Abstract
The problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential operations is discussed based on the general theory of self-adjoint extensions of symmetric operators outlined in the previous Chap. 4. We describe various methods for specifying self-adjoint operators associated with self-adjoint differential operations by boundary conditions. An attention is focused on features peculiar to differential operators, among them a notion of natural domain and a representation of asymmetry forms of the adjoint operator in terms of boundary forms.
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Notes
- 1.
Our conventions about understanding this notion and the related terminology are explained in Sect. 2.3.1.
- 2.
The derivative of order k of a function ψ is commonly denoted by ψk.
- 3.
This restriction is natural, e.g., for radial Hamiltonians.
- 4.
This identification implies that we use a system of units where ℏ = 1.
- 5.
Quasiderivatives naturally emerge in this form when a product \(\overline{\chi }\check{f}\psi \) is integrated by parts. The representation (4.10) can be taken as an independent definition of quasiderivatives. A similar representation evidently holds for quasiderivative differential operations.
- 6.
In the physics literature, such integrals are called overlap integrals. The formula (4.19) that follows implies that the overlap integrals for solutions of the eigenvalue problem are determined by the asymptotic behavior of the eigenfunctions at the endpoints of the interval.
- 7.
In the case under consideration, the usual Wronskian coincides with the quasi-Wronskian.
- 8.
- 9.
Although u is generally not square-integrable, the symbol , for the scalar product in (4.26) is correct because of the compactness of the support of the function χ.
- 10.
The more so, since \(\hat{{f}}^{{_\ast}}\) proves to be the adjoint of a symmetric operator; see below.
- 11.
As for any differential operator associated with an s.a. differential operation.
- 12.
- 13.
Another way to make sure that this is correct is to note that \(\hat{{H}}^{{_\ast}} =\) \(\widehat{{\mathcal{H}}}^{{_\ast}} +\hat{ V }\), where \(\hat{V } = V (x)\) is a bounded operator defined everywhere.
- 14.
“Hamiltonians” in the language of physics.
- 15.
We mean the reductions of the total Hamiltonian to the subspaces of partial waves with fixed l and m.
- 16.
Curiously, in these cases, we have ψ lm (r) → 0 as r → 0, but ψ lm ′(r) → ∞.
- 17.
In fact, this is the operator \(\hat{{V }}^{{_\ast}}\) associated with the operation \(\check{V } = V \left (x\right )\). For simplicity, we do not write here the superscript ∗ .
- 18.
It appears convenient to replace ξ ∗ in (2.24) by ψ ∗ , see immediately below, while ξ is naturally replaced with φ.
- 19.
In particular, the above-discussed additional boundary conditions on the wave functions belonging to the domain of the Hamiltonian \(\hat{H}\), which are justified by physical arguments, actually define s.a. restrictions of the non-s.a. operator \(\hat{{H}}^{{_\ast}}\) defined on the natural domain.
- 20.
In conventional units, a certain degree of length or momentum (or energy).
- 21.
We are following here a recent convention in the physics literature.
- 22.
In general, these sysyems are subsystems of the respective fundamental systems of solutions of (4.46), because the fundamental systems can contain non-square-integrable solutions.
- 23.
It is not obligatory to normalize the basis functions to unity; it is sufficient that their norms be the same.
- 24.
This is the starting point of the so-called method of splitting [116].
- 25.
We recall that the deficiency indices of a symmetric operator and those of its closure are the same.
- 26.
Of course taking into account the change of notation for some basic notions in this chapter in comparison with the previous one.
- 27.
It must be confessed that in this case we actually solve the inverse problem of finding a matrix U that yields periodic boundary conditions.
- 28.
Following the above convention for differential operators.
- 29.
These properties as applied to differential operators were already cited above.
- 30.
Of course, this condition is compatible with condition (4.92).
- 31.
- 32.
It is also easy to verify that the deficiency indices of the initial symmetric operator \(\widehat{\mathcal{H}}\) are minimum,\(\ {m}_{\pm } = n/2 = 1\) (see Sect. 6.2).
- 33.
We recall that this sum is direct, but not orthogonal.
- 34.
The so-called reduction to the principal axes of inertia by invertible linear transformations of the expansion coefficients.
- 35.
For example, we can take functions ψ ∗ α such that \({\psi }_{{_\ast}\alpha }^{(k-1)}({x}_{0}) = {\delta }_{\alpha }^{k}\), x 0 ∈ [a, a + ε].
- 36.
We clarify this point below by the example of a regular even s.a. differential operation.
- 37.
From this point on, we omit the subscript f in the symbol of boundary forms.
- 38.
For even s.a. differential operations, this is evident from (4.103).
- 39.
This is possible for a singular endpoint; examples are given below.
- 40.
We recall that by this, we actually mean the Hermitian form \(\frac{1} {2\mathrm{i}\kappa }{\Delta }_{{f}^{+}}\); this reduction is equivalent to reducing the Hermitian matrix ω to a canonical diagonal form.
- 41.
Which is partly done for future reference.
- 42.
The symbol a∕b means a and/or b depending on whether both endpoints a and b are regular or one of the endpoints, a or b, is regular.
- 43.
In some cases, the a.b. coefficients arise as numbers of the same nonzero dimension, in which case the factor 2i κ changes to a pure imaginary factor of appropriate dimension, while the matrix ω remains dimensionless.
- 44.
We could equivalently use representation (4.103) for the quadratic boundary form at a regular endpoint.
- 45.
Which can always be done by multiplying \(\check{f}\) by an appropriate inessential constant dimensional factor.
References
Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. National Bureau of Standards, New York (1964)
Adami, R., Teta, A.: On the Aharonov–Bohm effect. Lett. Math. Phys. 43, 43–54 (1998)
Alliluev, S.P.: The problem of collapse to the center in quantum mechanics. Sov. Phys. JETP 34, 8–13 (1972)
Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Pitman, Boston (1981)
Araujo, V.S., Coutinho, F.A.B., Perez, J.F.: Operator domains and self-adjoint operators. Amer. J. Phys. 72, 203–213 (2004)
Bawin, M., Coon, S.A.: Singular inverse square potential, limit cycles, and self-adjoint extensions. Phys. Rev. A 67(5) 042712 (2003)
Berezansky, Yu.M.: Eigenfunction Expansions Associated with Self-adjoint Operators. Naukova Dumka, Kiev (1965)
Berezin, F.A., Shubin, M.A.: Schrödinger Equation. Kluwer, New York (1991)
Dunford, N., Schwartz, J.T.: Linear operators, part II. Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers, New York (1963)
Faddeev, L.D., Maslov, B.P.: Operators in Quantum Mechanics. In: Krein, S.G. (ed.) Spravochnaya Matematicheskaya Biblioteka (Functional Analysis). Nauka, Moscow (1964)
Gelfand, I.M., Shilov, G.E.: Some problems of the theory of differential equations. Generalized functions, part 3. Fizmatgiz, Moscow (1958)
Gorbachuk, V.I., Gorbachuk, M.L., Kochubei, A.N.: Extension theory for symmetric operators and boundary value problems for differential equations. Ukr. Math. J. 41(10) 1117–1129 (1989); translation from Ukr. Mat. Zh. 41(10) 1299–1313 (1989)
Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht (1991)
Halperin, I.: Introduction to the Theory of Distributions. University of Toronto Press, Toronto (1952) (Based on the lectures given by Laurent Schwartz)
Hutson, V.C.L., Pym, J.S.: Applications of Functiomal Analysis and Operator Theory. Academic Press, London (1980)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)
Kostuchenko, A.G., Krein, S.G., Sobolev, V.I.: Linear Operators in Hilbert Space. In: Krein, S.G. (ed.) Spravochnaya Matematicheskaya Biblioteka (Functional Analysis). Nauka, Moscow (1964)
Kuzhel, A.V., Kuzhel, S.A.: Regular Extensions of Hermitian Operators. VSP, Utrecht (1998)
Levitan, B.M.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Gostechizdat, Moscow (1950) (in Russian)
Naimark, M.A.: Linear differential operators. Nauka, Moskva (1959) (in Russian). F. Ungar Pub. Co. New York (1967)
Narnhofer, H.: Quantum theory for 1 ∕ r 2 potentials. Acta Phys. Aust. 40, 306–322 (1974)
Perelomov A.M., Popov, V.S.: Fall to the center in quantum mechanics. Theor. Math. Phys. 4, 664–677 (1970)
Plesner, A.I.: Spectral Theory of Linear Operators. Nauka, Moscow (1965)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Functional Analysis. Academic Press, New York (1980)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators. Academic Press, New York (1978)
Richtmyer, R.D.: Principles of Advanced Mathematical Physics, vol. 1. Springer, New York (1978)
Riesz, F., Sz.-Nagy, B.: Lecons d’Analyse Fonctionnelle. Akademiai Kiado, Budapest (1972)
Shilov, G.E.: Mathematical Analysis. Second special course. Nauka, Moscow (1965)
Stone, M.H.: Linear Transformations in Hilbert space and their applications to analysis. Am. Math. Soc., vol. 15. Colloquium Publications, New York (1932)
Stepanov, V.V.: Course of differential equations. GIFML, Moskva (1959)
Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Clarendon Press, Oxford (1946)
Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Part II. Clarendon Press, Oxford (1958)
van Haeringen, H.: Bound states for r − 2 potentials in one and three dimensions. J. Math. Phys. 19, 2171–2179 (1978)
Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators.I. Russ. Phys. J. 50(1) 1–31 (2007)
Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Springer, Berlin (1987)
Weyl, H.: Über Gewöhnliche Lineare Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 37–64. Göttinger Nachrichten (1909)
Weyl, H.: Über Gewöhnliche differentialgleichungen mit singularitäten und zugehörigen entwicklungen willkürlicher funktionen. Math. Annal. 68, 220–269 (1910)
Weyl, H.: Über Gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 442–467. Göttinger Nachrichten (1910)
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Gitman, D.M., Tyutin, I.V., Voronov, B.L. (2012). Differential 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_4
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