Poincaré Inequality for Weighted First Order Sobolev Spaces on Loop Spaces

Abstract

dedicated to the memory of the second named author's father, otto peter ro cknerLet E be the loop space over a compact connected Riemannian manifold with a torsion skew symmetric connection. Let L D be the Ornstein Uhlenbeck operator on a nonempty connected component D of the loop space E and let V: D Ä R be the restriction on D of the potential in the logarithmic Sobolev inequality found by L. Gross on the loop group by S. Aida and by F. Z. Gong and Z. M. Ma on the loop space, respectively. We prove that the Schro dinger operator &L V := &L D +V always has a spectral gap at the bottom * 0 (V) of its spectrum and thus has its ground state transformed operator , &1 (&L V &* 0 (V)) ,, where , is the unique ground state of &L V . In particular, our result proves L. Gross's conjecture about the existence of a spectral gap for the ground state transform of the Schro dinger operator studied by him on the loop group. In addition, in all the above cases we identify the domain of the Dirichlet forms associated with the ground state transforms as weighted first order Sobolev spaces with weight given by , 2 , thus establishing a Poincare inequality for them. All these results are consequences from some new results in this paper on Dirichlet forms characterizing certain classes with spectral gaps and from results by S. Aida and M. Hino.

An Estimate of the Gap of Spectrum of Schrödinger Operators which Generate Hyperbounded Semigroups

“…Here we will assume that a weak Poincare inequality ( =WPI for short) holds for &L. The generator of the Dirichlet form E 0 is unitarily equivalent to the Schro dinger operator &L V &inf _(&L V ) by the unitary transformation U 0 : u Ä u0 &1 between L 2 (m) and L 2 (0 2 m). Actually our results are concerned with the gap of spectrum of &L V &inf _(&L V ) also the same as in [25]. In this respect, there is a nontrivial domain issue.…”

“…3) Clearly this is a generalization of the Poincare inequality and stronger than the irreducibility of E. WPI was introduced by Ro ckner and Wang [39] and they proved that it is equivalent to WSGP. Thus this paper will give a proof for the existence of spectral gap based on Dirichlet forms differently from [25] and [28]. Our proof gives a quantitative lower bound estimate on the gap of spectrum by using the function m(0 =) and !…”

“…For this conjecture, recently, Gong Ro ckner Wu [25] proved the existence of the spectral gap for the Dirichlet form E 0 (actually including more general setting) by using the Hino's result and the author's result.…”

On a Certain Semiclassical Problem on Wiener Spaces

“…By the result in [14], if −L V generates a hyperbounded semigroup, then E 0 is an eigenvalue. In addition to the assumption, if −L satisfies the weak spectral gap property (see [27], [4], [23], [13]), E 1 − E 0 > 0 holds which was proved recently by Gong, Röckner and Wu [12]. This is a generalization of a part of Theorem 4.5 in Simon and Hoegh-Krohn [33].…”

“…This is a generalization of a part of Theorem 4.5 in Simon and Hoegh-Krohn [33]. The results in [12] can be applied to Schrödinger type operator on loop spaces over compact Riemannian manifolds which were studied in [16], [3] and [11]. However, differently from finite dimensional cases, little is known about semiclassical analysis in infinite dimensional spaces.…”