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Schrödinger operator


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1: William P. Reinhardt
Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. …
2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. … …
Example 2: Radial 3D Schrödinger operators, including the Coulomb potential
See §18.39 for discussion of Schrödinger equations and operators. …
3: 18.39 Applications in the Physical Sciences
The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form …
§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
Analogous to (18.39.7) the 3D Schrödinger operator is … noting that the ψ p , l ( r ) are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). … Here tridiagonal representations of simple Schrödinger operators play a similar role. …
4: 18.36 Miscellaneous Polynomials
Completeness and orthogonality follow from the self-adjointness of the corresponding Schrödinger operator, Gómez-Ullate and Milson (2014), Marquette and Quesne (2013).
5: Bibliography S
  • B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.
  • 6: Bibliography C
  • H. L. Cycon, R. G. Froese, W. Krisch, and B. Simon (2008) Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry. Springer Verlag, New York.
  • 7: 18.38 Mathematical Applications
    Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. … A further operator, the so-called Casimir operator
    Dunkl Type Operators and Nonsymmetric Orthogonal Polynomials
    Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. …
    8: 31.17 Physical Applications
    We use vector notation [ 𝐬 , 𝐭 , 𝐮 ] (respective scalar ( s , t , u ) ) for any one of the three spin operators (respective spin values). …
    𝐻 s Ψ ( 𝐱 ) ( 2 𝐬 𝐭 ( 2 / a ) 𝐬 𝐮 ) Ψ ( 𝐱 ) = h s Ψ ( 𝐱 ) ,
    The operators 𝐉 2 and 𝐻 s admit separation of variables in z 1 , z 2 , leading to the following factorization of the eigenfunction Ψ ( 𝐱 ) : … Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). …
    9: Errata
    This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
  • Equation (3.3.34)

    In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

    Reported by Nico Temme on 2021-06-01

  • Subsections 1.15(vi), 1.15(vii), 2.6(iii)

    A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order α was more precisely identified as the Riemann-Liouville fractional integral operator of order α , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).