Eigenvalue Asymptotics of Schrödinger Operators with Only Discrete Spectrum

Tachizawa, Kazuya

doi:10.2977/prims/1195167733

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…Here dT (t, x, y) is the parabolic distance alluded at the beginning of the introduction, see (13) for the precise definition. By relating it to the Agmon distance we are able to obtain an effective bound on dT (t, x, y), which, when combined with the theorem above, yields the following corollary.…”

Section: Resultsmentioning

confidence: 99%

“…Then by Weyl's law (Cf. [13]), λ k ln(k). For such slowly growing eigenvalue distributions, it is unreasonable to consider the limit lim t→0 T r s (exp(−t f )).…”

Section: Weak Weyl Law For Witten Laplacianmentioning

confidence: 99%

“…This condition should be interpreted in terms of the (semiclassical) Weyl's law for Schrödinger operators in Euclidean space ( Cf. [11,13] ), which would guarantee the polynomial growth of the eigenvalues. However, though expected, we could not find in the literature such Weyl's law on manifolds.…”

Section: Introduction 1overviewmentioning

confidence: 99%

“…To focus our discussion on the asymptotic expansion of heat kernel and local index theorem, we only prove a weaker version of Weyl's law here under our assumption. In a separate paper we will prove such Weyl's law by using similar techniques in [13] and also give a treatment using heat kernel following the techniques developped here in Sections 3 and 4.…”

Section: Introduction 1overviewmentioning

confidence: 99%

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Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

Dai¹,

Yan²

2020

Preprint

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Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as C n with (quasi-homogeneous) polynomials potentials, it's already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li-Yau's famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. This is the second of our papers in understanding Landau-Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray-Singer torsion for Witten deformation in the non-compact setting.

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Section: Resultsmentioning

confidence: 99%

“…Then by Weyl's law (Cf. [13]), λ k ln(k). For such slowly growing eigenvalue distributions, it is unreasonable to consider the limit lim t→0 T r s (exp(−t f )).…”

Section: Weak Weyl Law For Witten Laplacianmentioning

confidence: 99%

Section: Introduction 1overviewmentioning

confidence: 99%

Section: Introduction 1overviewmentioning

confidence: 99%

See 2 more Smart Citations

Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

Dai¹,

Yan²

2020

Preprint

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show abstract

Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Dai

Yan

2022

Math. Z.

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Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as $${\mathbb {C}}^n$$ C n with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li–Yau’s famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. This is the second of our papers in understanding Landau–Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray–Singer torsion for Witten deformation in the non-compact setting.

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Some estimates for eigenvalues of Schrödinger operators

Tachizawa¹

1994

Proc. Japan Acad. Ser. A Math. Sci.

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Eigenvalue Asymptotics of Schrödinger Operators with Only Discrete Spectrum

Cited by 3 publications

References 18 publications

Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

Witten Deformation on Non-compact Manifold: Heat Kernel Expansion and Local Index Theorem

Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Some estimates for eigenvalues of Schrödinger operators

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