Semiclassical limit of the lowest eigenvalue of Schrödinger operators in path spaces over Riemannian manifolds

Shigeki Aida Osaka University

1 Introduction

Let (X, g) be a compact Riemannian manifold and E, V be smooth real-valued functions on X. Let us consider the weighted probability measure dμ_{λ} = Z_{λ}^{−1}e^{−λE}dx on X, where dx is the volume element and Z_{λ} is the normalizing constant. Let

e_{0}(λ) = inf {∫_{M} |∇f(x)|^{2}dμ_{λ}(x) + λ^{2} ∫_{M} V (x)f(x)^{2}dμ_{λ}(x) | ∥f∥_{L^{2}(μ_{λ})} = 1} .  (1.1)

We recall the asymptotic behavior of e_{0}(λ) as λ → ∞. Assume that (A1) U(x) = \frac{|∇E(x)|^{2}}{4} + V (x) is a nonnegative function which has finite zero point set S = \{c_{1}, …, c_{N}\}. (A2) The Hessian of U at c_{i} (1 ≤ i ≤ N) is strictly positive.

Then, by the semiclassical analysis (see [3]), we have

lim_{λ→∞} \frac{e_{0}(λ)}{λ} = min_{1≤i≤N} tr \{\sqrt{\frac{(∇^{2}U)(c_{i})}{2}} − \frac{∇^{2}E}{2} (c_{i})\} ,  (1.2)

where ∇^{2} denotes the second covariant derivative which is defined by the Levi-Civita connection. In this talk, we discuss the case where X = P_{m_{0}}(M), that is, a path space over a d-dimensional compact Riemannian manifold (M, g), where m_{0} ∈ M.

2 Path space case

More precisely, let μ_{λ} (we dare use the same notation as in the Section 1) be the Brownian motion measure on P_{m_{0}}(M) := C([0, 1] → M | γ(0) = m_{0}) such that

μ_{λ} (γ(t_{1}) ∈ dx_{1}, …, γ(t_{l}) ∈ dx_{l})
= (∏_{i=1}^{l} p (λ^{−1}(t_{i} − t_{i−1}), x_{i−1}, x_{i})) dx_{1} ⋯ dx_{l},  (2.1)

where t_{0} = 0, x_{0} = m_{0}, p(t, x, y) = e^{tΔ/2}(x, y) and Δ is the Laplace-Beltrami operator. The so-called Bismut tangent space along γ is defined by the parallel translation by the Levi-Civita connection and the H-derivative ∇ can be defined on P_{m_{0}}(M). Let V be a measurable function on P_{m_{0}}(M). Let −L_{λ} := ∇_{μ_{λ}}^{∗}∇ and −L_{λ,V} := −L_{λ} + λ^{2}V. Let e_{0}(λ) = inf σ(−L_{λ,V}). Formally,

dμ_{λ}(γ) = Z_{λ}^{−1} exp (−λE(γ)) dγ,  (2.2)

where E(γ) = \frac{1}{2} ∫_{0}^{1} |γ̇(t)|^{2}dt and dγ is the volume element. We expect that e_{0}(λ) behaves as in (1.2) formally. Of course, E(γ) is not well-defined in P_{m_{0}}(M). Instead of P_{m_{0}}(M), we consider a subset which consists of finite energy paths, P_{m_{0},f}(M) = {γ ∈ P_{m_{0}}(M) | E(γ) < ∞}. P_{m_{0},f}(M) is a Hilbert manifold whose Riemannian structure is given by identifying the tangent space at a path with the H := H_{0}^{1}([0, 1] → T_{m_{0}}(M)) using the parallel translation which is defined by the Levi-Civita connection on M. We already mentioned this structure in the case of P_{m_{0}}(M). We denote b(t) = ∫_{0}^{t} τ(γ)_{s}^{−1}γ̇(s)ds ∈ T_{m_{0}}(M), where τ(γ)_{s} denotes the parallel translation operator along γ ∈ P_{m_{0},f}(M). Let Ṽ ∈ C^{∞} (C([0, 1] → T_{m_{0}}) → ℝ) and define V (γ) = Ṽ (b) (γ ∈ P_{m_{0},f}(M)). Note that the definition of V (γ)(= Ṽ (b)) can be extended to γ ∈ P_{m_{0}}(M) a.s. by the Itô calculus which is our potential function. Note that b(t) which is defined by the Itô calculus is the Brownian motion on T_{m_{0}}(M) whose Cameron-Martin subspace is H. Note that |∇E(γ)|^{2} = 2E(γ) (γ ∈ P_{m_{0},f}(M)). So U(γ) = \frac{1}{2}E(γ) + V (γ) (γ ∈ P_{m_{0},f}(M)). We calculate the derivatives of E and U. Suppose U(γ) satisfies (A1) and (A2). We denote K_{i} = \frac{1}{2}(D^{2}Ṽ)(h_{i}), where h_{i}(t) = ∫_{0}^{t} τ(c_{i})_{s}^{−1}ċ_{i}(s)ds and D denotes the H-derivative on H. Then we have

\sqrt{\frac{(∇^{2}U)(c_{i})}{2}} − \frac{∇^{2}E}{2} (c_{i}) = \frac{1}{2} [\sqrt{U_{i}(I_{H} + 4K_{i})U_{i}^{∗}} − (I_{H} + \frac{1}{2}(S(c_{i}) + S(c_{i})^{∗}))]
(S(c_{i})h)(t) := ∫_{0}^{t} \overline{R(c_{i})_{s}}(h_{i}(s), h(s))(ḣ_{i}(s))ds
U_{i} := I_{H} + T_{i}
(T_{i}h)(t) := ∫_{0}^{t} (∫_{s}^{1} du \{∫_{u}^{1} dτ \overline{R(c_{i})_{u}}(ḣ(τ), ḣ_{i}(τ))\} (ḣ_{i}(u))) ds,

where R denotes the curvature tensor and ∇ denotes the Levi-Civita covariant derivative on P_{m_{0},f}(M). Using these and logarithmic Sobolev inequalities on P_{m_{0}}(M), we prove the following results.

Theorem 2.1 Assume inf Ṽ > −∞ and (A1) and (A2). Then we have

lim sup_{λ→∞} \frac{e_{0}(λ)}{λ} ≤ e_{0} := min_{1≤i≤N} \frac{1}{2} \{tr [\sqrt{U_{i}(I_{H} + 4K_{i})U_{i}^{∗}} − I_{H}] + \frac{1}{2} ∫_{0}^{1} (\overline{Ric(c_{i})_{s}}h_{i}(s), ḣ_{i}(s)) ds\}
lim inf_{λ→∞} \frac{e_{0}(λ)}{λ} > −∞

Note that \sqrt{U_{i}(I_{H} + 4K_{i})U_{i}^{∗}} − I_{H} is a trace class operator but S(c_{i}) is not a trace class operator on H. Nevertheless, it holds that the infinite series

∑_{n=1}^{∞} (S(c_{i})e_{n}, e_{n}) = −\frac{1}{2} ∫_{0}^{1} (\overline{Ric(c_{i})_{s}}h_{i}(s), ḣ_{i}(s)) ds  (2.3)

are absolutely convergent for any c.o.n.s.{e_{n}} of H such that

{e_{n}} = {f_{i} ⊗ ε_{j} | {f_{i}}_{i=1}^{∞}: c.o.n.s. in H^{1}([0, 1] → ℝ | h(0) = 0), {ε_{j}}_{j=1}^{d} : c.o.n.s. in T_{m_{0}}(M)}.

References

[1] S. Aida, Witten Laplacian on pinned path group and its expected semiclassical behavior, IDAQP, Vol.6, No.supp 01, (2003), 103–114.

[2] S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal.203, (2003), 401–424.

[3] B. Simon, Semiclassical Analysis of Low Lying Eigenvalues I. Nondegenerate Minima: Asymptotic Expansions, Ann. Inst. Henri Poincaré, Vol. XXXVIII, no. 4, (1983), 309–347.

[4] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom., 17 (1982), 661–692.