Abstract
We present the basics of the general spectral theory of self-adjoint operators and its application to the spectral analysis of self-adjoint ordinary differential operators. In finding spectrum and inversion formulas (eigenfunction expansion), we follow the Krein method of guiding functionals.
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Notes
- 1.
- 2.
Another name is “an (operator) spectral function.”
- 3.
Another name is “cyclic vector.”
- 4.
Nondegenerate in physics terminology.
- 5.
In the physics literature, the term “degenerate spectrum” is conventionaly used.
- 6.
Which is determined by the rank of the matrix σ ij (λ).
- 7.
We recall that coefficients of even s.a. differential operations are real-valued.
- 8.
An operator \(\hat{f}\) is called real if ξ ∈ D f implies that \(\overline{\xi } \in {D}_{f}\) and \(\hat{f}\xi = \eta \) implies that \(\hat{f}\overline{\xi } = \overline{\eta }\).
- 9.
The spectral function generally can contain the so-called singular, or singular continuous, term; see [97]. Such terms are absent in all the cases encountered in this book.
- 10.
In such a situation, the functions u j (x; λ) are conventionally called the generalized eigenfunctions of the operator \(\hat{f}\).
- 11.
For some λ, it may be less than m.
- 12.
For the continuous spectrum, these relations are symbolic in a sense.
- 13.
Treated in the sense of distributions.
- 14.
That is, the potential can be singular only at the endpoints of the interval.
- 15.
We note that possible mixed s.a. boundary conditions, like periodic ones, are beyond the scope of our consideration in this section, although such boundary conditions can yield a simple spectrum.
- 16.
Which is sufficient if the functional proves to be simple.
- 17.
- 18.
We note that a preliminary estimate of the asymptotic behavior of the function ψ (5.41) at the left endpoint a may be sufficient to assert that ψ ∈ L 2(a, b).
- 19.
This is the case in which p + (a) = p − (a) = 0, i.e., the left a.b. coefficients are equal to zero.
- 20.
This is the case in which p + (a) = p − (a) = 1.
- 21.
The constant κ0 is of dimension of inverse length, so that κ0δ is dimensionless.
- 22.
This is the case in which p + (a) = p − (a) = 1.
- 23.
By the function U(x; z), we here mean the specific function U for each operator \(\hat{{f}}_{\mathfrak e}\).
- 24.
Namely, for μ > 1 ∕ 2 and ν≠ ± π ∕ 2.
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Spectral Analysis of Self-adjoint Operators. In: Self-adjoint Extensions in Quantum Mechanics. Progress in Mathematical Physics, vol 62. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4662-2_5
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