Second-Order Differential Operators in the Limit Circle Case

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.


Setting the problem
It is a common wisdom that spectral properties of second-order differential operators and Jacobi operators are in many respects similar. Recently this analogy was used in articles [7,8] to recover known and obtain some new results for Jacobi operators with coefficients stabilizing at infinity. In this paper we move in the opposite direction and study differential operators in the limit circle case relying on an analogy with similar problems for Jacobi operators. Actually, we follow rather closely an approach developed for Jacobi operators in [9].
We consider second-order differential operators A defined by the formula and acting in the space L 2 (R + ). The scalar product in this space is denoted ·, · ; I is the identity operator. We always suppose that functions p(x) and q(x) are real. Then the operator A defined on the set C ∞ 0 (R + ) is symmetric, but to make it self-adjoint, one has to add boundary conditions at x = 0 and, eventually, for x → ∞. We suppose that conditions at these two points are separated. The boundary condition at the point x = 0 looks as where α =ᾱ. (1.2) This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html The value α = ∞ is not excluded. In this case (1.2) should be understood as the equality u(0) = 0. We always require condition (1.2), fix α and do not keep track of α in notation. Our objective is to study a singular case, where all solutions u of the equation Au = zu for z ∈ C are in L 2 (R + ). This instance is known as the limit circle (LC) case. In this case the operator A with boundary condition (1.2) has a one parameter family of self-adjoint realizations distinguished by some conditions for x → ∞. Their description can be performed in various terms. We here adopt an approach similar to the one used for Jacobi operators as presented in the book [5,Section 16.3] or in the survey [6, Section 2].

Structure of the paper
In Sections 2.1 and 2.2, we collect standard information about differential equations of secondorder and realizations of differential operators A in the space L 2 (R + ). We first define symmetric operators A min with minimal domains D(A min ). Their self-adjoint extensions A satisfy the condition In the LC case the operators A max are not symmetric. In Section 2.3, we recall the traditional procedure of constructing self-adjoint extensions of the operator A min in terms of some boundary conditions for x → ∞. Then we suggest in Section 2.4 an alternative approach to this problem, where self-adjoint extensions A t , t ∈ R ∪ {∞}, of A min are defined in a way analogous to the case of the Jacobi operators. Our main result, an explicit formula for the resolvents R t (z) = (A t − zI) −1 , is obtained in Section 3.2, Theorem 3.7. Previously, we construct in Section 3.1 (see Theorem 3.1) an operator R(z) playing, in some sense, the role of the resolvent of the maximal operator A max . The operator R(z), we call it the quasiresolvent, is the key element of our construction. Note that the operator valued function R(z) depends analytically on z ∈ C. Then, using the operator R(z), we prove Theorem 3.7. This also yields a representation (see Section 3.3) for spectral families E t (λ) of A t which is a modification of the Nevalinna formula in the theory of Jacobi operators; see the original paper [3] or [5,6].

Differential equations and associated operators
We refer to the books [2, Section 17] and [4, Section X.1] for necessary background information on the theory of symmetric differential operators. A lot of relevant results can also be found in the encyclopedic book [10]; see, in particular, Chapter 10.

Limit point versus limit circle
Let us consider a second-order differential equation associated with operator (1.1). To avoid inessential technical complications, we always suppose that p ∈ C 1 (R + ), q ∈ C(R + ) and the functions p(x), q(x) have finite limits as x → 0. More general conditions on the regularity of p(x) and q(x) are stated, for example, in [2, Section 15]. We assume that p(x) > 0 for x ≥ 0. The solutions of equation (2.1) exist, belong to C 2 (R + ) and they have limits u(+0) =: u(0), u (+0) =:

Minimal and maximal operators
We first define a minimal operator A 00 by the equality A 00 u = Au on domain D(A 00 ) that consists of functions u ∈ C 2 (R + ) such that u(x) = 0 for sufficiently large x, limits u(+0) =: u(0), u (+0) =: u (0) exist and condition (1.2) is satisfied. Thus, the boundary condition (1.2) at x = 0 is included in the definition of the operator A 00 so that its self-adjoint extensions are determined by conditions for x → ∞. The closure of A 00 will be denoted A min . This operator is symmetric in the space L 2 (R + ), but without additional assumptions on the coefficients p(x) and q(x) its domain D(A min ) does not admit an efficient description. The adjoint operator A * min =: A max is again given by the formula A max u = Au on a set D(A max ) that consists of functions u(x) belonging locally to the Sobolev space H 2 , satisfying boundary condition (1.2) and such that u ∈ L 2 (R + ), Au ∈ L 2 (R + ). In the LC case, the operator A max is not symmetric. Integrating by parts, we see that for all where the limit in the right-hand side exists but is not necessarily zero.
Recall that The operator A min is self-adjoint if and only if the LP case occurs. In this paper we are interested in the LC case when Since the operator A min commutes with the complex conjugation, its deficiency indices are equal, i.e., d + = d − =: d, and, so, A min admits self-adjoint extensions. For an arbitrary z ∈ C, all solutions of equation (2.1) with boundary condition (1.2) are given by the formula u(x) = cϕ z (x) for some c ∈ C. They belong to D(A max ) if and only if ϕ z ∈ L 2 (R + ). Therefore d = 0 if ϕ z ∈ L 2 (R + ) for Im z = 0; otherwise d = 1.

Boundary conditions at infinity
In this paper we are interested in the case, where equation (2.1) is in the limit circle (LC) case at infinity. This means that all solutions of this equation for some (and then for all) z ∈ C are in L 2 (R + ) or, equivalently, that relation (2.5) is satisfied. First, we briefly recall the traditional description of self-adjoint extensions of the minimal operator J min in terms of boundary conditions at infinity. We refer to the classical books [1, Chapter IX, Section 4] and [2, Section 18] for detailed presentations. We mention also the relatively recent book [5], where a concise exposition of the case of second-order differential operators is given. The results stated below can be found, for example, in [5,Proposition 15.14].
Let v j (x), j = 1, 2, be some real valued functions of x ∈ R + such that z (x) satisfying boundary condition (2.7). Then, for all h ∈ L 2 (R + ) and Im z = 0, one has is the Wronskian of the solutions ϕ z and f (s) z of equation (2.1). Note that the results stated above are obtained by approximating the problem on the halfaxis R + by regular problems on intervals (0, ) and studying the limit → ∞.

Self-adjoint extensions
The description of self-adjoint extensions of the operator A min given in the previous subsection seems to be not very efficient. In particular, it depends on a choice of the functions v j (x), j = 1, 2, satisfying condition (2.6). We suggest an alternative approach motivated by an analogy with Jacobi operators in Theorem 2.4 (cf. [5, Lemma 6.22 and Theorem 6.23] or [6, Theorem 2.6]). In the long run, it relies on von Neumann formulas but is adapted to operators (1.1) with real coefficients p(x) and q(x).
Our descriptions of various domains are given in terms of the solutions ϕ z (x) and θ z (x) of differential equation (2.1). Note that the function ϕ z (x) satisfies boundary condition (1.2) so that ϕ z ∈ D(A max ), but this is not the case for θ z (x). To get rid of this nuisance, we introduce a functionθ z (x) = ω(x)θ z (x), where the cut-off ω ∈ C ∞ (R + ), ω(x) = 0 for small x and ω(x) = 1 for large x; thenθ z ∈ D(A max ). A direct calculation shows that has a compact support. Now we are in a position to describe D(A max ). For a vector h ∈ L 2 (R + ), we denote by {h} the one dimensional subspace of L 2 (R + ) spanned by the vector h. The symbol denotes the direct sum of subspaces.
Since the difference of functionsθ 0 corresponding to two different cut-offs ω(x) is in D(A min ), the direct sum in (2.10) does not depend on a particular choice ofθ 0 .
Proof . We start a proof of Theorem 2.1 with a direct calculation.
Proof . Let us calculate Using (2.8), we see that Similarly, we find that Comparing the last two equalities and taking into account that the functions ψ 0 , ϕ 0 andθ 0 are real, we see that Using definition (2.9) and integrating by parts, it is easy to calculate Since {ϕ 0 , θ 0 } = 1, this integral equals 1. Therefore identity (2.13) can be rewritten as (2.12).
Obviously, the right-hand side of (2.10) is contained in its left-hand side. Actually, there is the equality here because the operator A min has deficiency indices (1, 1) so that the dimension of the factor space D(A max )/D(A min ) equals 2. This concludes the proof of Theorem 2.1.
All self-adjoint extensions A t of the operator A min are parametrized by numbers t ∈ R and t = ∞. Let sets D(A t ) ⊂ D(A max ) be distinguished by conditions Proof . We proceed from Lemma 2.3. Let u, v ∈ D(A max ) so that equalities (2.11) are satisfed. If u, v ∈ D(A t ), then according to (2.14) or (2.15) we have α 1 = tα 2 , β 1 = tβ 2 if t ∈ R and α 2 = β 2 = 0 if t = ∞. Therefore it follows from relation (2.12) that A t u, v = u, A t v , and hence the operators A t are symmetric.

Resolvents of self-adjoint extensions
Our goal in this section is to construct resolvents of the operators A t . We start however with a construction of a similar object for the operator A max .

Quasiresolvent of the maximal operator
Recall that in the LC case inclusions (2.4) are satisfied. Let us define, for all z ∈ C, a bounded operator R(z) in the space L 2 (R + ) by the equality We prove (see Theorem 3.1) that, in a natural sense, R(z) can be considered as a quasiresolvent of the operator A max . It plays the role of the resolvent of the operator A max . Let us enumerate some simple properties of the operator R(z). Obviously, the operator R(z) belongs to the Hilbert-Schmidt class. It depends analytically on z ∈ C and R(z) * = R(z). Differentiating definition (3.1), we see that for all h ∈ L 2 (R + ). In particular, it follows from relations (3.1) and (3.2) that where ϕ z (0) and ϕ z (0) are defined by equalities (2.2) or (2.3). A proof of the following statement is close to the construction of the resolvent for essentially self-adjoint Schrödinger operators.
Since the Wronskian {ϕ z , θ z } = 1, the last term in the right-hand side equals −h(x). Putting now equalities (3.1) and (3.7) together and using equation (2.1) for the functions ϕ z (x) and θ z (x), we obtain the equation Taking also into account boundary condition (1.2), we see that A max u − zu = h. Since h ∈ L 2 (R + ), this yields both (3.5) and (3.6).
Note that solutions u(x) of differential equation (2.1) satisfying condition (1.2) are given by the formula u(x) = Γϕ z (x) for some Γ ∈ C. Therefore we can state Corollary 3.3. All solutions of the equation where z ∈ C and h ∈ L 2 (R + ), for u ∈ D(A max ) are given by the formula A relation below is a direct consequence of definition (3.1) and condition (2.4): This asymptotic formula can be supplemented by the following result. in L 2 (R + ) as k → ∞. Let us take a sequence of functions h (k) with compact supports in R + such that h (k) → h and set Then, as was already shown, u (k) ∈ D(A 00 ) and u (k) → u as k → ∞ because the operator R(z) is bounded. It follows from formula (3.6) that The right-hand side equals (A − z)u by formula (3.6) and definition (3.9). This proves relations (3.10) whence u ∈ D(A min ).

Resolvent representation
First, we find a link between the solutions ϕ z , θ z of equation (2.1) for an arbitrary z ∈ C and for z = 0.
Putting together Lemma 3.5 with Proposition 3.4 (for z = 0), we can also state the following result.
Lemma 3.6. For all z ∈ C, we have Now we are in a position to construct the resolvents of the self-adjoint operators A t . Theorem 3.7. Let inclusions (2.4) hold true. For all z ∈ C with Im z = 0 and all h ∈ L 2 (R + ), the resolvent R t (z) = (A t − zI) −1 of the operator A t is given by an equality for all h ∈ L 2 (R + ). It follows that f (t) z = γ t (z)ϕz for some γ t (z) ∈ C. This yields representation (3.15), where the constant γ t (z) is determined by the condition (3.18) Let us show that this inclusion leads to expressions (3.16) or (3.17) for γ t (z). By definitions (2.14) or (2.15) of the set D(A t ), inclusion (3.18) means that for some number X = X t (z) ∈ C. On the other hand, it follows from relations (3.9) and (3.15) that Comparing (3.19) and (3.20), we see that (3.18) is equivalent to inclusions Note that h, ϕz = 0 because the sum in (2.10) is direct and set Y = h, ϕz −1 X. It follows from Lemma 3.6 that inclusion (3.21) is equivalent to an equality Comparing here the coefficients at ϕ 0 andθ 0 , we obtain equations Solving this equation with respect to γ t (z), we arrive at formula (3.16). Similarly, using again Lemma 3.6, we see that inclusion (3.22) is equivalent to an equality Inclusion (3.22) holds true if and only if the coefficient atθ 0 equals zero. This yields formula (3.17).
Corollary 3.8. If z ∈ C is a regular point of the operator A t , then its resolvent R t (z) is in the Hilbert-Schmidt class. In particular, the spectra of all operators A t are discrete.
The result of this corollary is well known. It follows, for example, from Theorem 1 in Section 19.1 of the book [2].
We emphasize that, for different t, the resolvents R t (z) of the operators A t differ from each other only by the coefficient γ t (z) at the rank one operator ·, ϕz ϕ z . This is consistent with the fact that the operator A min has deficiency indices (1,1). Observe also that γ t (z) = γ t (z).

Spectral measure
In view of the spectral theorem, Theorem 3.7 yields a representation for the Cauchy-Stieltjes transform of the spectral measure dE t (λ) of the operator A t . Theorem 3.9. Let inclusions (2.4) hold true. Then for all z ∈ C with Im z = 0 and all h ∈ L 2 (R + ), we have an equality (3.23) Recall that the operators R(z) are defined by formula (3.1). Therefore (R(z)h, h) are entire functions of z ∈ C, and the singularities of the integral in (3.23) are determined by the function γ t (z). Thus, (3.23) can be considered as a modification of the classical Nevanlinna formula (see his original paper [3] or, for example, formula (7.6) in the book [5]) for the Cauchy-Stieltjes transform of the spectral measure in the theory of Jacobi operators. We mention however that, for Jacobi operators acting in the space 2 (Z + ), there is the canonical choice of a generating vector and of a spectral measure. This is not the case for differential operators in L 2 (R + ).
Let us discuss spectral consequences of Theorem 3.7. Since the functions ϕ z , ϕ 0 and ϕ z , θ 0 are entire, it again follows from (3.16) and (3.17) that the spectra of the operators A t are discrete. Theorem 3.7 yields also an equation for their eigenvalues.
This assertion is a modification of a R. Nevanlinna's result obtained by him for Jacobi operators.
We finally note an obvious fact: if λ is an eigenvalue of an operator A t , then the corresponding eigenfunction equals cϕ λ (x), where c ∈ C. In particular, this implies that all eigenvalues of the operators A t are simple.
B. Starting from Theorem 2.1, we can everywhere replace the functions ϕ 0 and θ 0 by ϕ ζ and θ ζ , where ζ is an arbitrary real fixed number. Then the construction of the paper remains unchanged if the factor z in formulas (3.16) and (3.17) for γ t (z) is replaced by z −ζ. The simplest way to see this is to apply the results obtained above to the operator A − ζI instead of A.
C. Finally, we compare resolvent formulas in the LP and LC cases. In the LP case the resolvent R(z) of a self-adjoint operator A = A min is given by the relation where f z (x) is a unique (up to a constant factor) solution of equation (2.1) belonging to L 2 (R + ). It can be chosen in a form f z = θ z + w(z)ϕ z , where w(z) is known as the Weyl function. Substituting this expression into (3.28), we see that formally R(z) = w(z) ·, ϕz ϕ z + R(z), (3.29) where R(z) is given by equality (3.1). This relation looks algebraically similar to (3.15), where γ t (z) plays the role of the Weyl function w(z). Note, however, that in the LP case w(z) is determined uniquely by the condition θ z + w(z)ϕ z ∈ L 2 (R + ), while in the LC case γ t (z) depends on the choice of a self-adjoint extension of the operator A min . We also emphasize that relation (3.29) is only formal because ϕ z and θ z are not in L 2 (R + ) in the LP case.