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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Self-Adjoint and Symmetric Operators
One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner. An essential feature of such symmetric operators is that their eigenvalues λ are real, and eigenfunctions …
Self-adjoint extensions of a symmetric Operator
We have a direct sum of linear spaces: 𝒟 ( T ) = 𝒟 ( T ) + N i + N i . …
2: 35.2 Laplace Transform
35.2.3 f 1 f 2 ( 𝐓 ) = 𝟎 < 𝐗 < 𝐓 f 1 ( 𝐓 𝐗 ) f 2 ( 𝐗 ) d 𝐗 .
3: 18.38 Mathematical Applications
Define operators K 0 and K 1 acting on symmetric Laurent polynomials by K 0 = L ( L given by (18.28.6_2)) and ( K 1 f ) ( z ) = ( z + z 1 ) f ( z ) . … The Dunkl type operator is a q -difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial R n ( z ; a , b , c , d | q ) and the ‘anti-symmetric’ Laurent polynomial z 1 ( 1 a z ) ( 1 b z ) R n 1 ( z ; q a , q b , c , d | q ) , where R n ( z ) is given in (18.28.1_5). …
4: 1.3 Determinants, Linear Operators, and Spectral Expansions
Real symmetric ( 𝐀 = 𝐀 T ) and Hermitian ( 𝐀 = 𝐀 H ) matrices are self-adjoint operators on 𝐄 n . …
5: 18.39 Applications in the Physical Sciences
§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
6: 21.7 Riemann Surfaces
Define the operation
21.7.12 T 1 T 2 = ( T 1 T 2 ) ( T 1 T 2 ) .
21.7.14 𝜼 ( T 1 T 2 ) = 𝜼 ( T 1 ) + 𝜼 ( T 2 ) ,
21.7.15 4 𝜼 1 ( T ) 𝜼 2 ( T ) = 1 2 ( | T U | g 1 ) ( mod 2 ) ,
7: 2.6 Distributional Methods
2.6.32 0 f ( t ) ( t + z ) ρ d t , ρ > 0 ,
An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …
8: 32.2 Differential Equations
32.2.13 z ( 1 z ) 𝐼 ( w d t t ( t 1 ) ( t z ) ) = w ( w 1 ) ( w z ) ( α + β z w 2 + γ ( z 1 ) ( w 1 ) 2 + ( δ 1 2 ) z ( z 1 ) ( w z ) 2 ) ,
where
32.2.14 𝐼 = z ( 1 z ) d 2 d z 2 + ( 1 2 z ) d d z 1 4 .
§32.2(v) Symmetric Forms
9: Bibliography G
  • I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.
  • V. X. Genest, L. Vinet, and A. Zhedanov (2016) The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.
  • J. S. Geronimo, O. Bruno, and W. Van Assche (2004) WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
  • 10: 3.1 Arithmetics and Error Measures
    Rounding
    Symmetric rounding or rounding to nearest of x gives x or x + , whichever is nearer to x , with maximum relative error equal to the machine precision 1 2 ϵ M = 2 p . … The elementary arithmetical operations on intervals are defined as follows: …