orthonormal
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11: 18.39 Applications in the Physical Sciences
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►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation
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►With the normalization factor the are orthonormal in .
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►The orthonormal stationary states and corresponding eigenvalues are then of the form
…The finite system of functions is orthonormal in , see (18.34.7_3).
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►with an infinite set of orthonormal
eigenfunctions
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12: 3.5 Quadrature
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►The corresponding orthonormal polynomials satisfy the recurrence relation
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►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi).
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►The monic version and orthonormal version of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by respectively , with and as in Table 18.3.1.
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13: Bibliography K
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Orthonormal polynomials with generalized Freud-type weights.
J. Approx. Theory 121 (1), pp. 13–53.
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14: 18.9 Recurrence Relations and Derivatives
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►They imply the recurrence coefficients for the orthonormal versions of the classical OP’s as well, see again §3.5(vi).
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15: Errata
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►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.
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Subsection 33.14(iv)
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Just below (33.14.9), the constraint described in the text “ when ,” was removed. In Equation (33.14.13), the constraint was added. In the line immediately below (33.14.13), it was clarified that is times a polynomial in , instead of simply a polynomial in . In Equation (33.14.14), a second equality was added which relates to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions , , do not form a complete orthonormal system.