥(c − L)^{−1}V_{−}∥_{op}∥g∥_{2}.

Since c was chosen so that ∥(c − L)^{−1}V_{−}∥_{op} < 1, we have g = 0. This shows (2.4) and the proof is complete.

We now have to give a sufficient condition for (2.3). To do this, we make use of the logarithmic Sobolev inequality. It is usually written in terms of bilinear form, but the generator itself is involved in our case. So we have to rewrite it a bit.

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}. (2.6)

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 (2.7)

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 2.2. Asume thet the logarithmic Sobolev (2.6) 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}. (2.8)

Proof. Let k be a function on ℝ such that k is a concave and even function of class C^{2} with k(0) = 0 and

k(x) = x log^{1/2} x, x ≥ e.

Since k′(x) = log^{1/2} x + \frac{1}{2 log^{1/2} x}, we have

log k(x) = log x + log log^{1/2} x,
k′(x)^{2} = log x + \frac{1}{4 log x} + 1.

Hence

k′(x)^{2} ≤ log k(x) (2.9)

for sufficiently large x. We also note that

k(x)k′(x) = x(\frac{1}{2} + log |x|) (2.10)

for |x| ≥ e. By the logarithmic Sobolev inequality (2.6), we have

∫_{B} g^{2} log |g| dμ ≤ −α ∫_{B} gLg dμ + β∥g∥_{2}^{2} + ∥g∥_{2}^{2} log ∥g∥_{2}.

Set g = k(f). Then, by the local property of ℰ,

Lg = k′(f)Lf + k′′(f)Γ(f, f).

Hence

∫_{B} k(f)^{2} log k(f) dμ

≤ −α ∫_{B} k(f){k′(f)Lf + k′′(f)Γ(f, f)} dμ + β∥k(f)∥_{2}^{2} + ∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}

≤ α ∫_{B} k(f)|k′(f)||Lf| dμ + β∥k(f)∥_{2}^{2} + ∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}.

Now, for large x, by using (2.9),

k(x)^{2} log k(x) ≥ k(x)^{2}k′(x)^{2}.

For small x, by takiking C_{1} > 0 large enough, we have

k(x)^{2} log k(x) ≥ k(x)^{2}k′(x)^{2} − C_{1}.

Combining them gives

∫_{B} k(f)^{2}k′(f)^{2} dμ − C_{1} ≤ ∫_{B} k(f)^{2} log k(f) dμ

≤ α ∫_{B} k(f)|k′(f)||Lf| dμ + β∥k(f)∥_{2}^{2} + ∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}

≤ \frac{1}{2} ∫_{B} k(f)^{2}|k′(f)|^{2} dμ + \frac{α^{2}}{2} ∫_{B} |Lf|^{2} dμ

+ β∥k(f)∥_{2}^{2} + ∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}.

Here, in the last line, we used the inequality ab ≤ \frac{1}{2}(a^{2} + b^{2}). Thus

∫_{B} k(f)^{2}k′(f)^{2} dμ ≤ 2C_{1} + α^{2} ∫_{B} |Lf|^{2} dμ + 2β∥k(f)∥_{2}^{2} + 2∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}. Recall that k(x)k′(x) = x(\frac{1}{2} + log |x|) for large x. Then, by changing constants, we have

∫_{B} f^{2}(\frac{1}{2} + log |f|)^{2} dμ ≤ C_{2} + α^{2} ∫_{B} |Lf|^{2} dμ + 2β∥k(f)∥_{2}^{2} + 2∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}.

Again, by changing constant, we have ∫_{B} f^{2} log_{+}^{2} |f| dμ ≤ C_{3} + α^{2} ∫_{B} |Lf|^{2} dμ + 2β∥k(f)∥_{2}^{2} + 2∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}. (2.11) The essential part has been done. It remains to estimate ∥k(f)∥_{2}^{2} and ∥k(f)∥_{2}^{2} log ∥k(f)∥_{2}. We first need to compute ∥k(f)∥_{2}^{2}.

∥k(f)∥_{2}^{2} ≤ E[f^{2} log_{+} |f|] + C_{4}.

Here we denote the integral with respect to μ by E[ ]. We use this convention from now on. We may assume C_{4} ≥ 1. If r ≥ 1, then log(x+r) ≤ log_{+} x+log 2r (x ≥ 0). Therefore

∥k(f)∥_{2}^{2} log ∥k(f)∥_{2} ≤ E[f^{2} log_{+} |f| + C_{4}] log E[f^{2} log_{+} |f| + C_{4}]
≤ E[f^{2} log_{+} |f| + C_{4}]{log E[f^{2} log_{+} |f|] + log(2C_{4})}

≤ 2E[f^{2} log_{+} |f|] log_{+} E[f^{2} log_{+} |f|] + C_{5}.

By the Schwarz inequality,

E[f^{2} log_{+} |f|] = E[f · f log_{+} |f|] ≤ {E[f^{2}]E[f^{2} log_{+}^{2} |f|]}^{1/2}.

Combining this with the inequality x log_{+} x ≤ x^{3/2} leads

E[f^{2} log_{+} |f|] log_{+} E[f^{2} log_{+} |f|] ≤ {E[f^{2}]E[f^{2} log_{+}^{2} |f|]}^{1/2} log_{+}{E[f^{2}]E[f^{2} log_{+}^{2} |f|]}^{1/2}
≤ {E[f^{2}]E[f^{2} log_{+}^{2} |f|]}^{3/4}
= E[f^{2}]^{3/4}E[f^{2} log_{+}^{2} |f|]^{3/4}.

Further, by usting the inequality

xy ≤ \frac{x^{4}}{4} + \frac{y^{4/3}}{\frac{4}{3}} ,

we have, for δ > 0,

E[f^{2} log_{+} |f|] log_{+} E[f^{2} log_{+} |f|] ≤ \frac{E[f^{2}]^{3}}{4δ^{4}} + \frac{δ^{4/3}E[f^{2} log_{+}^{2} |f|]}{\frac{4}{3}}.

Then, for any ε > 0, there exist constants C_{6}, C_{7} such that

∥k(f)∥_{2}^{2} log ∥k(f)∥_{2} ≤ εE[f^{2} log_{+}^{2} |f|] + C_{6} + C_{7}E[f^{2}]^{3}.

We can esitmate ∥k(f)∥_{2}^{2} in a similar way. By combining this with (2.11), we can get the desired inequality.

Set Φ(x) = x log_{+}^{2} x. Then

ϕ(x) = Φ′(x) = log_{+}^{2} x + 2 log_{+} x.

Further

ϕ(e^{\sqrt{x+1}−1}) = x,

which leads that the inverse function ψ of ϕ is given by

ψ(x) = e^{\sqrt{x+1}−1}.

From this, we can easily see that ψ(x) ≤ e^{\sqrt{x}}. Let Ψ be an integral of ψ. Φ is called a complimentary function. Φ can be estimated as

Ψ(x) = ∫_{0}^{x} ψ(y)dy ≤ ∫_{0}^{x} e^{\sqrt{y}}dy = ∫_{0}^{x} 2\sqrt{y}(e^{\sqrt{y}})′dy

= 2\sqrt{x}e^{\sqrt{x}} − ∫_{0}^{x} \frac{1}{\sqrt{y}} e^{\sqrt{y}}dy ≤ 2\sqrt{x}e^{\sqrt{x}}.

Thus, by the Hausdorff-Young inequality, we have

xy ≤ Φ(x) + Ψ(y) ≤ x log_{+}^{2} x + 2\sqrt{y}e^{\sqrt{y}} (2.12)

Now we are ready to give a sufficient condition for the inequality (2.3) by means of the logarithmic Sobolev inequality.

Theorem 2.3. Assume the same conditions in Theorem 2.2. Suppose that a non-nagetive function v satisfies

e^{v} ∈ L^{2α+} = ⋃_{p>2α} L^{p}. (2.13)

Then, there exist positive constant a < 1 and b such that

∥vf∥_{2} ≤ a∥Lf∥_{2} + b∥f∥_{2}. (2.14)

Proof. Take ε > 0 so that e^{v} ∈ L^{2(1+2ε)α}. From the inequality (2.12), we have

4(1 + ε)^{2}α^{2}∥vf∥_{2}^{2} = E[4(1 + ε)^{2}α^{2}v^{2}f^{2}]
≤ E[f^{2} log_{+}^{2} f^{2} + 4(1 + ε)αve^{2(1+ε)αv}]
≤ 4E[f^{2} log_{+}^{2} f] + 4(1 + ε)αE[ve^{2(1+ε)αv}].

Now the inequality (2.7) leads to

4(1 + ε)^{2}α^{2}∥vf∥_{2}^{2} ≤ 4α^{2}(1 + ε)∥Lf∥_{2}^{2} + 4K_{1} + 4K_{2}∥f∥_{2}^{6} + 4(1 + ε)αE[ve^{2(1+ε)αv}].

By taking f/∥f∥_{2} instead of f, we have

4(1 + ε)^{2}α^{2} \frac{∥vf∥_{2}^{2}}{∥f∥_{2}^{2}} ≤ 4α^{2}(1 + ε)\frac{∥Lf∥_{2}^{2}}{∥f∥_{2}^{2}} + 4K_{1} + 4K_{2} + 4(1 + ε)αE[ve^{2(1+ε)αv}].

Thus

∥vf∥_{2}^{2} ≤ \frac{1}{1 + ε}∥Lf∥_{2}^{2} + \{\frac{K_{1} + K_{2}}{(1 + ε)^{2}α^{2}} + \frac{1}{(1 + ε)α}E[ve^{2(1+ε)αv}]\}∥f∥_{2}^{2}.

Taking the sqaure and using the inequality \sqrt{x + y} ≤ \sqrt{x} + \sqrt{y}, we have

∥vf∥_{2} ≤ \frac{1}{\sqrt{1 + ε}}∥Lf∥_{2} + \{\frac{K_{1} + K_{2}}{(1 + ε)^{2}α^{2}} + \frac{1}{(1 + ε)α}E[ve^{2(1+ε)αv}]\}^{1/2}∥f∥_{2},

which is the desired result.

On an abstract Wiener space, it is known that the inequality (2.6) holds for α = 1, β = 0 (Gross’ inequality):

∫_{B} |f|^{2} log(|f|/∥f∥_{2}) dμ ≤ ∫_{B} |∇f|^{2} dμ. (2.15)

Now, using the previous result, we have

Theorem 2.4. 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 ℱ C_{0}^{∞}.

3. The domain of a Schrödinger operator

In this section, we consider an issue of the domain of a Schrödinger operator of the form 𝔄 = L − V + W. In Section 2, we have decomposed the potential into positive part and negative part. 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}.

Though the way of decomposing the postential is different from the one in previous section, it is not hard to see that we can apply the result of Section 2 and so we have that 𝔄 = L − V + W is essentially self-adjoint on ℱ C_{0}^{∞}. We fix the constant α in (ii) throughout this section. We only remark that a can be chosen as close as 1.

To determine the domain, we use the intertwining property of operators; to be precise, we make use of an operator A satisfying

\sqrt{V} 𝔄 = A\sqrt{V} . (3.1)

Let us give a definition of an operator A. Assuming the differentiability, we define a vector field b by

b = \frac{∇V}{2V} = \frac{1}{2}∇ log V. (3.2)

We further define a bilinear form as

ℰ_{A}(f, g) = (∇f, ∇g) + (b · ∇f, g) − (f, b · ∇g) + ((V − W − |b|^{2})f, g). (3.3)

Note that this bininear form is not symmetric. By the following formal computation

(f, b · ∇g) = (∇^{∗}(fb), g) = (f∇^{∗}b − b · ∇f, g),

an associated generator is given by

A = L − 2b · ∇ + (∇^{∗}b − V + W + |b|^{2}). (3.4)

We remark that this is not a regorous expression because we do not assume the differentiability of b and so ∇^{∗}b is not well-defined. (3.4) is merely a formal expression. A is defined through the bilinear form (3.3).

We now give an sufficient condition so that ℰ_{A} defines a closed bilinear form whose symmetric part is bounded frm below. The problem is what kind of regularity of b should be imposed. We consider the symmetric part and skew-symmetric part separately. The symmetric part is given by

ℰ̂_{A}(f, g) = (∇f, ∇g) + ((V − W − |b|^{2})f, g)

and the skew-symmetric part is

ℰ̌_{A}(f, g) = (b · ∇f, g) − (f, b · ∇g).

We denote a bilinear form added by λ times the inner product of L^{2} by

ℰ̂_{A−λ}(f, g) = (∇f, ∇g) + ((V − W − |b|^{2})f, g) + λ(f, g).

We first consider the symmetric part (∇f, ∇g) + ((V − W − |b|^{2})f, g). We use a general theory of perturbation: the KLMN theorem for bilinear form (see, e.g., Reed-Simon [4, Theorem X.17]). We regard W + |b|^{2} as a perturbation of a bilinear form associated to L − V:

ℰ_{L−V} (f, g) = (∇f, ∇g) + (V f, g).

The domain of this bilenear form is Dom(∇) ∩ Dom(\sqrt{V}). It is well-known that this form is a Dirichlet form.

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}. (3.5)

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

|b|^{2} ≤ αV + C. (3.6)

Then we have the following

Proposition 3.1. In addition to (A.1), (A.2) we assume that either (3.5) or (3.6) is fulfilled. Then there exists a constant β such that

(W + |b|^{2}f, f) ≤ αℰ_{L−V} (f, f) + β(f, f) (3.7)

and hence

(1 − α)ℰ_{L−V} (f, f) ≤ ℰ_{A}(f, f) + β(f, f) ≤ (1 + α)ℰ_{L−V} (f, f) + β(f, f). (3.8)

Therefore ℰ̂_{A} with the domain Dom(ℰ_{L−V}) = Dom(∇) ∩ Dom(\sqrt{V}) is a closed bilinear form that is bounded from below.

Proof. First we assume (3.5). By the logarithmic Sobolev inequality (2.15) and the Housdorff-Young xy ≤ x log x − x + e^{y}, we have

(2/α)((W + |b|^{2})f, f) = (2/α)E[(W + |b|^{2})f^{2}]
≤ E[f^{2} log f^{2}] − E[f^{2}] + E[e^{2(W +|b|^{2})/α}]
≤ 2E[|∇f|^{2}] + 2∥f∥_{2}^{2} log ∥f∥_{2}^{2} − ∥f∥_{2}^{2} + E[e^{2(W +|b|^{2})/α}].

Multiplying the both hand by α/2, we have

(W + |b|^{2}f, f) ≤ αE[|∇f|^{2}] + α∥f∥_{2}^{2} log ∥f∥_{2}^{2} − \frac{α}{2}∥f∥_{2}^{2} + \frac{α}{2}E[e^{2(W +|b|^{2})/α}].

Take f/∥f∥_{2} instead of f. Then we get

(W + |b|^{2}f, f) ≤ αE[|∇f|^{2}] + \frac{α}{2}E[e^{2(W +|b|^{2})/α} − 1]∥f∥_{2}^{2}

≤ αℰ_{L−V} (f, f) + \frac{α}{2}E[e^{2(W +|b|^{2})/α} − 1]∥f∥_{2}^{2}.

Thus we have obtained (3.7) and the rest is easy form the KLMN theorem.

Second we assume (3.6). By the logarithmic Sobolev inequality and the assumption (A.2), we have

(W f, f) ≤ αE[|∇f|^{2}] + \frac{α}{2}E[e^{2W/α} − 1]∥f∥_{2}^{2}.

Further, by (3.6),

(|b|^{2}f, f) ≤ αE[V |f|^{2}] + C∥f∥_{2}^{2}.

By combining them, we can deduce

(W + |b|^{2}f, f) ≤ αℰ_{L−V} (f, f) + (\frac{α}{2}E[e^{2W/α} − 1] + C)∥f∥_{2}^{2}.

The rest is the same as before.

We can give a sufficinet condition which is a mixture of (3.5) and (3.6), but we do not go into details.

Proposition 3.2. ℱ C_{0}^{∞} is dense in Dom(ℰ̂_{A}).

Proof. By Proposition 3.1, Dom(ℰ̂_{A}) is the same as the domain of the bilinear form associated with L − V. Recall that L − V is essentially self-adjoint in ℱ C_{0}^{∞}, i.e., ℱ C_{0}^{∞} is dense in Dom(L − V). Hence ℱ C_{0}^{∞} is dense in the domain of the bilinear form. This completes the proof.

Let us proceed to the estimate of skew-symmetric part. Assume the inequality (3.7). We impose one of the following conditions on b:

e^{|b|^{2}} ∈ L^{0+} (3.9)

or there exists a constant C > 0 such that

|b|^{2} ≤ CV. (3.10)

Proposition 3.3. In addition to (3.7), we assume either (3.9) or (3.10). Then, for sufficiently large λ, there exists a constant K > 0 such that

|ℰ̌_{A}(f, g)| ≤ Kℰ̂_{A−λ}(f, f)^{1/2}ℰ̂_{A−λ}(g, g)^{1/2}. (3.11)

Therefore ℰ_{A} satisfies the sector condition.

Proof. We first consider the case that (3.9) is satisfied. We use the logarithmic Sobolev inequality. We only need to estimate (b · ∇f, g). Take ε > 0 so that e^{ε|b|^{2}} ∈ L^{1}. Then, by the Schwarz inequality,

|(b · ∇f, g)| ≤ E[\frac{1}{ε}|∇f|^{2}]^{1/2}E[ε|b|^{2}g^{2}]^{1/2}.

Further, by using the Housdorff-Young xy ≤ x log x − x + e^{y}, we have

E[ε|b|^{2}g^{2}] ≤ E[g^{2} log g^{2}] − E[g^{2}] + E[e^{ε|b|^{2}}]
≤ 2E[|∇g|^{2}] + ∥g∥_{2}^{2} log ∥g∥_{2}^{2} − ∥g∥_{2}^{2} + E[e^{ε|b|^{2}}].

Now replacing g with g/∥g∥_{2}, we get

E[ε|b|^{2}g^{2}] ≤ 2E[|∇g|^{2}] + E[e^{ε|b|^{2}} − 1]∥g∥_{2}^{2}.

Hence

|(b · ∇f, g)| ≤ \frac{1}{\sqrt{ε}}E[|∇f|^{2}]^{1/2}\{2E[|∇g|^{2}] + E[e^{ε|b|^{2}} − 1]∥g∥_{2}^{2}\}^{1/2}.

The right hand side can be estimated the norm in ℰ̂_{A} by (3.8) and so can obtain the desired result.

We can also prove it when (3.10) is fulfilled. In fact, it is enough to notice that in the proof above E[|b|^{2}g^{2}] can be dominated by E[CV g^{2}].

In the sequel, we always assume the conditions (A.1), (A.2). Further we assume either (3.5) or (3.6) so that ℰ_{A} is well-defined as a closed bilinear form. Establising the ℰ_{A}, we proceed to an issue of intertwining property. Our next task is to show the following intertwining propert:

\sqrt{V} 𝔄 = A\sqrt{V} .

But A in (3.4) is merely a formal expression, we prove this in the following form:

ℰ_{𝔄}(f, \sqrt{V} g) = ℰ_{A}(\sqrt{V} f, g). (3.12)

Proposition 3.4. (3.12) holds for f,g ∈ ℱ C_{0}^{∞}. Moreover, we have, for f ∈ Dom(𝔄), g ∈ Dom(A^{∗}),

(𝔄f, \sqrt{V} g) = (\sqrt{V} f, A^{∗}g). (3.13)

Proof. We take any f,g ∈ ℱ C_{0}^{∞}. Then

[∇, \sqrt{V}]f = ∇(\sqrt{V} f) − \sqrt{V} ∇f = (∇\sqrt{V})f = \frac{∇V}{2\sqrt{V}} f = \sqrt{V} fb,

which means

[∇, \sqrt{V}]f = \sqrt{V} fb (3.14)

Using this, we will investigate the commutation relation between L and \sqrt{V}.

(∇f, ∇(\sqrt{V} g)) = (∇f, [∇, \sqrt{V}]g) + (∇f, \sqrt{V} ∇g)
= (∇f, \sqrt{V} gb) + (\sqrt{V} ∇f, ∇g)

= (\sqrt{V} ∇f, gb) + (\sqrt{V} ∇f, ∇g)

= ([\sqrt{V} , ∇]f, gb) + (∇(\sqrt{V} f), gb) + ([\sqrt{V} , ∇]f, ∇g) + (∇(\sqrt{V} f), ∇g)

= −(\sqrt{V} fb, gb) + (b · ∇(\sqrt{V} f), g) − (\sqrt{V} fb, ∇g) + (∇(\sqrt{V} f), ∇g)

= −(|b|^{2}\sqrt{V} f, g) + (b · ∇(\sqrt{V} f), g) − (\sqrt{V} f, b · ∇g) + (∇(\sqrt{V} f), ∇g).

It is also clear that \sqrt{V} and V −W are commutative. Let ℰ_{𝔄} be a bilinear form associated with 𝔄:

ℰ_{𝔄}(f, g) = (∇f, ∇g) + ((V − W)f, g). (3.15)

Then the commutation relation between \sqrt{V} and 𝔄 is obtained as

ℰ_{𝔄}(f, \sqrt{V} g) = (∇(\sqrt{V} f), ∇g) + (b · ∇(\sqrt{V} f), g)
− (\sqrt{V} f, b · ∇g) − (|b|^{2}\sqrt{V} f, g) + ((V − W)\sqrt{V} f, g)
= ℰ_{A}(\sqrt{V} f, g).

So far, f and g are assumed to be taken form ℱ C_{0}^{∞}. We will show that, by taking limit, this holds for f ∈ Dom(𝔄), g ∈ Dom(A^{∗}). To do this, we first fix f ∈ ℱ C_{0}^{∞}. Then we have

−(𝔄f, \sqrt{V} g) = ℰ_{A}(\sqrt{V} f, g) (3.16)

for g ∈ ℱ C_{0}^{∞}. Using Proposition 3.2 and taking limit, we can show that (3.16) holds for g ∈ Dom(ℰ_{A}). In particular, if g ∈ Dom(A^{∗}), then

(𝔄f, \sqrt{V} g) = (\sqrt{V} f, A^{∗}g) ∀f ∈ ℱ C_{0}^{∞}.

Now, by noting the essential self-adjointness, we can see that this identity holds for all f ∈ Dom(𝔄). This completes the proof.

We are ready to determin the domain of the Schrödinger operator 𝔄. The method here is to use the intertwining property of operators, which was developped in [8, 9]. The main result in this section is as follows:

Theorem 3.5. We assume the same assumptions as before. Then we have 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}. (3.17)

Proof. We use the intertwining proerty among (𝔄, L^{2}(B)), (A, L^{2}(B)) and \sqrt{V}. For f ∈ Dom(𝔄), we define a linear functional on Dom(ℰ_{A}) as

Φ(g) = ((λ − 𝔄)f, \sqrt{V} g).

Then, by Proposition 3.1, we have

|Φ(g)| ≤ ∥(λ − 𝔄)f∥_{2}∥\sqrt{V} g∥_{2} ≤ C∥(λ − 𝔄)f∥_{2}ℰ_{A−λ}(g, g)^{1/2}.

Now, by the Lax-Milgram theorem, there exist k ∈ Dom(ℰ_{A}) such that

Φ(g) = ℰ_{A−λ}(k, g).

We also note that the norm of k is estimated as

ℰ_{A−λ}(k, k)^{1/2} ≤ C∥(λ − 𝔄)f∥_{2}. (3.18)

Take any g ∈ Dom(A). By using the intertwining property, we have

(\sqrt{V} f, (λ − A^{∗})g) = ((λ − 𝔄)f, \sqrt{V} g) = Φ(g) = ℰ_{A−λ}(k, g) = (k, (λ − A^{∗})g).

Since g can run through Dom(A), we can get \sqrt{V} f = k. Now noting Proposition 3.1 and (3.18), we can deduce that

∥V f∥_{2} = ∥\sqrt{V} k∥_{2} ≤ ℰ_{L−V} (k, k)^{1/2} ≤ C_{1}ℰ_{A−λ}(k, k)^{1/2} ≤ C_{2}∥(λ − 𝔄)f∥_{2}

We also note that, by Theorem 2.3, for α < α′ < 1,

∥W f∥_{2} ≤ α′∥Lf∥_{2} + β∥f∥_{2}. (3.19)

Hence, if we take f ∈ ℱ C_{0}^{∞}, we have

∥Lf∥_{2} ≤ ∥(L − V + W − λ)f∥_{2} + ∥V f∥_{2} + ∥W f∥_{2} + λ∥f∥_{2}
≤ ∥(𝔄 − λ)f∥_{2} + C_{2}∥(λ − 𝔄)f∥_{2} + α′∥Lf∥_{2} + β∥f∥_{2} + λ∥f∥_{2},

which leads to

(1 − α′)∥Lf∥_{2} ≤ (1 + C_{2})∥(λ − 𝔄)f∥_{2} + (β + λ)∥f∥_{2}.

This, combined with the essential self-adjointness of 𝔄, yields that Dom(𝔄) ⊂ Dom(L). Thus we have obtained that if f ∈ Dom(𝔄), then f ∈ Dom(𝔄) ∩ Dom(V).

The reversed inclusion Dom(L) ∩ Dom(V) ⊆ Dom(𝔄) can be seen as

∥(L − V + W − λ)f∥_{2} ≤ ∥Lf∥_{2} + ∥V f∥_{2} + ∥W f∥_{2} + λ∥f∥_{2}
≤ ∥Lf∥_{2} + ∥V f∥_{2} + α′∥Lf∥_{2} + β∥f∥_{2} + λ∥f∥_{2},

which completes the proof.

4. Spectral gap of a Schrödinger oerator

In this section, we investigate the spectrum of a Schöodinger 𝔄 = L − V on an abstract Wiener space (B, H, μ). We first consider the case that V is bounded. In this case, L − V is a bounded perturbation of L and so it is essentially self-adjoint on ℱ C_{0}^{∞}. We denote the spectrum of 𝔄 by σ(𝔄). We have the following

Theorem 4.1. l = sup σ(𝔄) is a point spectrum of multiplicity one and the associated eigenfunction can be chosen to be positive. Moreover, the spectrum is discrete on (l−1, l], i.e., it consists of point spectrums of finite multiplicity.

Proof. The spectrum of L is well-known: σ(L) = {0, −1, −2, … } and the maximal eigenvalue 0 is a point spectrum of multiplicity one and the associated eigenfunction Therefore, if (B, H, μ) is of finite dimension, then the spectrum of L is discrete and the resolvent operators are compact. Moreover 𝔄 = L − V has a compact resolvent if V is bounded and so the spectrum of 𝔄 is discrete.

To deal with infinite dimensional case, we use an approximation method. Let {φ_{i}}_{i=1}^{∞} ⊆ B^{∗} be a c.o.n.s of H^{∗}. We set ℱ_{n} = σ(φ_{1}, φ_{2}, …, φ_{n}) and define V_{n} = E[V |ℱ_{n}]. Then V_{n} is uniformly bounded and converges to V a.s. as n → ∞. Let {h_{i}}_{i=1}^{∞} be a dual basis of {φ_{i}}, i.e., {h_{i}}_{i=1}^{∞} is a c.o.n.s. of H and satisfies ⟨h_{i}, φ_{j}⟩ = δ_{ij} where δ_{ij} is the Kronecker delta. We set H_{n} = span{h_{1}, h_{2}, …, h_{n}} and B_{n} = {x ∈ B; ⟨x, φ_{i}⟩ = 0, i = 1, 2, …, n}. Then B can be demcomposed as a direct sum B = H_{n} ⊕ B_{n}. According to this decomposition, L − V_{n} can also be decomposed as follows. On H_{n}, it has of the form L_{n} − V_{n}, and on B_{n} it is just the Ornstein-Uhlenbeck operator on B_{n}. This reveals that the spectrum of L − V_{n} is discrete on (λ(V_{n}) − 1, λ(V_{n})] where λ(V_{n}) = sup σ(L − V_{n}).

It remains to show that L − V_{n} converges to L − V as n → ∞ in the norm resolvent sense.

We denote the resolvents of L − V_{n} and L − V by G^{(n)} and G, respectively, i.e., G^{(n)} = (λ − L + V_{n})^{−1} and G = (λ − L + V)^{−1}. We take λ to be large enough. Then

G − G^{(n)} = G^{(n)}(V − V_{n})G.

Since α is large, we may assume that ∥G^{(n)}∥_{op} ≤ 1 and so it suffices to show that ∥(V − V_{n})G∥_{op} → 0 as n → ∞. By the Hausdorff-Young xy ≤ x log x−x+e^{y} and the logarithmic Sobolev inequality, we have

∥(V − V_{n})Gf∥_{2}^{2} = E[(V − V_{n})^{2}(Gf)^{2}]

= \frac{1}{N} E[N(V − V_{n})^{2}(Gf)^{2}]

≤ \frac{1}{N} E[(Gf)^{2} log(Gf)^{2} − (Gf)^{2} + e^{N(V −V_{n})^{2}}]

≤ \frac{1}{N} \{2E[|∇Gf|^{2}] + ∥Gf∥_{2}^{2} log ∥Gf∥_{2}^{2} − ∥Gf∥_{2}^{2} + E[e^{N(V −V_{n})^{2}}]\}.

Now replacing f with f/∥Gf∥_{2}, we get

∥(V − V_{n})Gf∥_{2}^{2} ≤ \frac{1}{N} \{2E[|∇Gf|^{2}] + E[e^{N(V −V_{n})^{2}} − 1]∥Gf∥_{2}^{2}\}

≤ \frac{1}{N} \{−2E[(LGf)Gf] + E[e^{N(V −V_{n})^{2}} − 1]∥f∥_{2}^{2}\}

≤ \frac{1}{N} \{2E[((λ − L + V)Gf)Gf] − λE[(Gf)^{2}] − E[V (Gf)^{2}]

+ E[e^{N(V −V_{n})^{2}} − 1]∥f∥_{2}^{2}}

≤ \frac{1}{N} \{2E[fGf] − λE[(Gf)^{2}] + E[|V |(Gf)^{2}] + E[e^{N(V −V_{n})^{2}} − 1]∥f∥_{2}^{2}\}

≤ \frac{1}{N} \{E[f^{2}] + E[(Gf)^{2}] + E[|V |(Gf)^{2}] + E[e^{N(V −V_{n})^{2}} − 1]∥f∥_{2}^{2}\}

≤ \frac{1}{N} \{(2 + ∥V ∥_{∞})∥f∥_{2}^{2} + E[e^{N(V −V_{n})^{2}} − 1]∥f∥_{2}^{2}\}.

Hence

∥(V − V_{n})G∥_{op}^{2} ≤ \frac{1}{N} \{2 + ∥V ∥_{∞} + E[e^{N(V −V_{n})^{2}} − 1]\}.

Since V − V_{n} is uniformly bounded and converges to 0 a.s., we can use the Lebesgue dominated convergence theorem and so we obtain

lim sup_{n→∞} ∥(V − V_{n})G∥_{op}^{2} ≤ \frac{2 + ∥V ∥_{∞}}{N} .

Since N is arbitrary, it follows that

lim_{n→∞} ∥(V − V_{n})G∥_{op} = 0.

Thus we have shown that G^{(n)} converges to G in norm sence, which we wanted. Using the Feynman-Kac formula, we can show that the multiplicity of the maximal eigenvalue is simple and the associated eigenfunction can be taken to be positive by a stadard argument. This completes the proof.

Now we proceed to the general case. We consider a operator of the form 𝔄 = L−V +W. We assume the assumptions of the previous section to use Theorem 3.5.

Theorem 4.2. Assume that V, W sasisfy the conditions (A.1), (A.2) in Section 3. We also assume that either (3.5) or (3.6) is fulfilled. Then the spectrum is discrete on (l−1, l], i.e., it consists of point spectrums of finite multiplicity.

Proof. Define functions V_{n}, W_{n} that approximate V, W as follows. Let ψ_{n}: ℝ ↦ ℝ be a function satisfying

ψ_{n}(x) = \begin{cases}n + 1, & x ≥ n + 2 \\ x, & |x| ≤ n \\ −n − 1, & x ≤ −n − 2 \end{cases}

and that 0 ≤ ψ′_{n} ≤ 1. We set

V_{n} = ψ_{n}(V), W_{n} = ψ_{n}(W).

As in the case of bounded potential, we set G^{(n)} = (λ − L + V_{n} − W_{n})^{−1}, G = (λ − L + V − W)^{−1} and note that

G − G^{(n)} = G(V − V_{n} − W + W_{n})G^{(n)}.

We need show the norm convergence of G − G^{(n)} to 0 as n → ∞. By taking λ to be large enugh, we may assume that ∥G^{(n)}∥_{op} ≤ 1, ∥G∥_{op} ≤ 1. Take any f, g ∈ L^{2}. Then

((G − G^{(n)})f, g) = (G(V − V_{n} − W + W_{n})G^{(n)}f, g)
= ((V − V_{n} − W + W_{n})G^{(n)}f, Gg)

= (\frac{V − V_{n} − W + W_{n}}{V + W} G^{(n)}f, (V + W)Gg).

We first estimate the term (V + W)Gg. By using (3.10) and (3.19), we have

∥(V + W)Gg∥_{2} ≤ ∥V Gg∥_{2} + ∥W Gg∥_{2}
≤ K_{2}∥(𝔄 − λ)Gg∥_{2} + α′∥LGg∥_{2} + β∥Gg∥_{2}
≤ K_{2}∥g∥_{2} + K_{2}α′∥(𝔄 − λ)Gg∥_{2} + β∥Gg∥_{2}
≤ K_{2}(1 + α′)∥g∥_{2} + β∥g∥_{2}.

Further, by the Hausdorff-Young inequality, \frac{V −V_{n}−W +W_{n}}{V +W} G^{(n)}f can be estimated as

∥\frac{V − V_{n} − W + W_{n}}{V + W} G^{(n)}f∥_{2}^{2}

= E[\frac{(V − V_{n} − W + W_{n})^{2}}{(V + W)^{2}} (G^{(n)}f)^{2}]

≤ \frac{1}{N} E[(G^{(n)}f)^{2} log(G^{(n)}f)^{2} − (G^{(n)}f)^{2} + e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}}]

≤ \frac{1}{N} \{E[|∇G^{(n)}f|^{2}] + ∥G^{(n)}f∥^{2} log ∥G^{(n)}f∥^{2}

− ∥G^{(n)}f∥_{2}^{2} + E[e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}}]}.

Replacing f with f/∥G^{(n)}f∥_{2}, we have

∥\frac{V − V_{n} − W + W_{n}}{V + W} G^{(n)}f∥_{2}^{2}

≤ \frac{1}{N} \{2E[|∇G^{(n)}f|^{2}] + E[e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}} − 1]∥G^{(n)}f∥_{2}^{2}\}

≤ \frac{1}{N} \{2E[|∇G^{(n)}f|^{2}] + E[e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}} − 1]∥f∥_{2}^{2}\}.

Further, by using (3.17) again, we get

E[|∇G^{(n)}f|^{2}] = −(LG^{(n)}f, G^{(n)}f)
≤ ∥LG^{(n)}f∥_{2}∥G^{(n)}f∥_{2}
≤ K_{2}∥(L − V_{n} + W_{n} − λ)G^{(n)}f∥_{2}∥f∥_{2}
≤ K_{2}∥f∥_{2}^{2}.

Here, we must be careful about that (3.10) holds uniformly in n. To see this, note that constants in (3.10) depend on the integrability of e^{W_{n}} and the estimate of 2b_{n} = ∇V_{n}/V_{n}. As for e^{W_{n}}, their L^{p} norm can be easily estimated uniformly since W_{n} ≤ W. Let us consider b_{n}. Since b_{n} = 0 when V ≥ n + 2, and b_{n} = b when V ≥ n + 2, it is clear that they can be estimated uniformly. When n ≤ V ≤ n + 2, note that

|b_{n}| = |∇V_{n}|/V_{n} = ϕ′_{n}|∇V |/V_{n} ≤ (|∇V |/V)(V/V_{n}) ≤ (|∇V |/V)(n + 2)/n.

Then the uniform estimate can be deduced easily.

Thus we have

|((G − G^{(n)})f, g)| ≤ {K_{2}(1 + α) + β}∥g∥_{2}

× \frac{1}{N} \{2K_{2} + E[e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}} − 1]\}∥f∥_{2}

and hence

∥G − G^{(n)}∥_{op} ≤ \frac{1}{N} \{K_{2}(1 + α) + β\}\{2K_{2} + E[e^{N(V −V_{n}−W +W_{n})^{2}/(V +W)^{2}} − 1]\}.

Now we let n → ∞ and, by the Lebesgue bounded convergence theorem, we obtain

lim sup_{n→∞} ∥G − G^{(n)}∥_{op} ≤ \frac{2}{N} \{K_{2}(1 + α) + β\}K_{2}.

Since N is arbitrary, we eventually get

lim_{n→∞} ∥G − G^{(n)}∥_{op} = 0,

which is what we wanted. The rest is the same as the previous theorem The proof is completed.

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