Existence and uniqueness of physical ground states

Gross, Leonard

doi:10.1016/0022-1236(72)90057-2

Cited by 191 publications

(100 citation statements)

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“…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].…”

Section: §1 Introductionmentioning

confidence: 94%

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On a Certain Semiclassical Problem on Wiener Spaces

Aida¹

2003

Publ. Res. Inst. Math. Sci.

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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

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Section: §1 Introductionmentioning

confidence: 94%

“…By [33] and [14], L λ V has the lowest eigenvalue and the ground state. Let us denote by E 0 (λ) and Ω(λ, φ) the lowest eigenvalue and the positive ground state of −L λ V .…”

Section: So (336) Holds §4 Asymptotics Of Lowest Eigenvalue Of −L λ Vmentioning

confidence: 99%

On a Certain Semiclassical Problem on Wiener Spaces

Aida¹

2003

Publ. Res. Inst. Math. Sci.

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“…In this description we follow, in particular, L. Gross [6] and I. E. Segal [10]. Let H be a complex Hilbert space with an inner product (·, ·).…”

Section: Description Of a II 1 -Factormentioning

confidence: 99%

TAYLOR MAP ON GROUPS ASSOCIATED WITH A II₁-FACTOR

Gordina

2002

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract. A notion of the heat kernel measure is introduced for the L 2 completion of a hyperfinite II 1 -factor with respect to the trace. Some properties of this measure are derived from the corresponding stochastic differential equation. Then the Taylor map is studied for a space of holomorphic functions square integrable with respect to the heat kernel measure. We also define a skeleton map from this space to a Hilbert space of holomorphic functions on a certain Cameron-Martin group. This group is a subgroup of the group of invertible elements of the II 1 -factor.

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“…The assumption (1) in the theorem implies in particular the operator estimate XI < H. Hence the conclusion states "existence of ground states" [4] and, if P is a priori known to be one-dimensional, also uniqueness. In the special case where H is known to be selfadjoint and X = inf a(H) the theorem yields that X is in fact an eigenvalue.…”

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confidence: 91%

Point-spectrum of semibounded operator extensions

Jørgensen¹

1981

Proc. Amer. Math. Soc.

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Abstract. Let H denote the Friedrichs extension of a given semibounded operator H in a Hubert space. Assume XI < H, and X € a(H). If for a finite-dimensional projection P in the Hubert space we have I -P < Coast.(H -XI), then it follows that A is an eigenvalue of H, and the corresponding eigenspace is contained in the range of P. Using this, together with the known order structure on the family of selfadjoint extensions, with given lower bound 0, of minus the Laplace-Beltrami operator, we establish the identity Ug(\) =1 for all g B G for the following problem.U is a unitary representation of a Lie group G, and acts on the Hubert space L2(Q) for some Nikodym-domain 8cC.Moreover U is obtained as a certain normalized integral for the left-G-invariant vector fields on Í2, that is, for each such vector field X, the skew-adjoint operator dU(X) is an extension of X when regarded as a skew-symmetric operator in L2(ß) with domain Cg°(Q).1. Introduction. Recently, two results, related to the spectral theory of the Friedrichs extension, have appeared. In [7] it was shown that the SegalFuglede integration problem for constant coefficient partial differential operators [1] has an affirmative answer in a more general setting of Lie groups and symmetric spaces. The answer relates the geometry of a given finite-volume domain ß to the integration problem for Lie algebras of invariant vector fields on ñ. However, the spectral theory of the corresponding Laplace-Beltrami operators (and their extensions) is still not well understood. (Independently,in [6] it is established that the Friedrichs extension of a semibounded operator H has derivation properties with respect to a given operator algebra if it is known that the initial nonselfadjoint operator H has these properties.)The following theorem has implications for both of the above-mentioned problems, and it seems to be of independent interest (in view of the scarcity of general results on the point spectrum of the Friedrichs extension). The idea of the proof is based on that of Fuglede's main lemma [1], which is in fact about families of selfadjoint nonsemibounded operators. The results however have little in common.

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Existence and uniqueness of physical ground states

Cited by 191 publications

References 20 publications

On a Certain Semiclassical Problem on Wiener Spaces

On a Certain Semiclassical Problem on Wiener Spaces

TAYLOR MAP ON GROUPS ASSOCIATED WITH A II₁-FACTOR

Point-spectrum of semibounded operator extensions

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Existence and uniqueness of physical ground states

Cited by 191 publications

References 20 publications

On a Certain Semiclassical Problem on Wiener Spaces

On a Certain Semiclassical Problem on Wiener Spaces

TAYLOR MAP ON GROUPS ASSOCIATED WITH A II1-FACTOR

Point-spectrum of semibounded operator extensions

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TAYLOR MAP ON GROUPS ASSOCIATED WITH A II₁-FACTOR