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Albeverio, Sergio; Makarov, Konstantin A.

doi:10.1023/a:1005734807664

Cited by 7 publications

(1 citation statement)

References 25 publications

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“…[21][22][23][24][25][26][27][28][29][30]. As to the study of spectral problems of Hamiltonians involving pseudodifferential operators, see, e.g., [31][32][33]. Relativistic kinematics (thus involving pseudo-differential operators) also occurs in a Nelson's type of quantum fields (see [34,35]); for relativistic quantum fields constructed from pseudo-differential generators related to Lévy processes, see [16].…”

Section: Introductionmentioning

confidence: 99%

The discrete spectrum of the spinless one-dimensional Salpeter Hamiltonian perturbed byδ-interactions

Albeverio¹,

Fassari²,

Rinaldi³

2015

J. Phys. A: Math. Theor.

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We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive of strength centred at the origin, by explicitly providing its resolvent. Our approach is based on a ‘coupling constant renormalization’, a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited antisymmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength and the separation distance. With regard to the latter, a remarkable phenomenon is observed: differently from the well-behaved Schrödinger case, the 1D-Salpeter Hamiltonian with two identical Dirac distributions symmetrically situated with respect to the origin does not converge, as the separation distance shrinks to zero, to the one with a single centred at the origin having twice the strength. However, the expected behaviour in the limit (in the norm resolvent sense) can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself.

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Section: Introductionmentioning

confidence: 99%

The discrete spectrum of the spinless one-dimensional Salpeter Hamiltonian perturbed byδ-interactions

Albeverio¹,

Fassari²,

Rinaldi³

2015

J. Phys. A: Math. Theor.

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Models of Zero-Range Interaction for the Bosonic Trimer at Unitarity

Gallone

Michelangeli

2022

Springer Monographs in Mathematics

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The Morse and Maslov indices for Schrödinger operators

Latushkin

Sukhtaiev

Sukhtayev

2018

JAMA

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We study the spectrum of Schrödinger operators with matrix valued potentials utilizing tools from infinite dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the multidimensional Schrödinger operators with periodic potentials. For quasi convex domains in R n we recast the results connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.

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Cited by 7 publications

References 25 publications

The discrete spectrum of the spinless one-dimensional Salpeter Hamiltonian perturbed byδ-interactions

The discrete spectrum of the spinless one-dimensional Salpeter Hamiltonian perturbed byδ-interactions

Models of Zero-Range Interaction for the Bosonic Trimer at Unitarity

The Morse and Maslov indices for Schrödinger operators

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