A Feynman-Kac Formula for Unbounded Semigroups

Barry Simon

Abstract. We prove a Feynman-Kac formula for Schrödinger operators with potentials V (x) that obey (for all ε > 0)

V (x) ≥ −ε|x|^{2} − C_{ε}.

Even though e^{−tH} is an unbounded operator, any φ, ψ ∈ L^{2} with compact support lie in D(e^{−tH}) and ⟨φ, e^{−tH}ψ⟩ is given by a Feynman-Kac formula.

1. Introduction

One of the most useful tools in the study of Schrödinger operators, both conceptually and analytically, is the Feynman-Kac formula. All the standard proofs, (see, e.g., [7]) assume the Schrödinger operator H is bounded below, so the Schrödinger semigroup e^{−tH} is bounded. This means, for example, that Stark Hamiltonians are not included.

But the restriction to semibounded H is psychological, not real. We deal with unbounded H’s all the time, so why not unbounded e^{−tH}? Once one considers the possibility, the technical problems are mild, and it is the purpose of this note to show that.

The form of the Feynman-Kac formula we will discuss is in terms of the Brownian bridge (Theorem 6.6 of [7]). Once one has this, it is easy to extend to the various alternate forms of the Feynman-Kac formula.

The ν-dimensional Brownian bridge consists of ν jointly Gaussian processes, {α_{i}(t)}_{i=1;0≤t≤1}^{ν} with covariance

E(α_{i}(t)α_{j}(s)) = δ_{ij} min(t, s)[1 − max(t, s)]
E(α_{i}(t)) = 0.

If b is Brownian motion, then α(s) = b(s) − sb(1) is an explicit realization of the Brownian bridge.

1991 Mathematics Subject Classification. Primary: 81S40, 47D08; Secondary: 60J65.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-9707661. The Government has certain rights in this material.

To appear in Proceedings of the International Conference on Infinite Dimensional (Stochastic)

Analysis and Quantum Physics, Leipzig, 1999.

For any real function V on ℝ^{ν} and t > 0, define (the expectation may be infinite):

Q(x, y; V, t) = E (exp (− ∫_{0}^{t} V ((1 − \frac{s}{t}) x + \frac{s}{t} y + \sqrt{t} α (\frac{s}{t})) ds)) . (1.1)

Throughout this paper, let

H_{0} = −\frac{1}{2}Δ

on L^{2}(ℝ^{ν}), so

e^{−tH_{0}}(x, y) = (2πt)^{−ν/2} exp (−\frac{|x − y|^{2}}{2t} ) . (1.2)

The Feynman-Kac formula I’ll start with — one of many in [7] — is

Theorem 1.1. Suppose V is a continuous function on ℝ^{ν} which is bounded from below. Let H = H_{0} + V. Then for any t > 0 and φ, ψ ∈ L^{2}(ℝ^{ν}):

⟨φ, e^{−tH}ψ⟩ = ∫ \overline{φ(x)} ψ(y) e^{−tH_{0}}(x, y)Q(x, y; V, t). (1.3)

In this paper, we will consider potentials V (x) for which for any ε > 0, there is C_{ε} so that

V (x) ≥ −ε|x|^{2} − C_{ε}. (1.4)

It is known (see [5], Theorem X.38) that for such V, H = H_{0} + V is essentially self-adjoint on C_{0}^{∞}(ℝ^{ν}), so we can use the functional calculus to define e^{−tH} which might be unbounded. Our main goal here is to prove:

Theorem 1.2. Suppose V is a continuous function which obeys (1.4). Then for all x, y ∈ ℝ^{ν}, t > 0, (1.1) is finite. Let φ, ψ ∈ L^{2}(ℝ^{ν}) have compact support. Then for all t > 0, φ, ψ ∈ D(e^{−tH}) and (1.3) holds.

Remarks. 1. It isn’t necessary to suppose that φ, ψ have compact support. Our proof shows that it suffices that e^{εx^{2}}ψ, e^{εx^{2}}φ ∈ L^{2} for some ε > 0. In particular, φ, ψ can be Gaussian.

2. Using standard techniques [1],[3],[7], one can extend the proof to handle V = V_{1} + V_{2} where V_{1} obeys (1.4) but is otherwise in L_{loc}^{1} and V_{2} is in the Kato class, K_{ν}.

3. If one only has V (x) ≥ −C_{1} − C_{2}x^{2} for a fixed C_{2}, our proof shows that the Feynman-Kac formula holds for t sufficiently small. It may not hold if t is large since it will happen if V (x) = −x^{2} that E(exp(− ∫_{0}^{t} V (α(s)) ds)) will diverge if t is large.

As for applications of Theorem 1.2, one should be able to obtain various regularity theorems as in [6]. Moreover, for H = −Δ+F ·x, one can compute e^{−tH}(x, y) explicitly and so obtain another proof of the explicit formula of Avron and Herbst [2].

Dedication. Sergio Albeverio has been a master of using and extending the notion of path integrals. It is a pleasure to dedicate this to him on the occasion of his 60th birthday.

2. A Priori Bounds on Path Integrals

Our goal in this section is to prove

Theorem 2.1. Let V obey (1.4) and let Q be given by (1.1). Then, for each t > 0 and δ > 0, we have that

Q(x, y; V, t) ≤ D exp(δx^{2} + δy^{2}),

where D depends only on t, δ and the constants {C_{ε}}.

Lemma 2.2. Let X be a Gaussian random variable. Suppose ε Exp(X^{2}) < \frac{1}{2}. Then E(exp(εX^{2})) < ∞ (and is bounded by a function of ε Exp(X^{2}) alone).

Proof. A direct calculation. Alternately, we can normalize X so Exp(X^{2}) = 1. Then E(exp(εX^{2})) = (2π)^{−1/2} ∫ exp((ε − \frac{1}{2})x^{2}) dx < ∞.

Proof of Theorem 2.1. Note that if 0 < θ < 1, and x, y, α ∈ ℝ^{ν}, then

|θx + (1 − θ)y + α|^{2} ≤ 2|θx + (1 − θ)y|^{2} + 2|α|^{2}
≤ 2(x^{2} + y^{2} + |α|^{2}).

Thus, by (1.4),

Q(x, y; V, t) ≤ E (exp (C_{ε}t + 2εt(x^{2} + y^{2}) + 2ε ∫_{0}^{1} t^{2}α(s)^{2} ds)) . (2.1)

By Jensen’s inequality,

E (exp (2 ∫_{0}^{1} εt^{2}α(s)^{2} ds)) ≤ ∫_{0}^{1} E(exp(2εt^{2}α(s)^{2}) ds). (2.2)

Since E(α(s)^{2}) is maximized at s = \frac{1}{2} when it is \frac{1}{4}, we see that

RHS of (2.2) ≤ E(exp(2εt^{2}α(\frac{1}{2})^{2}))

is finite if εt^{2} < 1, so we can pick ε = δ_{0}/t^{2} with δ_{0} < 1 and find (using the explicit value of E(exp(X^{2})) in that case

Q(x, y; V, t) ≤ \sqrt{2} (1 − δ_{0})^{−1/2} exp(C_{ε}t + 2δ_{0}(x^{2} + y^{2})/t),

which proves Theorem 2.1.

3. A Convergence Lemma

In this section, we will prove:

Theorem 3.1. Let A_{n}, A be self-adjoint operators on a Hilbert space H so that A_{n} → A in strong resolvent sense. Let f be a continuous function on ℝ and ψ ∈ H with ψ ∈ D(f(A_{n})) for all n. Then

(i) If sup_{n} ∥f(A_{n})ψ∥ < ∞, then ψ ∈ D(f(A)).

(ii) If sup_{n} ∥f(A_{n})^{2}ψ∥ < ∞, then f(A_{n})ψ → f(A)ψ.

Remark. Let H = L^{2}(0, 1), ψ(x) ≡ 1, A_{n} = multiplication by n^{1/2} times the characteristic function [0, 1/n], and A ≡ 0. Then A_{n} → A in strong resolvent sense and sup_{n} ∥A_{n}ψ∥ < ∞, but A_{n}ψ does not converge to Aψ so one needs more than sup_{n} ∥f(A_{n})ψ∥ < ∞ to conclude that f(A_{n})ψ → f(A)ψ. The square is overkill. We need only sup_{n} ∥F (f(A_{n}))ψ∥ < ∞ for some function F : ℝ → ℝ with lim_{|x|→∞} |F (x)|/x = ∞.

Proof. Suppose that sup_{n} ∥f(A_{n})ψ∥ < ∞. Let

f_{m}(x) = \begin{cases}m & if f(x) ≥ m \\ f(x) & if |f(x)| ≤ m \\ −m & if f(x) ≤ −m. \end{cases}

Then ([4], Theorem VIII.20) for each fixed m, f_{m}(A_{n}) → f_{m}(A) strongly. It follows that

∥f_{m}(A)ψ∥ = lim_{n} ∥f_{m}(A_{n})ψ∥

≤ sup_{n} ∥f_{m}(A_{n})ψ∥ ≤ sup_{n} ∥f(A_{n})ψ∥.

Thus, sup_{m} ∥f_{m}(A)ψ∥ < ∞, which implies that ψ ∈ D(f(A)).

Now suppose sup_{n} ∥f(A_{n})^{2}ψ∥ < ∞. Then

∥(f(A_{n}) − f_{m}(A_{n}))ψ∥ ≤ \frac{1}{m} ∥f(A_{n})^{2}ψ∥.

Thus f_{m}(A_{n})ψ → f(A_{n})ψ uniformly in n which, given that f_{m}(A_{n})ψ → f_{m}(A)ψ, implies that f(A_{n})ψ → f(A)ψ.

4. Putting It Together

We are now ready to prove Theorem 1.2. Let V be continuous and obey (1.4). Let V_{n}(x) = max(V (x), −n). Then V_{n} is bounded from below, so Theorem 1.1 applies, and so (1.3) holds. Let φ ∈ L^{2} with compact support. By Theorem 2.1, we have

sup_{n} ∥ exp(−tH_{n})φ∥ < ∞

for each t positive.

By the essential self-adjointness of H on C_{0}^{∞}(ℝ^{ν}) and (V_{n} − V)η → 0 for any η ∈ C_{0}^{∞}, we see that H_{n} converges to H in strong resolvent sense. Hence setting A_{n} = H_{n}, A = H, f(x) = e^{−tx}, and ψ = φ, we can use Theorem 3.1 to see that φ ∈ D(exp(−tH)) and ∥[exp(−tH_{n}) − exp(−tH)]φ∥ → 0. Thus as n → ∞, the left-hand side of the Feynman-Kac formula converges. By the a priori bound in Theorem 2.1 and the dominated convergence theorem, the right-hand side converges. So Theorem 1.2 is proven.

References

[1] S. Albeverio and R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Springer, New York, 1976.

[2] J.E. Avron and I. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Commun. Math. Phys. 52 (1977), 239–254.

[3] K.L. Chung and Z. Zhao, From Brownian Motion to Schrödinger’s Equation, Springer, New York, 1995.

[4] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I. Functional Analysis (rev. ed.), Academic Press, London, 1980.

[5] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness, Academic Press, London, 1975.

[6] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.

[7] B. Simon, Functional Integration and Quantum Physics, Academic Press, London, 1979.

Division of Physics, Mathematics, and Astronomy, 253-37, California Institute of Technology, Pasadena, CA 91125, USA.

E-mail address: bsimon@caltech.edu