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V. S. Serov 《Mathematical Notes》2000,67(5):639-645
We obtain sharp conditions for the absolute uniform convergence of Fourier series in the eigenfunctions of the Schrödinger operator with Kato potential in a bounded domain for functions lying in the domains of generalized fractional powers of the original Schrödinger operator or in generalized Besov classes with a sharp exponent. 相似文献
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A. I. Karol' 《Journal of Mathematical Sciences》1993,64(6):1341-1348
A meromorphic extension to the entire plane is obtained for the -function of a Schrödinger operator with a potential that increases at infinity in a power-like manner.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 176–187, 1990. 相似文献
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G. D. Raikov 《Inventiones Mathematicae》1992,110(1):75-93
Summary We consider the Schrödinger operatorH=–+W+V acting inL
2(
m
),m2, with periodic potentialW perturbed by a potentialV which decays slowly at infinity. We study the asymptotic behaviour of the discrete spectrum ofH near any given boundary point of the essential spectrum.Oblatum 1-VII-1991 & 20-I-1992Partly supported by the Bulgarian Science Foundation under contract No MM 8/1991 相似文献
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M. V. Buslaeva 《Journal of Mathematical Sciences》1987,37(1):797-798
The ideas of scattering theory are applied to the construction of a unitary operator realizing the similarity of the operator - id/d in L2() with a one-dimensional Schrödinger operator on the semiaxis with potential v(x), admitting at infinity the estimate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 10–12, 1985. 相似文献
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Yu. P. Chuburin 《Theoretical and Mathematical Physics》1999,120(2):1045-1057
We consider the Schrödinger operator with a potential that is periodic with respect to two variables and has the shape of a small step perturbed by a function decreasing with respect to a third variable. We show that under certain conditions on the magnitudes of the step and the perturbation, a unique level that can be an eigenvalue or a resonance exists near the essential spectrum. We find the asymptotic value of this level. 相似文献
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I. V. Kachkovskiy 《Functional Analysis and Its Applications》2013,47(2):104-112
We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δ Σ(x,y), where Σ is a periodic system of hypersurfaces. We establish that there are no eigenvalues in the spectrum of this operator, provided that Σ is sufficiently smooth and σ ∈ L p,loc(Σ), p > d ? 1. 相似文献
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Summary. The electronic Schrödinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry. The present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the Pauli principle. It is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to N+1 with N the number of electrons, across the singularities of the interaction potentials. The result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces. It indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic Schrödinger equation and that it may even be possible to reduce the computational complexity of an N-electron problem to that of a one-electron problem.Mathematics Subject Classification (1991): 35J10, 35B65, 41A63, 65D99 相似文献
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This article contains the justification of the quasi-classical asymptotic formula (usually attributed to N. Bohr) for the counting function N(λ) of the 1-D Schrödinger operator with potential V increasing at infinity. Results were known under strong regularity conditions on V. Using direct methods based on the phase formalism, we give much wider sufficient conditions on V for the applicability of the Bohr formula. Several counterexamples show that these conditions cannot be significantly improved. 相似文献
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《Comptes Rendus Mathematique》2008,346(11-12):635-640
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Yu. P. Chuburin 《Theoretical and Mathematical Physics》2009,158(1):96-104
We consider a two-dimensional periodic Schrödinger operator perturbed by the interaction potential of two one-dimensional particles. We prove that quasilevels (i.e., eigenvalues or resonances) of the given operator exist for a fixed quasimomentum and a small perturbation near the band boundaries of the corresponding periodic operator. We study the asymptotic behavior of the quasilevels as the coupling constant goes to zero. We obtain a simple condition for a quasilevel to be an eigenvalue. 相似文献
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A. G. Smirnov 《Theoretical and Mathematical Physics》2016,187(2):762-781
We consider the one-dimensional Schrödinger equation -f″ + qκf = Ef on the positive half-axis with the potential qκ(r) = (κ2 - 1/4)r-2. For each complex number ν, we construct a solution uνκ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions uνκ(E) determine a unitary eigenfunction expansion operator Uκ,ν: L2(0,∞) → L2(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?r2 + qκ(r) for the Hamiltonian is diagonalized by the operator Uκ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures Vκ,ν and prove their continuity in κ and ν. 相似文献
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Yu. B. Orochko 《Mathematical Notes》1976,20(4):877-881
We examine the operators=–+v, v L2, loe (R
n
), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L2(Rn).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976. 相似文献
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V. Zh. Sakbaev I. V. Volovich 《P-Adic Numbers, Ultrametric Analysis, and Applications》2017,9(1):39-52
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states. 相似文献
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L. V. Kritskov 《Mathematical Notes》1999,65(4):454-461
Suppose thatА is a nonnegative self-adjoint extension to {
} of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {
} or the condition {
} in which the nonnegative function itχ(r) is such that {
}. For each α∈(0, 2], we establish an estimate of the generalized Fourier transforms of an arbitrary function {
} of the form {
} If, in addition, {
}, then, along with this estimate, a similar lower bound is established.
Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 542–551, April, 1999. 相似文献