On a Certain Semiclassical Problem on Wiener Spaces

Abstract

We study asymptotic behavior of the spectrum of a Schrödinger type operator L λ V = L − λ 2 V on the Wiener space as λ → ∞. Here L denotes the OrnsteinUhlenbeck operator and V is a nonnegative potential function which has finitely many zero points. For some classes of potential functions, we determine the divergence order of the lowest eigenvalue. Also tunneling effect is studied. §1. Introduction Let ∆ be the Laplace operator on R n and consider a Schrödinger operator We recall basic results in semiclassical problem on R n . Now we assume that V has finitely many zero points and the Hessian is nondegenerate there and lim inf |x|→∞ V (x) > 0. Then the divergence order of the low lying eigenvalues is the same as λ and the coefficients are determined by the harmonic oscillators which are obtained by approximating V by the quadratic functions near the zero points. Also the ground state (= positive normalized eigenfunction corresponding to the lowest eigenvalue) localizes near zero points as λ → ∞. If the

Tunneling for spatially cut-off $P(ϕ)_2$-Hamiltonians

“…For any w ∈ W , there exists h ∈ H such that w + h ∈ O which implies ρ W u (w, O) < ∞. The measurability follows from the same argument as in the proof of Lemma 3.2 in [7]. We prove the latter half of the statement.…”

“…The expression of the second and third equation on the right-hand side may be rough but the final estimate is true by an approximation argument. (7) We need only to prove h − c(t c) . This implies the desired estimate.…”

Tunneling for spatially cut-off P(ϕ)2-Hamiltonians

“…For any w ∈ W , there exists h ∈ H such that w + h ∈ O which implies ρ W u (w, O) < ∞. The measurability follows from the same argument as in the proof of Lemma 3.2 in [5]. We prove the latter half of the statement.…”

Witten Laplacian on Pinned Path Group and Its Expected Semiclassical Behavior

“…Take a positive number R 0 such that { 1≤i≤n |λ i | 2 } 1/2 ≤ R 0 . By (2), for any large number R, except finite number of geodesics, l(c) ≥ R holds. For these geodesics, it holds that ( 1≤i≤n (2πk i ) 2 ) 1/2 ≥ R − R 0 .…”

“…Then taking the unitary equivalence between λ and ν λ into accounts, one may consider ν λ on L 2 (∧T * X, dν λ ) as the mathematically well-defined Hodge-Kodaira-Witten Laplacian. Motivated by this, the author [2,3] studied semiclassical behaviors of spectrum of Schrödinger type operators on Wiener spaces. Note that this problem is related with semiclassical problems in Euclidean field theory [7].…”