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

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.

Section: Acknowledgmentsmentioning

confidence: 96%

Poincaré Inequality for Weighted First Order Sobolev Spaces on Loop Spaces

Gong

Röckner

2001

“…Poincaré inequalities with potentials and a number of modified Poincaré inequalities have been intensively studied for which we refer to the recent work of Gong-Ma [29]. See also Gong-Röckner-Wu who showed the existence of the gap of spectrum for a Schrödinger operators with the potential function given in Gross's paper, for which Aida [4] gave an estimate. Their work builds on the results of Aida [2] and M. Hino [33] on exponential decay estimates of the associated semi-group .…”

Section: The Theoremmentioning

confidence: 97%

A Poincaré inequality on loop spaces

Chen

2010

“…To prove this result, we need to identify the domain of the Dirichlet form which is obtained by the ground state transformation by Ω 0,λ . We refer the reader to [3] for this problem in a setting of hyperbounded semi-group. By taking a trial function f which satisfies the assumption of the right-hand side of the above, we prove (5.16).…”

Section: Remark 53mentioning

confidence: 99%

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

Aida

2012

Self Cite

We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off P (φ) 2 -Hamiltonian in the semi-classical limit. We determine the semi-classical limit of the lowest eigenvalue of the Hamiltonian in terms of the Hessian of the potential function of the corresponding classical equation. Moreover, we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast in the semi-classical limit when the potential function is double well type. In fact, we prove that the exponential decay rate is greater than or equal to the Agmon distance between two zero points of the symmetric double well potential function. Also we study basic properties of the Agmon distance and instanton.