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1: 10.17 Asymptotic Expansions for Large Argument
10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) ) ,
where 𝒱 denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that | t | changes monotonically. Bounds for 𝒱 z , i ( t ) are given by
10.17.15 𝒱 z , i ( t ) { | z | , 0 ph z π , χ ( ) | z | , 1 2 π ph z 0  or  π ph z 3 2 π , 2 χ ( ) | z | , π < ph z 1 2 π  or  3 2 π ph z < 2 π ,
The bounds (10.17.15) also apply to 𝒱 z , i ( t ) in the conjugate sectors. …
2: 10.40 Asymptotic Expansions for Large Argument
10.40.11 | R ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ( t ) exp ( | ν 2 1 4 | 𝒱 z , ( t 1 ) ) ,
where 𝒱 denotes the variational operator2.3(i)), and the paths of variation are subject to the condition that | t | changes monotonically. Bounds for 𝒱 z , ( t ) are given by
10.40.12 𝒱 z , ( t ) { | z | , | ph z | 1 2 π , χ ( ) | z | , 1 2 π | ph z | π , 2 χ ( ) | z | , π | ph z | < 3 2 π ,
3: 1.4 Calculus of One Variable
1.4.33 𝒱 a , b ( f ) = sup j = 1 n | f ( x j ) f ( x j 1 ) | ,
If 𝒱 a , b ( f ) < , then f ( x ) is of bounded variation on ( a , b ) . In this case, g ( x ) = 𝒱 a , x ( f ) and h ( x ) = 𝒱 a , x ( f ) f ( x ) are nondecreasing bounded functions and f ( x ) = g ( x ) h ( x ) . …
1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x ,
Lastly, whether or not the real numbers a and b satisfy a < b , and whether or not they are finite, we define 𝒱 a , b ( f ) by (1.4.34) whenever this integral exists. …
4: 2.7 Differential Equations
2.7.23 | ϵ j ( x ) | , 1 2 f 1 / 2 ( x ) | ϵ j ( x ) | exp ( 1 2 𝒱 a j , x ( F ) ) 1 , j = 1 , 2 ,
provided that 𝒱 a j , x ( F ) < . …and 𝒱 denotes the variational operator2.3(i)). …
2.7.25 𝒱 a j , x ( F ) = | a j x | 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) g ( t ) f 1 / 2 ( t ) | d t | .
Assuming also 𝒱 a 1 , a 2 ( F ) < , we have …
5: 2.8 Differential Equations with a Parameter
In addition, 𝒱 𝒬 j ( A 1 ) and 𝒱 𝒬 j ( A n ) must be bounded on 𝚫 j ( α j ) . … These results are valid when 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , α 2 ( | ξ | 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 0 , α 2 ( ξ 1 / 2 B 0 ) and 𝒱 0 , α 2 ( ξ 1 / 2 B n 1 ) are finite. … These results are valid when 𝒱 α 1 , 0 ( | ξ | 1 / 2 B 0 ) and 𝒱 α 1 , 0 ( | ξ | 1 / 2 B n 1 ) are finite. …
6: 2.3 Integrals of a Real Variable
In both cases the n th error term is bounded in absolute value by x n 𝒱 a , b ( q ( n 1 ) ( t ) ) , where the variational operator 𝒱 a , b is defined by
2.3.6 𝒱 a , b ( f ( t ) ) = a b | f ( t ) | d t ;
7: Errata
  • Equation (2.7.25)
    2.7.25 𝒱 a j , x ( F ) = | a j x | 1 f 1 / 4 ( t ) d 2 d t 2 ( 1 f 1 / 4 ( t ) ) g ( t ) f 1 / 2 ( t ) | d t |

    The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value for 𝒱 a j , x ( F ) .

  • Equation (2.3.6)
    2.3.6 𝒱 a , b ( f ( t ) ) = a b | f ( t ) | d t

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Juan Luis Varona on 2021-02-08

  • Equation (1.4.34)
    1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Tran Quoc Viet on 2020-08-11

  • Equation (10.17.14)
    10.17.14 | R ± ( ν , z ) | 2 | a ( ν ) | 𝒱 z , ± i ( t ) exp ( | ν 2 1 4 | 𝒱 z , ± i ( t 1 ) )

    Originally the factor 𝒱 z , ± i ( t 1 ) in the argument to the exponential was written incorrectly as 𝒱 z , ± i ( t ) .

    Reported 2014-09-27 by Gergő Nemes.

  • 8: Bibliography S
  • C. Schwartz (1961) Variational calculations of scattering. Ann. Phys. 16, pp. 36–50.
  • I. J. Schwatt (1962) An Introduction to the Operations with Series. 2nd edition, Chelsea Publishing Co., New York.
  • B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.
  • B. Simon (1995) Operators with Singular Continuous Spectrum: I. General Operators. Annals of Mathematics 141 (1), pp. 131–145.
  • 9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Bounded and Unbounded Linear Operators
    Self-Adjoint Operators
    Spectrum of an Operator
    10: 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 … Analogous to (18.39.7) the 3D Schrödinger operator is …where L 2 is the (squared) angular momentum operator (14.30.12). … The radial operator (18.39.28) … While s in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). …