Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces

Лаптев, Ари

doi:10.1006/jfan.1997.3155

Cited by 149 publications

(150 citation statements)

References 13 publications

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“…The proof of Theorem 3.1 relies on a lifting technique, which was introduced in [25], see also [26], [8], [33], and [9] for further developments and applications.…”

Section: Remark 32 Let Us Definementioning

confidence: 99%

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Geometrical Versions of improved Berezin–Li–Yau Inequalities

Geisinger¹,

Лаптев²,

Weidl³

2011

J. Spectr. Theory

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Abstract. We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set inIn particular, we derive upper bounds on Riesz means of order 3=2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.Mathematics Subject Classification (2010). Primary 35P15; Secondary 47A75.

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“…The proof of Theorem 3.1 relies on a lifting technique, which was introduced in [25], see also [26], [8], [33], and [9] for further developments and applications.…”

Section: Remark 32 Let Us Definementioning

confidence: 99%

“…Thus by taking the limit " ! 0 we find X From (9) we obtain similar bounds on higher eigenvalues using a method introduced in [25]. …”

Section: Lower Bounds On Individual Eigenvaluesmentioning

confidence: 99%

Geometrical Versions of improved Berezin–Li–Yau Inequalities

Geisinger¹,

Лаптев²,

Weidl³

2011

J. Spectr. Theory

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“…We outline a short proof of (13) from [16]. Let {^} be an ortho-normed eigenfunctions of H^, which we continue by 0 on R^O.…”

Section: Fact For All a > 1 A > 0 D G N And Any Open Domain 0 It Hmentioning

confidence: 99%

Recent results on Lieb-Thirring inequalities

Лаптев¹,

Weidl²

2000

Journées Équations Aux Dérivées Partielles

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“…Next, we shall show that our bounds for d = 2 can be applied to deduce analogous bounds for d = 3. This argument is in the spirit of the lifting argument from [La1,La2,LaWe]. We denote byĤ…”

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confidence: 99%

Eigenvalue bounds for Schrödinger operators with complex potentials

Frank

2011

Bulletin of the London Mathematical Society

149

349

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Abstract. We prove Lieb-Thirring inequalities for Schrödinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.

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Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces

Abstract: We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L1997 Academic Press

Cited by 149 publications

References 13 publications

Geometrical Versions of improved Berezin–Li–Yau Inequalities

Geometrical Versions of improved Berezin–Li–Yau Inequalities

Recent results on Lieb-Thirring inequalities

Eigenvalue bounds for Schrödinger operators with complex potentials

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