variational operator
(0.001 seconds)
10 matching pages
1: 10.17 Asymptotic Expansions for Large Argument
…
►
10.17.14
►where denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that changes monotonically.
Bounds for are given by
►
10.17.15
…
►The bounds (10.17.15) also apply to in the conjugate sectors.
…
2: 10.40 Asymptotic Expansions for Large Argument
…
►
10.40.11
►where denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that changes monotonically.
Bounds for are given by
►
10.40.12
…
3: 1.4 Calculus of One Variable
…
►
1.4.33
…
►If , then is of bounded
variation on .
In this case, and are nondecreasing bounded functions and .
…
►
1.4.34
…
►Lastly, whether or not the real numbers and satisfy , and whether or not they are finite, we define
by (1.4.34) whenever this integral exists.
…
4: 2.7 Differential Equations
…
►
2.7.23
,
►provided that .
…and denotes the variational operator (§2.3(i)).
…
►
2.7.25
►Assuming also , we have
…
5: 2.8 Differential Equations with a Parameter
…
►In addition, and must be bounded on .
…
►These results are valid when and are finite.
…
►These results are valid when and are finite.
…
►These results are valid when and are finite.
…
6: 2.3 Integrals of a Real Variable
…
►In both cases the th error term is bounded in absolute value by , where the variational
operator
is defined by
►
2.3.6
…
7: Errata
…
►
Equation (2.7.25)
…
►
Equation (2.3.6)
…
►
Equation (1.4.34)
…
►
Equation (10.17.14)
…
2.7.25
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 .
2.3.6
The integrand has been corrected so that the absolute value does not include the differential.
Reported by Juan Luis Varona on 2021-02-08
1.4.34
The integrand has been corrected so that the absolute value does not include the differential.
Reported by Tran Quoc Viet on 2020-08-11
10.17.14
Originally the factor in the argument to the exponential was written incorrectly as .
Reported 2014-09-27 by Gergő Nemes.
8: Bibliography S
…
►
Variational calculations of scattering.
Ann. Phys. 16, pp. 36–50.
►
An Introduction to the Operations with Series.
2nd edition, Chelsea Publishing Co., New York.
…
►
The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions.
Annals of Physics 97 (2), pp. 279–288.
…
►
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 is the (squared) angular momentum operator (14.30.12).
…
►The radial operator (18.39.28)
…
►While 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).
…