Generalized pointlike interactions in R3 and related models with rational S matrix

Shondin, Yuri

doi:10.1007/bf01018356

Cited by 10 publications

(21 citation statements)

References 3 publications

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“…So in [5] an extension theory of Hermitian operators in spaces with the same inner product was constructed at the algebraic level by Berezin, but he did not consider the completion of this space. The completion of this space and of the operators acting there was obtained in [6]. The paper [7] deals with general models of self-adjoint opcrators acting in spaces with the indefinite inner product (1).…”

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confidence: 99%

Extensions of Laguerre operators in indefinite inner product spaces

Derkach¹

1998

Math Notes

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ABSTRACT. The Laguerre-Sonin polynomials L (a) are orthogonal in linear spaces with indefinite inner product if a < -1. We construct the completion II(a) of this space and describe self-adjoint extensions of the Laguerre operator l(y) = zy" + (1 + ~ -z)y ~ , a < -1, in the space II(a). In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctions coincide with the Laguerre-Sonin polynomials and form an orthogonal basis in II(c~).KEY WORDS: self-adjoint operator, indefinite inner product space, Laguerre operator, Laguerre-Sonin polynomial.w Introduction For a > -1, the Laguerre-Sonin polynomials = . 9 z+, L(.~)(x)are the eigenfunctions of some self-adjoint operator A~ corresponding to the differential expression t~(y) = xy" + (1 + ~ -z)y'and form an orthonormal system in the Hilbert space L2(R+,wa), where wa = xae -~ 9 For a < -1, the operator generated in L2 (R+, wa) by the differential expression la is also a self-adjoint operator (and even unitarily equivalent to the operator A_h in the space L2(R+, w,~)). However, the Laguerre-Sonin polynomials satisfying the equation l~(L~ ~)) = -nL(n ~) , n 6 Z+ no longer belong to the space L2(R+, wa) (for a < -1). It was pointed out in [1] that for -n-1 < ~ < -n the Laguerre-Sonin polynomials L~ =) are orthogonal in the space with indefinite inner productIn view of this fact, we naturally wish to construct the self-adjoint operator generated by the differential expression la in the Pontryagin space equipped with the indefinite inner product (1) rather than in the space L~(R+,wa). This problem was considered in [2,3]; the same problem for polynomials of the Jacobi type was studied in [4]. However, it should be noted that [2] describes an erroneous procedure for completing the space with metric (1). A similar problem arises in the theory of generalized point interactions; the divergence in this theory is eliminated by introducing a renormed inner product similar to the metric (1). So in [5] an extension theory of Hermitian operators in spaces with the same inner product was constructed at the algebraic level by Berezin, but he did not consider the completion of this space. The completion of this space and of the operators acting there was obtained in [6]. The paper [7] deals with general models of self-adjoint opcrators acting in spaces with the indefinite inner product (1).

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confidence: 99%

Extensions of Laguerre operators in indefinite inner product spaces

Derkach¹

1998

Math Notes

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“…Statement (2) of Theorem 4.1 was obtained (in the Hilbert case) in [5]. Note that the extension problems for Hermitian linear relations on Pontryagin spaces naturally arise in quantum models of point interactions [31]. …”

Section: Thus T(k) E N~(7~) It Follows From the Relation ~-(R + M(mentioning

confidence: 92%

On generalized resolvents of Hermitian relations in Krein spaces

Derkach¹

1999

J Math Sci

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“…It was shown there that spaces with indefinite metric appear in this model. Development and new applications are reflected in [11,[19][20][21][22][23]. In [11], Hamiltonians are obtained for these models and stationary wave operators and a scattering matrix are found.…”

Section: {X} = {Xl~i} Where Xli(x) = Bt~(io)6(x -Xi)mentioning

confidence: 99%

Generalized pointlike interactions in R3 and related models with rational S matrix

Cited by 10 publications

References 3 publications

Extensions of Laguerre operators in indefinite inner product spaces

Extensions of Laguerre operators in indefinite inner product spaces

On generalized resolvents of Hermitian relations in Krein spaces

Perturbations of elliptic operators on high codimension subsets and the extension theory on an indefinite metric space

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