Identities Related to Variational Principles

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Abstract
The existence of a well‐known identity associated with variational principles for any scattering parameter Q, and often serving as the starting point for the development of a variational bound on Q, strongly suggests that it might be useful to construct identities associated with variational principles for quantities other than scattering parameters. An identity associated with the variational principle for the determination of inner products of the linear form g†φ, a generalization of the aforementioned identity, is presented. Here, g is a known function, and φ is an unknown function satisfying Mφ = ω and specified boundary conditions, where M is a known linear operator and ω is a known function. The generalized identity is obtained from a variational principle for g†φ, this variational principle being itself a generalization of the usual Kohn variational principle for scattering amplitudes and phase shifts. An identity associated with a variational principle for the quadratic form φ†Wφ, with φ as above and W a known linear operator, is also obtained. Finally, we obtain an identity for φ(∞), where φ is defined as the solution of a nonlinear differential equation. The generalized identity related to g†φ is verified for a simple exactly solvable problem.

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