Uniqueness of the self-adjoint extension of singular elliptic differential operators

Ikebe, Teruo; Kato, Tosio

doi:10.1007/bf00253334

Cited by 175 publications

(68 citation statements)

References 7 publications

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“…As for Theorem 1 we have only to prove (iii), since it is known that (i) holds by Ikebe-Kato [5] and Jorgens [6], and (ii) by Jorgens [6] and Zislin [11]. Hereafter we assume (2.…”

Section: Remarkmentioning

confidence: 99%

Finiteness of the Number of Discrete Eigenvalues of the Schrödinger Operator for a Three Particle System

Uchiyama¹

1969

Publ. Res. Inst. Math. Sci.

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“…As for Theorem 1 we have only to prove (iii), since it is known that (i) holds by Ikebe-Kato [5] and Jorgens [6], and (ii) by Jorgens [6] and Zislin [11]. Hereafter we assume (2.…”

Section: Remarkmentioning

confidence: 99%

Finiteness of the Number of Discrete Eigenvalues of the Schrödinger Operator for a Three Particle System

Uchiyama¹

1969

Publ. Res. Inst. Math. Sci.

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“…A first Sears-type result for H b,V with locally singular V was obtained by T. Ikebe and T. Kato [37] (still with a radial minorant for V ). The paper [77] extends this result to complete Riemannian manifolds, allows non-radial minorant of V with Oleinik-type completeness condition, though requires V to be locally bounded.…”

Section: Appendix C Stummel and Kato Classesmentioning

confidence: 95%

Essential self-adjointness of Schrödinger-type operators on manifolds

2002

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Abstract. We obtain several essential self-adjointness conditions for the Schrödinger type operator HV = D * D + V , where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV . These results generalize the theorems of E. C. Titchmarsh, D. B. Sears, F. S. Rofe-Beketov, I. M. Oleinik, M. A. Shubin and M. Lesch. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in R n with the Lebesgue measure. We also allow singular potentials V . In particular, we obtain a new self-adjointness condition for a Schrödinger operator on R n whose potential has the Coulomb-type singularity and is allowed to fall off to −∞ at infinity.For a specific case when the principal symbol of D * D is scalar, we establish more precise results for operators with singular potentials. The proofs are based on an extension of the Kato inequality which modifies and improves a result of

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“…The earliest version of this is Ikebe and Kato. 130 My favorite theorem of this genre is due to Faris and Lavine 80 ͑see Reed and Simon, 211 Theorem X.38͒. In particular, Stark Hamiltonians where V ϭc"xϩV 0 are essentially self-adjoint for suitable V 0 .…”

Section: Remarksmentioning

confidence: 99%