Self-adjoint differential operators assosiated with self-adjoint differential expressions

B.L. Voronov; D.M. Gitman; I.V. Tyutin

REVIEW

Self-adjoint differential operators assosiated with self-adjoint differential expressions

B.L. Voronov, D.M. Gitman, I.V. Tyutin

Quantum Physics · Submitted 21 Mar 2006 · 33 references

Pith's verdict: Useful review with correct mathematics, but the advertised shortcut is not a shortcut—the boundary-condition method reproduces deficiency-subspace data in disguise.

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arXiv IDquant-ph/0603187 (version 2)

Pith IDYAKQCKHX, Pith's permanent record of this paper and its review

CitationB.L. Voronov, D.M. Gitman, I.V. Tyutin. "Self-adjoint differential operators assosiated with self-adjoint differential expressions." arXiv:quant-ph/0603187 (2006). https://pith.science/paper/YAKQCKHX

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Editor's note

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Bottom line: this is a solid, useful review paper, but the advertised advantage is overblown. The alternative boundary-condition method in Theorem 15 is a repackaging of von Neumann's deficiency-subspace theory, not an independent shortcut.

What the paper does well: it gives a self-contained account of self-adjoint extensions for ordinary differential operators using asymmetry forms rather than the usual Cayley transform, and it works through concrete examples—momentum on the line, half-line, and finite interval; free Hamiltonian on a finite interval and half-line—with explicit U(1) and U(2) families of boundary conditions. That is genuinely handy for physicists who need to see what the abstract extension theorems mean in practice. The examples check out, the exposition is careful, and the paper is honest enough to state limitations in Sec. 3.1 and Sec. 3.7.

The soft spot is the central claim. The abstract says the method avoids evaluating deficiency subspaces and deficiency indices. Theorem 15 requires a set {w_k} in the adjoint domain that is linearly independent modulo the closure domain and satisfies the asymmetry-form condition, with m equal to the deficiency index. To produce those vectors you need boundary-behavior information of exactly the kind you would need to find the deficiency subspaces. The paper itself concedes the method is not universal and depends on how well you can establish boundary behavior, and Sec. 3.7 restricts to the case where the deficiency indices are already known and equal. For singular boundaries, the w_k "generally have an asymptotic form," but getting that form is the same analytic problem as solving the equations whose square-integrable solutions define D_+ and D_-. So the comparative advantage stated in the abstract does not hold. This is a framing problem rather than a mathematical error: Theorem 15 is fine, the examples are correct, but the novelty claim should be revised.

Who is this for? Anyone who wants a clearly organized map of how self-adjoint boundary conditions arise from extension theory, and a reference for explicit boundary-condition parametrizations of momentum and Hamiltonian on intervals. The paper deserves a serious referee; I would send it to review, with a request to rewrite the abstract and soften the "avoiding deficiency subspaces" language.

Referee report

Significance

If the main claim were fully established, the proposed alternative would be a useful practical shortcut for quantum mechanics with boundaries and singular potentials. The paper has genuine strengths: it is largely self-contained, it gives a coherent treatment of asymmetry forms, it converts the von Neumann description into explicit s.a. boundary conditions for several nontrivial examples, and it correctly reproduces known results such as the U(1) family for momentum on an interval and the one-parameter family of self-adjoint Hamiltonians on the semiaxis. The examples are concrete and checkable, and the exposition is pedagogically valuable. However, the load-bearing advertised advantage—that the new method avoids evaluation of deficiency subspaces and deficiency indices—is not supported by the paper's own hypotheses and limitations. The paper therefore needs substantial revision to bring the abstract and conclusions in line with what Theorem 15 actually delivers.

Major objections (4)

[Abstract and §3.7, Theorem 15] The advertised comparative advantage is not established. Theorem 15 requires the deficiency indices to be known and equal, m_+=m_-=m>0, and requires a set {w_k}_{k=1}^m in D_f+ that is linearly independent modulo Dbar_f and satisfies ω*(w_k,w_l)=0. The theorem provides no independent criterion for determining m; in differential-operator examples, m is obtained by solving the equations (f-check ∓ iκ)ψ=0 and counting square-integrable solutions, which is exactly the deficiency-subspace computation described in (100)–(101). The paper itself concedes in §3.7 that the construction applies only 'provided that the deficient indices are known and equal.' Thus the claim that the method avoids evaluating deficiency indices goes beyond what Theorem 15 supports.
[§3.4, §3.7, and Lemma 8] Verifying the hypotheses of Theorem 15 still requires the boundary-behavior analysis that the method claims to bypass. Lemma 8 characterizes the closure domain Dbar_f by the boundary conditions [ψ_*, ψ](a)=[ψ_*, ψ](b)=0 for all ψ_* in D_*, given in (108)–(109). Consequently, checking that the w_k are linearly independent modulo Dbar_f requires knowledge of asymptotic boundary values of the type expressed in (88)–(91). For singular endpoints, the proposed w_k 'generally have an asymptotic form,' but obtaining that form is precisely the analytic problem of determining the asymptotic behavior of solutions near the singular points. The method is therefore best described as a reformulation of the standard deficiency-subspace construction in terms of boundary conditions, not an independent shortcut.
[§3.7, semiaxis example after Theorem 14] The illustrative example for the free Hamiltonian on the semiaxis does not demonstrate avoidance of deficiency subspaces. The construction of the boundary matrix E_{1/2,θ}(0) and of the boundary condition ψ'_θ(0)=λψ_θ(0) uses the explicit functions e_± = (2κ)^{1/4} exp[(±i−1)√(κ/2)x], which are obtained by solving −ψ'' = ±iκψ and selecting the square-integrable solution. These are exactly the deficiency-subspace basis vectors. The example is correct and useful as an illustration of the boundary-condition formalism, but it undercuts the abstract's claim that the method avoids evaluating deficient subspaces.
[§3.1 and §2.8] The paper's own limitations statements are in tension with the abstract. Section 3.1 explicitly says the method 'is not universal at present' and that its applicability depends on 'to what extent we can establish the boundary behavior of functions involved.' Section 2.8, in the general step-by-step programme, also lists evaluation of deficiency subspaces as a required step. These caveats should be reflected in the abstract and in the concluding claims; as written, the abstract's unqualified statement of a 'comparative advantage' is not supported by the body of the paper.

Minor objections (4)

[Title] The title contains a typo: 'assosiated' should be 'associated.'
[Throughout] The phrase 'deﬁcient indices' appears in several places, e.g., in §3.7; the standard term is 'deficiency indices' and should be used consistently.
[Abstract] The abstract would be more accurate if it stated the method's limitation explicitly, e.g., 'when the deficiency indices are known and equal, the formulation can avoid explicit construction of deficiency-subspace bases.'
[§3.4, natural-domain discussion] The discussion of ψ_*(x) not necessarily vanishing at infinity is correct and important, but the wording around the statement 'we show this later' could be cross-referenced with the relevant equation number (the proof appears in the discussion preceding (95) and in §3.5).

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The paper

The paper claims

Self-adjoint extensions of differential operators can be specified directly by boundary conditions, skipping the usual deficiency-subspace calculation.

The reading

Formally self-adjoint differential expressions like momentum and the Hamiltonian define operators only after a domain is chosen, and on intervals with boundaries or singular potentials different domains produce genuinely different quantum observables. This paper argues that the family of self-adjoint operators associated with such an expression—a $U(m)$ family when the deficiency indices are equal and nonzero—can be specified directly by explicit self-adjoint boundary conditions. The central tool is the asymmetry form of the adjoint operator, which for differential expressions reduces to the boundary values of a local form in the functions and their derivatives. The paper's alternative method starts from $m$ vectors in the adjoint domain that are linearly independent modulo the closure domain and have mutually vanishing asymmetry form; such vectors define a self-adjoint extension without first computing deficiency subspaces and indices. This matters because boundary conditions are the natural physical language for fixing how a quantum system behaves at boundaries and singularities.

What carries the argument

The load-bearing object is the sesquilinear asymmetry form $\omega_*(\eta_*,\xi_*)=(\eta_*,\hat f^+\xi_*)-(\hat f^+\eta_*,\xi_*)$ on the domain of the adjoint, together with its quadratic version $\Delta_*(\xi_*)=\omega_*(\xi_*,\xi_*)$. For a symmetric operator the form vanishes exactly on the closure domain, and a self-adjoint extension is a maximal domain on which it vanishes. For ordinary differential expressions the integration-by-parts identity converts it into the boundary term $[\chi_*,\psi_*]|_a^b$ of a local form in the functions and their (quasi)derivatives, which is what turns abstract extension theory into boundary conditions. The key modification, stated as an addition to the main theorem, is that a self-adjoint extension is specified by any $m$ vectors $w_k$ in the adjoint domain that are linearly independent modulo the closure domain and satisfy $\omega_*(w_k,w_l)=0$; no explicit diagonalization of the adjoint into deficiency subspaces is needed.

Extended reading notes

Core claim

The paper establishes that every self-adjoint extension of a symmetric differential operator with equal nonzero deficiency indices $m_+=m_-=m$ can be written as a restriction of the adjoint to the domain $D_{f_U}=\{\psi_U\in D_{f^+}:\omega_*(w_k,\psi_U)=0,\ k=1,\ldots,m\}$, where the $w_k$ lie in the adjoint domain, are linearly independent modulo the closure domain, and satisfy $\omega_*(w_k,w_l)=0$. Conversely, any such $m$-tuple defines a self-adjoint extension. Because for an ordinary differential expression the sesquilinear asymmetry form $\omega_*(\chi_*,\psi_*)$ equals the boundary term $[\chi_*,\psi_*]|_a^b$ of the local form, these abstract conditions become explicit boundary conditions, interpreted asymptotically at singular ends. The paper demonstrates the route on momentum on a segment, which yields a one-parameter $U(1)$ family with boundary condition $\psi(l)=e^{i\vartheta}\psi(0)$, and on the free Hamiltonian on a half-line, which yields the $U(1)$ family $\psi'(0)=\lambda\psi(0)$ with $\lambda\in[-\infty,\infty]$.

Load-bearing premise

The shortcut assumes one can explicitly construct the $m$ boundary-behavior vectors $w_k$ in the adjoint domain, linearly independent modulo the closure domain and with mutually vanishing asymmetry form, without first solving the deficiency equations; the paper states this is not guaranteed for all singular boundaries.

Editorial extensions

If this is right

For a differential expression of order $n$ on a finite interval with two regular ends, the deficiency indices are $n$, so the self-adjoint operators form a $U(n)$ family specified by boundary conditions; the free Hamiltonian on a segment is a four-parameter $U(2)$ family containing Dirichlet, Neumann, periodic, and other boundary conditions.
For an even-order real-coefficient expression the deficiency indices are always equal, so an associated self-adjoint operator always exists, though generally not uniquely; for odd expressions such as $\mathrm i\,d/dx$ on a half-line the indices differ and no self-adjoint momentum operator exists.
The alternative boundary-condition method produces the extension family in the form physicists use for spectral problems, avoiding the intermediate construction of the closure domain and the solution of the deficiency eigenvalue equations $\hat f^+\psi_\pm=\pm\mathrm i\kappa\psi_\pm$.
Parameters absent from the formal expression enter through the extension choice, as the dimensional parameter $\kappa$ does in the half-line Hamiltonian example, showing how quantization of a system with boundaries can break a naive scale invariance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's worked examples, the criterion of the addition to the main theorem suggests that extension classification for strongly singular potentials, such as $V=-\alpha/r^\beta$ with $\beta>2$, could reduce to a finite linear algebra problem in the space of asymptotic boundary values, once those values are known.
The method offers a practical diagnostic for essential self-adjointness: evaluate the quadratic asymmetry form on a candidate domain; if a function with nonzero boundary value exists, the natural domain is not self-adjoint, as the paper's $V=-x^4$ example shows, and an extension problem is genuinely present.
A likely natural next step is to apply the same asymmetry-form boundary conditions to systems of differential equations, such as radial Dirac or spinor Hamiltonians, where the paper states matrix generalizations are direct but does not develop them in detail.

Original abstract

Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,229 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.89 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem. We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators. The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators. The main attention is given to a specification of s.a. extensions by s.a. boundary conditions. All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian. In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries. A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices. The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Considerable attention has been recently focused on quantum-mechanical systems with boundaries and/or singular potentials for which the construction of physical observables as self-adjoint (s.a.) operators is a nontrivial problem.	background 1.00	0.04	0.03	asserted 1.00
s1	We present a comparative review of various methods of specifying ordinary s.a. differential operators generated by formally s.a. differential expressions based on the general theory of s.a. extensions of symmetric operators.	contribution 0.98	0.81	0.10	asserted 0.99
s2	The exposition is untraditional and is based on the concept of asymmetry forms generated by adjoint operators.	method 0.90	0.65	0.05	asserted 1.00
s3	The main attention is given to a specification of s.a. extensions by s.a. boundary conditions.	method 0.64	0.47	0.16	asserted 0.99
s4	All the methods are illustrated by examples of quantum-mechanical observables like momentum and Hamiltonian.	method 0.75	0.76	0.05	asserted 1.00
s5	In addition to the conventional methods, we propose a possible alternative way of specifying s.a. differential operators by explicit s.a. boundary conditions that generally have an asymptotic form for singular boundaries.	contribution 0.97	0.91	0.35	hedged possibility 0.98
s6	A comparative advantage of the method is that it allows avoiding an evaluation of deficient subspaces and deficiency indices.	contribution 0.40	0.71	0.21	asserted 0.95
s7	The effectiveness of the method is illustrated by a number of examples of quantum-mechanical observables.	result 0.37	0.87	0.11	asserted 1.00

What kind of paper the classifier reports this is: review 0.71 · evidence from proof or derivation 0.56.

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References

Works cited (33)

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How the literature uses it

5 arXiv papers cite this in 13 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 12 sentences without a confident label

Cites it as background 1 paper

(see [ref] , Appendix to Section 10.1), which was widely discussed in the literature in the context of one-dimensional potential (the so-called Calogero potential) $g/x^2$ on the semi-axis or on the whole axis (e.g., see [ref] , [ref] , [ref] ).

New Exactly Solvable Two-Dimensional Quantum Model Not Amenable to Separation of Variables · Separation of Variables for $H(p=1)$. · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Berlin 1932)

View on Pith 1932

[2] [2]

L.D. Faddeev, O.A. Yakubovsky, Lectures on Qunatum Mechanic s (Leningrad State Uni- versity Press, Leningrad 1980)

View on Pith 1980

[3] [3]

F.A. Berezin and M.A. Shubin, Schr¨ odinger Equation(Kluwer Publ., New York, Amster- dam 1991)

View on Pith 1991

[4] [4]

F.A. Berezin, L.D. Faddeev, A remark on Schr¨ odinger’s equation with a singular potential, Sov. Math. Dokl. 2 (1961) 372-375

View on Pith 1961

[5] [5]

J. von Neumann, , Allgemaine Eigenwerttheorie Hermitischer Functional Ope ratoren, Matematische Annalen, 102 (1929) 49-131

View on Pith 1929

[6] [6]

M.H. Stone, Linear Transformations in Hilbert Space and their Applicat ions to Analysis , Amer. Math. Soc., Colloquium Publications Vol. 15 (Amer. Math. Soc., N ew York 1932)

View on Pith 1932

[7] [7]

N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space (Pitman, Boston 1981)

View on Pith 1981

[8] [8]

M.A. Naimark, Theory of Linear Diﬀerential Operators (Nauka, Moscow 1969)

View on Pith 1969

[9] [9]

K.M. Case, Singular potentials , Phys. Rev. 80 (1950) 797-806

View on Pith 1950

[10] [10]

M.F. Mott, H.S.W. Massey, Theory of Atomic Collisions (Oxford University Press, Oxford 1933)

View on Pith 1933

[11] [11]

I.E. Tamm, Mesons in a Coulomb ﬁeld , Phys. Rev. 58 (1940) 952

View on Pith 1940

[12] [12]

H.C. Corben, J. Schwinger, The electromagnetic properties of mesotrons , Phys. Rev. 58 (1940) 953-968

View on Pith 1940

[13] [13]

M. Reed and B. Simon, Methods of Modern Mathematical Physics , vol. II (Academic Press, New York 1972)

View on Pith 1972

[14] [14]

R.D. Richtmyer, Principles of Advanced Mathematical Physics , Vol. 1(Springer-Verlag, New York 1978)

View on Pith 1978

[15] [15]

D. Shin, On quasidiﬀerential operators in Hilbert space , Matematicheskii sbornik 13 (55) (1943) 39-70

View on Pith 1943

[16] [16]

W.N. Everitt, Fourth order diﬀerential equations , Math. Ann. 149 (1963) 320-340

View on Pith 1963

[17] [17]

E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Di ﬀerential Equations (Clarendon Press, Oxford 1958)

View on Pith 1958

[18] [18]

A.G. Kostuchenko, S.G. Krein, and V.I. Sobolev, Linear Operators in Hilbert Space , in Spravochnaya Matematicheskaya Biblioteka (Functional Analysis) Ed. S.G. Krein (Nauka, Moscow 1964) 90

View on Pith 1964

[19] [19]

B.P. Maslov and L.D. Faddeev, Operators in Quantum Mechanics , in Spravochnaya Matematicheskaya Biblioteka (Functional Analysis) Ed. S.G. Krein (N auka, Moscow 1964)

View on Pith 1964

[20] [20]

Yu.M. Berezansky, Eigenfunction Expansions Associated with Self-adjoint Op erators, (Naukova Dumka, Kiev 1965)

View on Pith 1965

[21] [21]

T. Kato, Perturbation Theory for Linear Operators , (Springer Verlag, Berlin, Heidelberg, New York 1966)

View on Pith 1966

[22] [22]

H. Weyl, ¨Uber gew¨ onliche lineare Diﬀerentialgleichungen mit sing ul¨ aren Stellen und ihre Eigenfunctionen, G¨ otinger Nachrichten (1909) 37-64

View on Pith 1909

[23] [23]

H. Weyl, ¨Uber gew¨ onliche Diﬀerentialgleichungen mit Singularit¨aten und zugeh¨ origen Entwiklungen Willk¨ urlicher Funktionen, Math. Annalen 68 (1910) 220-269

View on Pith 1910

[24] [24]

H. Weyl, ¨Uber gew¨ onliche Diﬀerentialgleichungen mit singul¨ arenStellen und ihre Eigen- functionen, G¨ otinger Nachrichten, (1910) 442-467

View on Pith 1910

[25] [25]

E.C. Titchmarsh, Eigenfunction Expansions Assosiated with Second-order Di ﬀerential Equations, (Clarendon Press, Oxford 1946 )

View on Pith 1946

[26] [26]

B.M. Levitan, Eigenfunction Expansions Assosiated with Second-order Di ﬀerential Equa- tions, (Gostechizdat, Moscow 1950)

View on Pith 1950

[27] [27]

J. Weidmann, Spectral Theory of Ordinary Diﬀerential Operators (Springer-Verlag, Berlin, New York 1987)

View on Pith 1987

[28] [28]

S. Albeverio, F. Gesztesy, R. H / oegh-Krohn, H. Holden, Solvable models in quantum me- chanics (Springer-Verlag, Berlin 1988)

View on Pith 1988

[29] [29]

G.E. Shilov, Mathematical Analysis. Second Special Course , (Nauka, Moscow 1965)

View on Pith 1965

[30] [30]

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