The Krein-von Neumann extension revisited
Guglielmo Fucci, Fritz Gesztesy, Klaus Kirsten, Lance L. Littlejohn, Roger Nichols, Jonathan Stanfill
aa r X i v : . [ m a t h . F A ] F e b THE KREIN–VON NEUMANN EXTENSION REVISITED
GUGLIELMO FUCCI, FRITZ GESZTESY, KLAUS KIRSTEN, LANCE L. LITTLEJOHN,ROGER NICHOLS, AND JONATHAN STANFILL
Abstract.
We revisit the Krein–von Neumann extension in the case wherethe underlying symmetric operator is strictly positive and apply this to derivethe explicit form of the Krein–von Neumann extension for singular, general(i.e., three-coefficient) Sturm–Liouville operators on arbitrary intervals. Inparticular, the boundary conditions for the Krein–von Neumann extension ofthe strictly positive minimal Sturm–Liouville operator are explicitly expressedin terms of generalized boundary values adapted to the (possible) singularitystructure of the coefficients near an interval endpoint.
Contents
1. Introduction 12. The Basics of Weyl–Titchmarsh–Kodaira Theory 33. The Krein–von Neumann extension of T min > Introduction
While the principal objective of this paper is to derive the explicit form of theKrein–von Neumann extension for singular (three-coefficient) Sturm–Liouville op-erators on arbitrary intervals with strictly positive underlying minimal operator,we briefly pause and first describe the abstract Krein–von Neumann extension of anonnegative symmetric operator in complex, separable Hilbert space in a nutshell.A linear operator S : dom( S ) ⊆ H → H in some complex, separable Hilbertspace H is called nonnegative if( u, Su ) H > , u ∈ dom( S ) (1.1)(in this case S is symmetric). In addition, S is called strictly positive , if for some ε >
0, ( u, Su ) H > ε k u k H , u ∈ dom( S ). Next, we recall the order relation 0 A B for two nonnegative self-adjoint operators in H in the form (see, e.g., [27, SectionI.6], [44, Theorem VI.2.21])0 A B if and only if ( B + aI H ) − ( A + aI H ) − for all a >
0. (1.2)
Date : February 2, 2021.2020
Mathematics Subject Classification.
Primary: 34B09, 34B24, 34C10, 34L40; Secondary:34B20, 34B30.
Key words and phrases.
Krein–von Neumann extension, Singular Sturm–Liouville operators,Bessel and Jacobi-type differential operators.
In the following 0 S is a linear, unbounded, densely defined, nonnegativeoperator in H , and we assume that S has nonzero deficiency indices. In particular,def( S ) = dim(ker( S ∗ − zI H )) ∈ N ∪ {∞} , z ∈ C \ [0 , ∞ ) , (1.3)is well-known to be independent of z (and if S > εI H for some ε > S ) of z extends to z ∈ C \ [ ε, ∞ )). Moreover, since S andits closure S have the same self-adjoint extensions in H , we will without loss ofgenerality assume that S is closed in H .The following is a fundamental result that cements the extraordinary role playedby the Friedrichs and Krein–von Neumann extensions of S , to be found in M. Krein’scelebrated 1947 paper [46] : Theorem 1.1.
Assume that S is a densely defined, closed, nonnegative operatorin H . Then, among all nonnegative self-adjoint extensions of S , there exist twodistinguished ones, S K and S F , which are, respectively, the smallest and largest ( in the sense of order between nonnegative self-adjoint operators ) such extensions.Furthermore, a nonnegative self-adjoint operator e S is a self-adjoint extension of S if and only if e S satisfies S K e S S F . (1.4) In particular, (1.4) determines S K and S F uniquely.In addition, if S > εI H for some ε > , one has S F > εI H , and dom( S F ) = dom( S ) . + ( S F ) − ker( S ∗ ) , (1.5)dom( S K ) = dom( S ) . + ker( S ∗ ) , (1.6)dom( S ∗ ) = dom( S ) . + ( S F ) − ker( S ∗ ) . + ker( S ∗ )= dom( S F ) . + ker( S ∗ ) , (1.7) in particular, ker( S K ) = ker (cid:0) ( S K ) / (cid:1) = ker( S ∗ ) = ran( S ) ⊥ . (1.8)Here the operator inequalities in (1.4) are understood in the resolvent sense,( S F + aI H ) − (cid:0) e S + aI H (cid:1) − ( S K + aI H ) − for some (and hence for all ) a > S K and S F are distinguished self-adjoint extensions of S , representing, inparticular, extremal points of all nonnegative self-adjoint extensions e S > S .We will call the operator S K the Krein–von Neumann extension of S . See [46]and also the discussion in [3], [8]. It should be noted that the Krein–von Neumannextension was first considered by von Neumann [69] in 1929 in the case where S is strictly positive , that is, if S > εI H for some ε >
0. However, von Neumanndid not isolate the extremal property of this extension as described in (1.4) and(1.9). M. Krein [46], [47] was the first to systematically treat the general case S > S , illustrating thespecial role of the Friedrichs extension (i.e., the “hard” extension) S F of S and theKrein–von Neumann (i.e., the “soft”) extension S K of S as extremal cases when See also Theorems 2 and 5–7 in the English summary on page 492. His construction appears in the proof of Theorem 42 on pages 102–103.
HE KREIN–VON NEUMANN EXTENSION REVISITED 3 considering all nonnegative extensions of S . For more results on the Krein–vonNeumann extension of a strictly positive symmetric operator S > εI H we refer tothe beginning of Section 3.However, the principal aim of this paper are (three-coefficient) generally singularSturm–Liouville differential expressions of the type τ = 1 r ( x ) (cid:20) − ddx p ( x ) ddx + q ( x ) (cid:21) for a.e. x ∈ ( a, b ) ⊆ R , (1.10)on a general interval ( a, b ) ⊆ R and their various L (( a, b ); rdx )-realizations, withthe coefficients p, q, r satisfying Hypothesis 2.1. In particular, the minimal operator T min associated with τ (cf. (2.5)), assumed in addition to be strictly positive,plays the role of S above, and the corresponding maximal operator T max (cf. (2.2))represents S ∗ . The explicit forms of the Friedrichs and Krein extensions of T min arethen of the form (3.23) and (3.24), (3.25), (3.27), (3.28), respectively. Moreover, thecorresponding boundary conditions are explicitly expressed in terms of generalizedboundary values adapted to the (possible) singularity structure of the coefficientsnear an interval endpoint with the help of principal and nonprincipal solutions ofthe underlying Sturm–Liouville equation.Briefly turning to a sketch of the content of each section, we note that Section2 focuses on the basics of Sturm–Liouville operators in L (( a, b ); rdx ) and the un-derlying Weyl–Titchmarsh–Kodaira theory, including self-adjoint extensions andgeneralized boundary values (and conditions) in the singular case. Section 3 thencontains the bulk of the new material in this paper. After continuing a discussion ofthe abstract Krein–von Neumann extension of a symmetric, strictly positive opera-tor S > εI H , an elementary characterization of the Krein–von Neumann extension S K as the unique self-adjoint extension of S containing ker( S ∗ ) in its domain is de-rived in Lemma 3.2. This result is then applied to derive an explicit description ofthe Krein–von Neumann extension of a strictly positive minimal Sturm–Liouvilleoperator T min in terms of generalized boundary values. We conclude this paperwith three nontrivial and representative examples in Section 4, including a gener-alized Bessel operator, a singular operator relevant in the context of acoustic blackholes, and the Jacobi operator.Finally, a few remarks on the notation employed: Given a separable complexHilbert space H , ( · , · ) H denotes the scalar product in H (linear in the secondfactor), and I H represents the identity operator in H . The domain and range ofa linear operator T in H are abbreviated by dom( T ) and ran( T ). The closure ofa closable operator S is denoted by S . The kernel (null space) of T is denoted byker( T ). The spectrum, point spectrum (i.e., the set of eigenvalues), and resolventset of a closed linear operator in H will be abbreviated by σ ( · ), σ p ( · ), and ρ ( · ),respectively. If U and U are subspaces of a Banach space X , their direct sum isdenoted by U . + U . We also employ the shortcut N = N ∪ { } . If the underlying L -space is understood, we denote the corresponding identity operator simply by I . 2. The Basics of Weyl–Titchmarsh–Kodaira Theory
In this section, following [30] and [33, Ch. 13], we summarize the singular Weyl–Titchmarsh–Kodaira theory as needed to treat the Krein–von Neumann extensionfor singular, general Sturm–Liouville operators in the remainder of this paper.
G. FUCCI, F. GESZTESY, K. KIRSTEN, L. LITTLEJOHN, R. NICHOLS, AND J. STANFILL
Throughout this section we make the following assumptions:
Hypothesis 2.1.
Let ( a, b ) ⊆ R and suppose that p, q, r are ( Lebesgue ) measurablefunctions on ( a, b ) such that the following items ( i ) – ( iii ) hold: ( i ) r > a.e. on ( a, b ) , r ∈ L loc (( a, b ); dx ) . ( ii ) p > a.e. on ( a, b ) , /p ∈ L loc (( a, b ); dx ) . ( iii ) q is real-valued a.e. on ( a, b ) , q ∈ L loc (( a, b ); dx ) . Given Hypothesis 2.1, we study Sturm–Liouville operators associated with thegeneral, three-coefficient differential expression τ = 1 r ( x ) (cid:20) − ddx p ( x ) ddx + q ( x ) (cid:21) for a.e. x ∈ ( a, b ) ⊆ R , (2.1)and introduce maximal and minimal operators in L (( a, b ); rdx ) associated with τ in the usual manner as follows. Definition 2.2.
Assume Hypothesis . Given τ as in (2.1) , the maximal operator T max in L (( a, b ); rdx ) associated with τ is defined by T max f = τ f,f ∈ dom( T max ) = (cid:8) g ∈ L (( a, b ); rdx ) (cid:12)(cid:12) g, g [1] ∈ AC loc (( a, b )); (2.2) τ g ∈ L (( a, b ); rdx ) (cid:9) . The preminimal operator T min, in L (( a, b ); rdx ) associated with τ is defined by T min, f = τ f,f ∈ dom( T min, ) = (cid:8) g ∈ L (( a, b ); rdx ) (cid:12)(cid:12) g, g [1] ∈ AC loc (( a, b )); (2.3)supp ( g ) ⊂ ( a, b ) is compact; τ g ∈ L (( a, b ); rdx ) (cid:9) . One can prove that T min, is closable, and one then defines the minimal operator T min as the closure of T min, . The following facts then are well known:( T min, ) ∗ = T max , (2.4)and hence T max is closed and T min = T min, is given by T min f = τ f,f ∈ dom( T min ) = (cid:8) g ∈ L (( a, b ); rdx ) (cid:12)(cid:12) g, g [1] ∈ AC loc (( a, b )); (2.5)for all h ∈ dom( T max ) , W ( h, g )( a ) = 0 = W ( h, g )( b ); τ g ∈ L (( a, b ); rdx ) (cid:9) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) W ( h, g )( a ) = 0 = W ( h, g )( b ) for all h ∈ dom( T max ) (cid:9) . Moreover, T min, is essentially self-adjoint if and only if T max is symmetric, andthen T min, = T min = T max .Here the Wronskian of f and g , for f, g ∈ AC loc (( a, b )), is defined by W ( f, g )( x ) = f ( x ) g [1] ( x ) − f [1] ( x ) g ( x ) , x ∈ ( a, b ) , (2.6)with y [1] ( x ) = p ( x ) y ′ ( x ) , x ∈ ( a, b ) , (2.7)denoting the first quasi-derivative of a function y ∈ AC loc (( a, b )).The celebrated Weyl alternative then can be stated as follows: HE KREIN–VON NEUMANN EXTENSION REVISITED 5
Theorem 2.3 (Weyl’s Alternative) . Assume Hypothesis . Then the following alternative holds: Either, ( i ) for every z ∈ C , all solutions u of ( τ − z ) u = 0 are in L (( a, b ); rdx ) near b ( resp., near a ) ,or, ( ii ) for every z ∈ C , there exists at least one solution u of ( τ − z ) u = 0 which isnot in L (( a, b ); rdx ) near b ( resp., near a ) . In this case, for each z ∈ C \ R , thereexists precisely one solution u b ( resp., u a ) of ( τ − z ) u = 0 ( up to constant multiples ) which lies in L (( a, b ); rdx ) near b ( resp., near a ) . This yields the limit circle/limit point classification of τ at an interval endpointand links self-adjointness of T min (resp., T max ) and the limit point property of τ at both endpoints as follows. Definition 2.4.
Assume Hypothesis .In case ( i ) in Theorem , τ is said to be in the limit circle case at b ( resp., at a ) . ( Frequently, τ is then called quasi-regular at b ( resp., a ) . ) In case ( ii ) in Theorem , τ is said to be in the limit point case at b ( resp., at a ) .If τ is in the limit circle case at a and b then τ is also called quasi-regular on ( a, b ) . Theorem 2.5.
Assume Hypothesis , then the following items ( i ) and ( ii ) hold: ( i ) If τ is in the limit point case at a ( resp., b ) , then W ( f, g )( a ) = 0 ( resp., W ( f, g )( b ) = 0) for all f, g ∈ dom( T max ) . (2.8)( ii ) Let T min = T min, . Then n ± ( T min ) = dim(ker( T max ∓ iI ))= if τ is in the limit circle case at a and b , if τ is in the limit circle case at a and in the limit point case at b , or vice versa, if τ is in the limit point case at a and b. (2.9) In particular, T min = T max is self-adjoint ( i.e., T min, is essentially self-adjoint ) if and only if τ is in the limit point case at a and b . Next, we turn to a description of all self-adjoint extensions of T min . Theorem 2.6.
Assume Hypothesis and that τ is in the limit circle case at a and b ( i.e., τ is quasi-regular on ( a, b )) . In addition, assume that v j ∈ dom( T max ) , j = 1 , , satisfy W ( v , v )( a ) = W ( v , v )( b ) = 1 , W ( v j , v j )( a ) = W ( v j , v j )( b ) = 0 , j = 1 , . (2.10)( E.g., real-valued solutions v j , j = 1 , , of ( τ − λ ) u = 0 with λ ∈ R , such that W ( v , v ) = 1 . ) For g ∈ dom( T max ) we introduce the generalized boundary values e g ( a ) = − W ( v , g )( a ) , e g ( b ) = − W ( v , g )( b ) , e g ( a ) = W ( v , g )( a ) , e g ( b ) = W ( v , g )( b ) . (2.11) Then the following items ( i ) – ( iii ) hold: G. FUCCI, F. GESZTESY, K. KIRSTEN, L. LITTLEJOHN, R. NICHOLS, AND J. STANFILL ( i ) All self-adjoint extensions T α,β of T min with separated boundary conditions areof the form T γ,δ f = τ f, γ, δ ∈ [0 , π ) ,f ∈ dom( T γ,δ ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) sin( γ ) e g ( a ) + cos( γ ) e g ( a ) = 0; (2.12)sin( δ ) e g ( b ) + cos( δ ) e g ( b ) = 0 (cid:9) . ( ii ) All self-adjoint extensions T ϕ,R of T min with coupled boundary conditions areof the type T ϕ,R f = τ f,f ∈ dom( T ϕ,R ) = (cid:26) g ∈ dom( T max ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18)e g ( b ) e g ( b ) (cid:19) = e iϕ R (cid:18)e g ( a ) e g ( a ) (cid:19) (cid:27) , (2.13) where ϕ ∈ [0 , π ) , and R is a real × matrix with det( R ) = 1 ( i.e., R ∈ SL (2 , R )) . ( iii ) Every self-adjoint extension of T min is either of type ( i ) ( i.e., separated ) or oftype ( ii ) ( i.e., coupled ) .Remark . ( i ) If τ is in the limit point case at one endpoint, say, at the endpoint b , one omits the corresponding boundary condition involving δ ∈ [0 , π ) at b in(2.12) to obtain all self-adjoint extensions T γ of T min , indexed by γ ∈ [0 , π ). In thecase where τ is in the limit point case at both endpoints, all boundary values andboundary conditions become superfluous as in this case T min = T max is self-adjoint.( ii ) Assume the special case where τ is regular on the finite interval [ a, b ], that is,suppose that Hypothesis 2.1 is replaced by the more stringent set of assumptions: Hypothesis ( τ regular on [ a, b ].)Let ( a, b ) ⊂ R be a finite interval and suppose that p, q, r are (Lebesgue ) measurablefunctions on ( a, b ) such that the following items ( i ′ )–( iii ′ ) hold:( i ′ ) r > a, b ), r ∈ L (( a, b ); dx ).( ii ′ ) p > a, b ), 1 /p ∈ L (( a, b ); dx ).( iii ′ ) q is real-valued a.e. on ( a, b ), q ∈ L (( a, b ); dx ).In this case one chooses v j ∈ dom( T max ), j = 1 ,
2, such that v ( x ) = ( θ ( λ, x, a ) , for x near a,θ ( λ, x, b ) , for x near b, v ( x ) = ( φ ( λ, x, a ) , for x near a,φ ( λ, x, b ) , for x near b, (2.14)where φ ( λ, · , d ), θ ( λ, · , d ), d ∈ { a, b } , are real-valued solutions of ( τ − λ ) u = 0, λ ∈ R , satisfying the boundary conditions φ ( λ, a, a ) = θ [1]0 ( λ, a, a ) = 0 , θ ( λ, a, a ) = φ [1]0 ( λ, a, a ) = 1 ,φ ( λ, b, b ) = θ [1]0 ( λ, b, b ) = 0 , θ ( λ, b, b ) = φ [1]0 ( λ, b, b ) = 1 . (2.15)Then one verifies that e g ( a ) = g ( a ) , e g ( b ) = g ( b ) , e g ( a ) = g [1] ( a ) , e g ( b ) = g [1] ( b ) , (2.16)and hence Theorem 2.6 in the special regular case recovers the well-known situationof separated self-adjoint boundary conditions for three-coefficient regular Sturm–Liouville operators in L (( a, b ); rdx ). HE KREIN–VON NEUMANN EXTENSION REVISITED 7 ( iii ) In connection with (2.11), an explicit calculation demonstrates that for g, h ∈ dom( T max ), e g ( d ) e h ( d ) − e g ( d ) e h ( d ) = W ( g, h )( d ) , d ∈ { a, b } , (2.17)interpreted in the sense that either side in (2.17) has a finite limit as d ↓ a and d ↑ b . Of course, for (2.17) to hold at d ∈ { a, b } , it suffices that g and h lie locallyin dom( T max ) near x = d . ⋄ In the special case where T min is bounded from below, one can further analyzethe generalized boundary values (2.11) in the singular context by invoking principaland nonprincipal solutions of τ u = λu for appropriate λ ∈ R . This leads to naturalanalogs of (2.16) also in the singular case, and we will turn to this topic next.We start by reviewing some oscillation theory with particular emphasis on prin-cipal and nonprincipal solutions, a notion originally due to Leighton and Morse[49], Rellich [57], [58], and Hartman and Wintner [38, Appendix] (see also [19], [24,Sects 13.6, 13.9, 13.0], [37, Ch. XI], [52], [72, Chs. 4, 6–8]). Definition 2.8.
Assume Hypothesis . ( i ) Fix c ∈ ( a, b ) and λ ∈ R . Then τ − λ is called nonoscillatory at a ( resp., b ) ,if every real-valued solution u ( λ, · ) of τ u = λu has finitely many zeros in ( a, c )( resp., ( c, b )) . Otherwise, τ − λ is called oscillatory at a ( resp., b ) . ( ii ) Let λ ∈ R . Then T min is called bounded from below by λ , and one writes T min > λ I , if ( u, [ T min − λ I ] u ) L (( a,b ); rdx ) > , u ∈ dom( T min ) . (2.18)The following is a key result. Theorem 2.9.
Assume Hypothesis . Then the following items ( i ) – ( iii ) areequivalent: ( i ) T min ( and hence any symmetric extension of T min ) is bounded from below. ( ii ) There exists a ν ∈ R such that for all λ < ν , τ − λ is nonoscillatory at a and b . ( iii ) For fixed c, d ∈ ( a, b ) , c d , there exists a ν ∈ R such that for all λ < ν , τ u = λu has ( real-valued ) nonvanishing solutions u a ( λ, · ) = 0 , b u a ( λ, · ) = 0 inthe neighborhood ( a, c ] of a , and ( real-valued ) nonvanishing solutions u b ( λ, · ) = 0 , b u b ( λ, · ) = 0 in the neighborhood [ d, b ) of b , such that W ( b u a ( λ, · ) , u a ( λ, · )) = 1 , u a ( λ, x ) = o ( b u a ( λ, x )) as x ↓ a , (2.19) W ( b u b ( λ, · ) , u b ( λ, · )) = 1 , u b ( λ, x ) = o ( b u b ( λ, x )) as x ↑ b , (2.20) ˆ ca dx p ( x ) − u a ( λ, x ) − = ˆ bd dx p ( x ) − u b ( λ, x ) − = ∞ , (2.21) ˆ ca dx p ( x ) − b u a ( λ, x ) − < ∞ , ˆ bd dx p ( x ) − b u b ( λ, x ) − < ∞ . (2.22) Definition 2.10.
Assume Hypothesis , suppose that T min is bounded from be-low, and let λ ∈ R . Then u a ( λ, · ) ( resp., u b ( λ, · )) in Theorem iii ) is calleda principal ( or minimal ) solution of τ u = λu at a ( resp., b ) . A real-valued solu-tion e u a ( λ, · ) ( resp., e u b ( λ, · )) of τ u = λu linearly independent of u a ( λ, · ) ( resp., u b ( λ, · )) is called nonprincipal at a ( resp., b ) . G. FUCCI, F. GESZTESY, K. KIRSTEN, L. LITTLEJOHN, R. NICHOLS, AND J. STANFILL
Principal and nonprincipal solutions are well-defined due to Lemma 2.11 below.
Lemma 2.11.
Assume Hypothesis and suppose that T min is bounded frombelow. Then u a ( λ, · ) and u b ( λ, · ) in Theorem iii ) are unique up to ( nonvanish-ing ) real constant multiples. Moreover, u a ( λ, · ) and u b ( λ, · ) are minimal solutionsof τ u = λu in the sense that u ( λ, x ) − u a ( λ, x ) = o (1) as x ↓ a , (2.23) u ( λ, x ) − u b ( λ, x ) = o (1) as x ↑ b , (2.24) for any other solution u ( λ, · ) of τ u = λu ( which is nonvanishing near a , resp., b ) with W ( u a ( λ, · ) , u ( λ, · )) = 0 , respectively, W ( u b ( λ, · ) , u ( λ, · )) = 0 . Given these oscillation theoretic preparations, one can now revisit and comple-ment Theorem 2.6 as follows:
Theorem 2.12.
Assume Hypothesis and that τ is in the limit circle case at a and b ( i.e., τ is quasi-regular on ( a, b )) . In addition, assume that T min > λ I forsome λ ∈ R , and denote by u a ( λ , · ) and b u a ( λ , · ) ( resp., u b ( λ , · ) and b u b ( λ , · )) principal and nonprincipal solutions of τ u = λ u at a ( resp., b ) , satisfying W ( b u a ( λ , · ) , u a ( λ , · )) = W ( b u b ( λ , · ) , u b ( λ , · )) = 1 . (2.25) Then the following items ( i ) – ( iii ) hold: ( i ) Introducing v j ∈ dom( T max ) , j = 1 , , via v ( x ) = (b u a ( λ , x ) , for x near a , b u b ( λ , x ) , for x near b , v ( x ) = ( u a ( λ , x ) , for x near a ,u b ( λ , x ) , for x near b , (2.26) one obtains for all g ∈ dom( T max ) , e g ( a ) = − W ( v , g )( a ) = e g ( a ) = − W ( u a ( λ , · ) , g )( a )= lim x ↓ a g ( x ) b u a ( λ , x ) , e g ( b ) = − W ( v , g )( b ) = e g ( b ) = − W ( u b ( λ , · ) , g )( b )= lim x ↑ b g ( x ) b u b ( λ , x ) , (2.27) e g ′ ( a ) = W ( v , g )( a ) = e g ( a ) = W ( b u a ( λ , · ) , g )( a )= lim x ↓ a g ( x ) − e g ( a ) b u a ( λ , x ) u a ( λ , x ) , e g ′ ( b ) = W ( v , g )( b ) = e g ( b ) = W ( b u b ( λ , · ) , g )( b )= lim x ↑ b g ( x ) − e g ( b ) b u b ( λ , x ) u b ( λ , x ) . (2.28) In particular, the limits on the right-hand sides in (2.27) , (2.28) exist. ( ii ) All self-adjoint extensions T γ,δ of T min with separated boundary conditions areof the form T γ,δ f = τ f, γ, δ ∈ [0 , π ) ,f ∈ dom( T γ,δ ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) sin( γ ) e g ′ ( a ) + cos( γ ) e g ( a ) = 0; (2.29)sin( δ ) e g ′ ( b ) + cos( δ ) e g ( b ) = 0 (cid:9) . HE KREIN–VON NEUMANN EXTENSION REVISITED 9
Moreover, σ ( T γ,δ ) is simple. ( iii ) All self-adjoint extensions T ϕ,R of T min with coupled boundary conditions areof the type T ϕ,R f = τ f,f ∈ dom( T ϕ,R ) = (cid:26) g ∈ dom( T max ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) e g ( b ) e g ′ ( b ) (cid:19) = e iϕ R (cid:18) e g ( a ) e g ′ ( a ) (cid:19) (cid:27) , (2.30) where ϕ ∈ [0 , π ) , and R ∈ SL (2 , R ) . Moreover, under the hypotheses of Theorem 2.12, relation (2.5) implies that theminimal operator takes on the form T min f = τ f,f ∈ dom( T min ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) e g ( a ) = e g ′ ( a ) = 0 = e g ( b ) = e g ′ ( b ) (cid:9) . (2.31)The Friedrichs extension T F of T min now permits a particularly simple charac-terization in terms of the generalized boundary values e g ( a ) , e g ( b ) as derived by Kalf[42] and subsequently by Niessen and Zettl [52] (see also [58], [59] and the extensiveliterature cited in [30], [33, Ch. 13]): Theorem 2.13.
Assume Hypothesis and that τ is in the limit circle case at a and b ( i.e., τ is quasi-regular on ( a, b )) . In addition, assume that T min > λ I forsome λ ∈ R . Then the Friedrichs extension T F of T min is characterized by T F f = τ f, f ∈ dom( T F ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) e g ( a ) = 0 = e g ( b ) (cid:9) . (2.32) In particular, T F = T , .Remark . ( i ) As in (2.17), one readily verifies for g, h ∈ dom( T max ), e g ( d ) e h ′ ( d ) − e g ′ ( d ) e h ( d ) = W ( g, h )( d ) , d ∈ { a, b } , (2.33)again interpreted in the sense that either side in (2.33) has a finite limit as d ↓ a and d ↑ b . In particular, if τ is regular at an endpoint then Remark 2.7 ( ii ) showsthat the generalized boundary values in (2.27), (2.28) reduce to the canonical onesin (2.16).( ii ) While the principal solution at an endpoint is unique up to constant multiples(which we will ignore), nonprincipal solutions differ by additive constant multiplesof the principal solution. As a result, if b u a ( λ , · ) −→ b u a ( λ , · ) + Cu a ( λ , · ) , C ∈ R , then e g ( a ) −→ e g ( a ) , e g ′ ( a ) −→ e g ′ ( a ) − C e g ( a ) , (2.34)and analogously at the endpoint b . Hence, generalized boundary values e g ′ ( d ) atthe endpoint d ∈ { a, b } depend on the choice of nonprincipal solution b u d ( λ , · ) of τ u = λ u at d . However, the Friedrichs boundary conditions e g ( a ) = 0 = e g ( b ) areclearly independent of the choice of nonprincipal solution.( iii ) As always in this context, if τ is in the limit point case at one or both intervalendpoints, the corresponding boundary conditions at that endpoint are droppedand only a separated boundary condition at the other end point (if the latter is alimit circle endpoint for τ ) has to be imposed in Theorems 2.12 and 2.13. In thecase where τ is in the limit point case at both endpoints, all boundary values andboundary conditions become superfluous as in this case T min = T max is self-adjoint. ⋄ All results surveyed in this section can be found in [30] and [33, Ch. 13] whichcontain very detailed lists of references to the basics of Weyl–Titchmarsh theory.Here we just mention a few additional and classical sources such as [2, Sect. 129],[21, Chs. 8, 9], [24, Sects. 13.6, 13.9, 13.10], [41, Ch. III], [50, Ch. V], [52], [55,Ch. 6], [60, Ch. 9], [70, Sect. 8.3], [71, Ch. 13], [72, Chs. 4, 6–8].3.
The Krein–von Neumann extension of T min > T min under theassumption T min > εI for some ε >
0. We continue with some more abstract factson the Krein–von Neumann extension of strictly positive symmetric operators in acomplex Hilbert space and refer to [2, Sect. 109], [3], [4], [7], [8], [9], [10], [11], [12],[13], [15, Sect. 5.4], [16], [22], [23], [27, Part III], [34], [35, Sect. 13.2], [39], [40],[46], [47], [51], [56], [61], [62], [63], [64], [65], [67], [68], [69], and the references citedtherein for some of the basic literature in this context.Denote by n ± ( T ) = dim(ker( T ∗ ∓ iI H )) = dim (cid:0) ran( T ± iI H ) ⊥ (cid:1) ∈ N ∪ {∞} , (3.1)the deficiency indices of a densely defined, closed, symmetric operator T in thecomplex, separable Hilbert space H .The Krein–von Neumann and Friedrichs extension of a densely defined, closed,symmetric operator S with n ± ( S ) >
0, satisfying S > , (3.2)are denoted by S K and S F , respectively. If, in addition, S > εI H (3.3)for some ε >
0, then one also has n ± ( S ) = dim(ker( S ∗ )) , (3.4) S F > εI H , (3.5)and dom( S K ) = dom( S ) . + ker( S ∗ ) ,S K f = S ∗ f, f ∈ dom( S K ) . (3.6)For completeness we also recall that under hypothesis (3.3)dom( S F ) = dom( S ) . + ( S F ) − ker( S ∗ ) , S F f = S ∗ f, f ∈ dom( S F ) , (3.7)dom( S ∗ ) = dom( S ) . + ( S F ) − ker( S ∗ ) . + ker( S ∗ ) , (3.8)ker( S K ) = ker( S ∗ ) . (3.9)Here the notation . + addresses the direct (not orthogonal direct) sum in the sensethat if X , X are linear subspaces of a Banach space X , then X . + X denotesthe subspace of X given by X . + X = { x ∈ X | x = x + x , x j ∈ X j , j = 1 , } , (3.10)assuming X ∩ X = { } . (3.11) HE KREIN–VON NEUMANN EXTENSION REVISITED 11
Remark . If S > εI H , ε >
0, then dom( S ) . + ker( S ∗ ) is well-defined since f ∈ dom( S ) ∩ ker( S ∗ ) (3.12)implies 0 = S ∗ f = Sf , and hence f = 0 as S > εI H , ε > ⋄ Lemma 3.2.
Suppose S is densely defined, symmetric, and for some ε > , S > εI H . If e S is a self-adjoint extension of S such that dom (cid:0) e S (cid:1) ⊃ ker( S ∗ ) , then e S = S K .Proof. Without loss of generality we may assume that S is closed. By definition,dom (cid:0) e S (cid:1) ⊃ dom( S ), and by assumption, dom (cid:0) e S (cid:1) ⊃ ker( S ∗ ), hence dom (cid:0) e S (cid:1) ⊇ dom( S ) . + ker( S ∗ ) = dom( S K ). Since self-adjoint operators are maximal in thesense that S K has no proper symmetric extension, one concludes that e S = S K . (cid:3) Remark . ( i ) From the outset, (3.6) implies (3.9), that is, ker( S K ) = ker( S ∗ ).Lemma 3.2 now implies a converse in the sense that if ker( e S ) = ker( S ∗ ) (and hencelies in dom (cid:0) e S (cid:1) ), then e S = S K . One notes that Lemma 3.2 does not a priori assumethat e S is bounded from below.( ii ) The fact (3.9) has been isolated in [3] as uniquely identifying the Krein–vonNeumann extension (under the hypothesis S > εI H ) (see also [10, eq. (2.39)]), asa byproduct of an entirely different quadratic form approach that characterizes allnonnegative self-adjoint extensions of S . Here we derive this uniqueness aspectwith entirely elementary means only.( iii ) Applications to 2 m th order regular differential operators: Suppose T min = τ m (cid:12)(cid:12) C ∞ (( a,b )) , then T ∗ min = T max , that is, no boundary conditions are necessaryin dom( T max ). Then e T , a self-adjoint extension of T min in L (( a, b ); rdx ) equals T min,K , the Krein extension of T min in L (( a, b ); rdx ), if and only ifdim (cid:0) ker (cid:0) e T (cid:1)(cid:1) = 2 m. (3.13)Of course, dim(ker( T max )) = 2 m since we assumed T min to be regular.( iv ) Relation (3.13) only holds if T min is indeed minimally defined, that is, asthe closure of τ m (cid:12)(cid:12) C ∞ (( a,b )) . If some of the possible zero boundary conditions aremissing in dom( T min ) then they will reappear in dom( T max = T ∗ min ) and hence 2 m in (3.13) has to be diminished accordingly.( v ) The 2 m solutions giving rise to (3.13) (in the regular case) are simply generatedby solving the ordinary differential equation of 2 m th order τ m y = 0 (3.14)in the distributional sense. ⋄ To demonstrate that ε >
Example 3.4.
Let H = L ((0 , ∞ ); dx ) , and T (0) min f = − f ′′ ,f ∈ dom (cid:0) T (0) min (cid:1) = (cid:8) g ∈ L ((0 , ∞ ); dx ) (cid:12)(cid:12) for all R > : g, g ′ ∈ AC ([0 , R ]); (3.15) g (0) = g ′ (0) = 0; g ′′ ∈ L ((0 , ∞ ); dx ) (cid:9) , T (0) min = − d /dx (cid:12)(cid:12) C ∞ ((0 , ∞ )) . (3.16) Then T (0) min > but there is no ε > such that T (0) min > εI . Moreover, T (0) max f = (cid:0) T (0) min (cid:1) ∗ f = − f ′′ ,f ∈ dom (cid:0) T (0) max (cid:1) = (cid:8) g ∈ L ((0 , ∞ ); dx ) (cid:12)(cid:12) for all R > : g, g ′ ∈ AC ([0 , R ]); (3.18) g ′′ ∈ L ((0 , ∞ ); dx ) (cid:9) , and T (0) min,K f = T (0) N f = − f ′′ ,f ∈ dom (cid:0) T (0) min,K = T (0) N (cid:1) = (cid:8) g ∈ dom (cid:0) T (0) max (cid:1) (cid:12)(cid:12) g ′ (0) = 0 (cid:9) . (3.19) Furthermore, consider self-adjoint extensions T (0) α of T (0) min given by T (0) α f = − f ′′ , α ∈ [0 , ∞ ) ∪ {∞} ,f ∈ dom (cid:0) T (0) α (cid:1) = (cid:8) g ∈ dom (cid:0) T (0) max (cid:1) (cid:12)(cid:12) g ′ (0) = αg (0) (cid:9) . (3.20) Then ker (cid:0) T (0) α (cid:1) = ker (cid:0) T (0) min,K = T (0) N ≡ T (0) α =0 (cid:1) = ker (cid:0)(cid:0) T (0) min (cid:1) ∗ = T (0) max (cid:1) = { } , (3.21) and σ (cid:0) T (0) α (cid:1) = [0 , ∞ ) , α ∈ [0 , ∞ ) ∪ {∞} . (3.22) Thus, Lemma requires ε > . For the fact (3.19) see, for example, [29, Corollary5.6] ( choose γ = π/ , q ( x ) = 0 , and note that m W ( z ) = iz / , hence m W (0) = 0) .Of course, T (0) α = ∞ f = T (0) min,F f = T (0) D f = − f ′′ ,f ∈ dom (cid:0) T (0) D (cid:1) = (cid:8) g ∈ dom (cid:0) T (0) max (cid:1) (cid:12)(cid:12) g (0) = 0 (cid:9) , (3.23) represents the Friedrichs ( resp., Dirichlet ) extension of T (0) min . Combining [20] and [30] one can now extend the description of the Krein exten-sion from the known regular case to the singular case as follows:
Theorem 3.5.
In addition to Hypothesis , suppose that T min > εI for some ε > . Then the following items ( i ) and ( ii ) hold: ( i ) Assume that n ± ( T min ) = 1 and denote the principal solutions of τ u = 0 at a and b by u a (0 , · ) and u b (0 , · ) , respectively. If τ is in the limit circle case at a andin the limit point case at b , then the Krein–von Neumann extension T γ K of T min isgiven by T γ K f = τ f,f ∈ dom( T γ K ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) sin( γ K ) e g ′ ( a ) + cos( γ K ) e g ( a ) = 0 (cid:9) , (3.24)cot( γ K ) = − e u ′ b (0 , a ) / e u b (0 , a ) , γ K ∈ (0 , π ) . Similarly, if τ is in the limit circle case at b and in the limit point case at a , thenthe Krein–von Neumann extension T δ K of T min is given by T δ K f = τ f, HE KREIN–VON NEUMANN EXTENSION REVISITED 13 f ∈ dom( T δ K ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) sin( δ K ) e g ′ ( b ) + cos( δ K ) e g ( b ) = 0 (cid:9) , (3.25)cot( δ K ) = − e u ′ a (0 , b ) / e u a (0 , b ) , δ K ∈ (0 , π ) . ( ii ) Assume that n ± ( T min ) = 2 , that is, τ is in the limit circle case at a and b .Then, introducing a basis for ker( T max ) , denoted by u (0 , · ) , u (0 , · ) as follows, τ u j (0 , · ) = 0 , j = 1 , , e u (0 , a ) = 0 , e u (0 , b ) = 1 , (3.26) e u (0 , a ) = 1 , e u (0 , b ) = 0 , the Krein–von Neumann extension T ,R K of T min is given by T ,R K f = τ f,f ∈ dom( T ,R K ) = (cid:26) g ∈ dom( T max ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) e g ( b ) e g ′ ( b ) (cid:19) = R K (cid:18) e g ( a ) e g ′ ( a ) (cid:19) (cid:27) , (3.27) where R K = 1 e u ′ (0 , a ) (cid:18) − e u ′ (0 , a ) 1 e u ′ (0 , a ) e u ′ (0 , b ) − e u ′ (0 , b ) e u ′ (0 , a ) e u ′ (0 , b ) (cid:19) . (3.28) Proof. ( i ) It suffices to prove (3.24), the proof for (3.25) being entirely analogous.Since ker( T max ) is one-dimensional, and τ is in the limit point case at b , one con-cludes that only the smallest solution of τ u = 0 near b , that is, the principalsolution, u b (0 , · ), can lie in ker( T max ),ker( T max ) = lin . span { u b (0 , · ) } . (3.29)One also notes that by (2.27), (2.28), e u b (0 , b ) = 0 , e u ′ b (0 , b ) = 1 . (3.30)Since the Krein extension is now necessarily associated with separated boundaryconditions, in fact, a self-adjoint boundary condition at the point a only, (3.9)implies that u b (0 , · ) must necessarily satisfy the boundary condition at a whichdefines the Krein extension of T min in this case. Thus, the underlying boundarycondition parameter γ K ∈ [0 , π ) is determined viasin( γ k ) e u ′ b (0 , a ) + cos( γ K ) e u b (0 , a ) = 0 . (3.31)If e u b (0 , a ) = 0, then together with (2.32) and (3.5) this would imply that u b (0 , · ) ∈ dom( T F = T ) in the notation of (2.29) (one notes that no boundary conditionis required at b due to the limit point property of τ at b ). Thus, 0 ∈ σ p ( T F ),contradicting T F > εI . Hence γ K >
0, completing the proof of (3.24).( ii ) Introducing a basis for the null space of T max as in (3.26), one notes via (2.33)that W ( u (0 , · ) , u (0 , · )) = − e u ′ (0 , a ) = e u ′ (0 , b ) . (3.32)Next, introducing R K as in (3.28) and employing (3.26) and (3.32) one computesthat det( R K ) = − e u ′ (0 , b ) / e u ′ (0 , a ) = 1 , that is, R K ∈ SL (2 , R ) . (3.33)Thus, (3.27) represents one of the self-adjoint extensions of T min characterized by ϕ = 0 and R = R K according to Theorem 2.12 ( iii ), and hence it remains to showthat this extension is precisely the Krein–von Neumann extension of T min . For this purpose we turn to Lemma 3.2 next: Sincedom( T ,R K ) = dom( T min ) . + ker( T ∗ min ) = dom( T min ) . + ker( T max ) , (3.34)and since T min is characterized by the vanishing of all generalized boundary valuesas depicted in (2.31), it suffices to verify that u (0 , · ) and u (0 , · ) both satisfythe boundary conditions in (3.27), given (3.28) (see also Remark 3.3 ( ii ) with m =1). This verification reduces to the following elementary computations, employing(3.26) and (3.32) once more, R K (cid:18)e u (0 , a ) e u ′ (0 , a ) (cid:19) = (cid:18) e u ′ (0 , b ) (cid:19) = (cid:18)e u (0 , b ) e u ′ (0 , b ) (cid:19) ,R K (cid:18)e u (0 , a ) e u ′ (0 , a ) (cid:19) = (cid:18) e u ′ (0 , b ) (cid:19) = (cid:18)e u (0 , b ) e u ′ (0 , b ) (cid:19) , (3.35)completing the proof of (3.27). (cid:3) Relations (3.27), (3.28) extend [20, Example 3.3] in the regular context to thesingular one. 4.
Three Examples
In this section we illustrate Theorem 3.5 with three examples, including a gen-eralized Bessel and Jacobi-type operators.We start with the generalized Bessel operator following the analysis in [32, Sec-tion 6].
Example 4.1 (A Generalized Bessel Operator) . Let a = 0 and b ∈ (0 , ∞ ) in (2.1) ,and consider the concrete example p ( x ) = x β , r ( x ) = x α , q ( x ) = (2 + α − β ) γ − (1 − β ) x β − ,α > − , β < , γ > , x ∈ (0 , b ) . (4.1) Then τ α,β,γ = x − α (cid:20) − ddx x β ddx + (2 + α − β ) γ − (1 − β ) x β − (cid:21) ,α > − , β < , γ > , x ∈ (0 , b ) , (4.2) is singular at the endpoint since the potential, q is not integrable near x = 0) and is regular at x = b . Furthermore, τ α,β,γ is in the limit circle case at x = 0 if γ < and in the limit point case at x = 0 when γ > .Solutions of τ α,β,γ u = zu are given by ( cf. [43, No. 2.162, p. 440]) y ,α,β,γ ( z, x ) = x (1 − β ) / J γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ > ,y ,α,β,γ ( z, x ) = ( x (1 − β ) / J − γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ / ∈ N ,x (1 − β ) / Y γ (cid:0) z / x (2+ α − β ) / / (2 + α − β ) (cid:1) , γ ∈ N , γ > , (4.3) where J ν ( · ) , Y ν ( · ) are the standard Bessel functions of order ν ∈ R ( cf. [1, Ch. 9]) .In the following we assume that γ ∈ [0 ,
1) (4.4)
HE KREIN–VON NEUMANN EXTENSION REVISITED 15 to ensure the limit circle case at x = 0 . In this case it suffices to focus on thegeneralized boundary values at the singular endpoint x = 0 . For this purpose weintroduce principal and nonprincipal solutions u ,α,β,γ (0 , · ) and b u ,α,β,γ (0 , · ) of τ α,β,γ u = 0 at x = 0 by u ,α,β,γ (0 , x ) = (1 − β ) − x [1 − β +(2+ α − β ) γ ] / , γ ∈ [0 , , b u ,α,β,γ (0 , x ) = ( (1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / , γ ∈ (0 , , (1 − β ) x (1 − β ) / ln(1 /x ) , γ = 0 ,α > − , β < , x ∈ (0 , . (4.5) The generalized boundary values for g ∈ dom( T max,α,β,γ ) at x = 0 are then of theform e g (0) = − W ( u ,α,β,γ (0 , · ) , g )(0)= ( lim x ↓ g ( x ) (cid:14)(cid:2) (1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / (cid:3) , γ ∈ (0 , , lim x ↓ g ( x ) (cid:14)(cid:2) (1 − β ) x (1 − β ) / ln(1 /x ) (cid:3) , γ = 0 , (4.6) e g ′ (0) = W ( b u ,α,β,γ (0 , · ) , g )(0)= lim x ↓ (cid:2) g ( x ) − e g (0)(1 − β )[(2 + α − β ) γ ] − x [1 − β − (2+ α − β ) γ ] / (cid:3) × (cid:2) (1 − β ) − x [1 − β +(2+ α − β ) γ ] / (cid:3) − , γ ∈ (0 , , lim x ↓ (cid:2) g ( x ) − e g (0)(1 − β ) x (1 − β ) / ln(1 /x ) (cid:3) × (cid:2) (1 − β ) − x (1 − β ) / (cid:3) − , γ = 0 . (4.7) Choosing u (0 , x ) = u ,α,β,γ (0 , x ) /u ,α,β,γ (0 , b ) , γ ∈ [0 , ,u (0 , x ) = b u ,α,β,γ (0 , x ) − (1 − β ) [(2 + α − β ) γ ] − b − (2+ α − β ) γ u ,α,β,γ (0 , x ) ,γ ∈ (0 , , b u ,α,β, (0 , x ) − (1 − β ) ln(1 /b ) u ,α,β, (0 , x ) , γ = 0 ,α > − , β < , x ∈ (0 , b ) , (4.8) in (3.26) – (3.28) yields the Krein–von Neumann extension T ,R K ,α,β,γ of T min,α,β,γ in the form T ,R K ,α,β,γ f = τ α,β,γ, f, (4.9) f ∈ dom( T ,R K ,α,β,γ ) = (cid:26) g ∈ dom( T max,α,β,γ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) e g ( b ) e g ′ ( b ) (cid:19) = R K,α,β,γ (cid:18) e g (0) e g ′ (0) (cid:19) (cid:27) , where R K,α,β,γ = b [ β − − (2+ α − β ) γ ] / × − β (2 + α − β ) γ b − β − β b − β +(2+ α − β ) γ (1 − β ) α − β ) γ − − β (cid:20)
12 + (2 + α − β ) γ − β ) (cid:21) b (2+ α − β ) γ ,γ ∈ (0 , , (1 − β )ln(1 /b ) b (1 − β ) / − β b (1 − β ) / (1 − β ) ln(1 /b ) − − β )2 b ( β − / b ( β − / , γ = 0 . (4.10) One verifies that det ( R K,α,β,γ ) = 1 . For the Krein extension of the standard Bessel operator on the half-line (0 , ∞ )(i.e., α = β = 0, a = 0, b = ∞ ) we also refer to [17] (see also [5], [6]).Next, we turn to a singular operator relevant to the phenomenon of acousticblack holes, following [14]. Example 4.2 (Acoustic Black Hole) . Let a = 0 and b ∈ (0 , ∞ ) in (2.1) andconsider p ( x ) = p ( x ) x α , r ( x ) = r ( x ) x β , q ( x ) = 0 , x ∈ (0 , b ) , (4.11) where α, β ∈ R are fixed and p and r are continuous real-valued functions on (0 , b ) that satisfy for some m, M ∈ (0 , ∞ ) , m p ( x ) M, m r ( x ) M, x ∈ (0 , b ) . (4.12) Then τ p ,r ,α,β = − r ( x ) x β ddx p ( x ) x α ddx , α, β ∈ R , x ∈ (0 , b ) . (4.13) Linearly independent solutions y j (0 , · ) , j ∈ { , } , of the differential equation τ p ,r ,α,β y = 0 are given by y ,p ,r ,α,β (0 , x ) = 1 , y ,p ,r ,α,β (0 , x ) = ˆ bx dtp ( t ) t α , α, β ∈ R , x ∈ (0 , b ) , (4.14) and they satisfy W ( y ,p ,r ,α,β (0 , · ) , y ,p ,r ,α,β (0 , · )) = 1 . (4.15) Explicit calculations reveal that y ,p ,r ,α,β (0 , · ) , y ,p ,r ,α,β (0 , · ) ∈ L (cid:0) (0 , b ); r ( x ) x β dx (cid:1) , (4.16) that is, τ p ,r ,α,β is in the limit circle case at x = 0 , if and only if β > max {− , α − } . In addition, τ p ,r ,α,β is regular at x = b . To avoid the scenario where τ p ,r ,α,β is in the limit point case at x = 0 , we thus assume that α and β satisfy β > max {− , α − } . To avoid that τ p ,r ,α,β is also regular at x = 0 ( and henceregular on (0 , b )) , we now also assume that α > . ( There is no need to discussthe Krein–von Neumann extensions in the regular case as that can be found in [20,Example 3.3]) . Altogether, this means we are assuming α > and β > α − . (4.17) HE KREIN–VON NEUMANN EXTENSION REVISITED 17
Under the assumption α > , y (0 , · ) and y (0 , · ) are principal and nonprinci-pal solutions, respectively, of τ p ,r ,α,β u = 0 at x = 0 . Thus, in accordance withTheorem , one chooses u ,p ,r ,α,β (0 , x ) = 1 , b u ,p ,r ,α,β (0 , x ) = ˆ bx dtp ( t ) t α ,α > , β > α − , x ∈ (0 , b ) . (4.18) The generalized boundary values for g ∈ dom( T max,p ,r ,α,β ) are then of the form e g (0) = lim x ↓ g ( x ) (cid:20) ˆ bx dtp ( t ) t α (cid:21) − , (4.19) e g ′ (0) = lim x ↓ (cid:20) g ( x ) − e g (0) ˆ bx dtp ( t ) t α (cid:21) , (4.20) e g ( b ) = lim x ↑ b g ( x ) = g ( b ) , (4.21) e g ′ ( b ) = lim x ↑ b [ pg ′ ]( x ) = [ pg ′ ]( b ) = p ( b ) b α g ′ ( b ) . (4.22) A basis for ker( T max,p ,r ,α,β ) which satisfies (3.26) is given by u ,p ,r ,α,β (0 , x ) = 1 , u ,p ,r ,α,β (0 , x ) = ˆ bx dtp ( t ) t α ,α > , β > α − , x ∈ (0 , b ) , (4.23) and explicit calculations using (4.18) , (4.20) , and (4.22) reveal e u ′ ,p ,r ,α,β (0 ,
0) = 1 , e u ′ ,p ,r ,α,β (0 ,
0) = 0 , e u ′ ,p ,r ,α,β (0 , b ) = 0 , e u ′ ,p ,r ,α,β (0 , b ) = − . (4.24) Using (4.24) in (3.28) then yields R K,p ,r ,α,β = (cid:18) − (cid:19) , α > , β > α − , (4.25) and hence the Krein–von Neumann extension T ,R K ,p ,r ,α,β of T min,p ,r ,α,β ischaracterized by T ,R K ,p ,r ,α,β f = τ p ,r ,α,β f, (4.26)dom( T ,R K ,p ,r ,α,β ) = (cid:8) g ∈ dom( T max,p ,r ,α,β ) (cid:12)(cid:12) g ( b ) = e g ′ (0) , g [1] ( b ) = − e g (0) (cid:9) . Finally, we turn to the Jacobi operator referring to [31] for a much more detailedanalysis.
Example 4.3 (Jacobi Operator) . Let a = − , b = 1 ,p ( x ) = p α,β ( x ) = (1 − x ) α +1 (1 + x ) β +1 , q ( x ) = q α,β ( x ) = 0 , (4.27) r ( x ) = r α,β ( x ) = (1 − x ) α (1 + x ) β , x ∈ ( − , , α, β ∈ R ( see, e.g., [1, Ch. 22] , [18] , [25, Sect. 23] , [26] , [28] , [36] , [45] , [48] , [54, Ch. 18] , [66,Ch. IV]) and consider the Jacobi differential expression τ α,β = − (1 − x ) − α (1 + x ) − β ( d/dx ) (cid:0) (1 − x ) α +1 (1 + x ) β +1 (cid:1) ( d/dx ) ,x ∈ ( − , , α, β ∈ R . (4.28) To decide the limit point/limit circle classification of τ α,β at the interval endpoints ± , it suffices to note that if y is a given solution of τ y = 0 , then a 2nd linearlyindependent solution y of τ y = 0 is obtained via the standard formula y ( x ) = y ( x ) ˆ xc dx ′ p ( x ′ ) − y ( x ′ ) − , c, x ∈ ( a, b ) . (4.29) Returning to the concrete Jacobi case at hand, one notices that y ( x ) = 1 , x ∈ ( − , ,y ( x ) = ˆ x dx ′ (1 − x ′ ) − − α (1 + x ′ ) − − β , x ∈ ( − , , (4.30) and hence y ( x ) (4.31)= − − α β − (1 + x ) − β [1 + O (1 + x )] + O (1) , α ∈ R , β ∈ R \{ } , as x ↓ − , − − − α ln(1 + x ) + O (1) , α ∈ R , β = 0 , as x ↓ − , − − β α − (1 − x ) − α [1 + O (1 − x )] + O (1) , α ∈ R \{ } , β ∈ R , as x ↑ +1 , − − − β ln(1 − x ) + O (1) , α = 0 , β ∈ R , as x ↑ +1 . Thus, an application of Theorem , Definition , and Remark ii ) impliesthe classification, τ α,β is regular at − if and only if α ∈ R , β ∈ ( − , ,in the limit circle case at − if and only if α ∈ R , β ∈ [0 , ,in the limit point case at − if and only if α ∈ R , β ∈ R \ ( − , ,regular at +1 if and only if α ∈ ( − , , β ∈ R ,in the limit circle case at +1 if and only if α ∈ [0 , , β ∈ R ,in the limit point case at +1 if and only if α ∈ R \ ( − , , β ∈ R . (4.32) The fact (4.30) naturally leads to principal and nonprincipal solutions u ± ,α,β (0 , x ) and b u ± ,α,β (0 , x ) of τ α,β y = 0 near ± as follows : u − ,α,β (0 , x ) = ( − − α − β − (1 + x ) − β [1 + O (1 + x )] , β ∈ ( −∞ , , , β ∈ [0 , ∞ ) , b u − ,α,β (0 , x ) = , β ∈ ( −∞ , , − − α − ln((1 + x ) / , β = 0 , − α − β − (1 + x ) − β [1 + O (1 + x )] , β ∈ (0 , ∞ ) , α ∈ R , (4.33) and u +1 ,α,β (0 , x ) = ( − β − α − (1 − x ) − α [1 + O (1 − x )] , α ∈ ( −∞ , , , α ∈ [0 , ∞ ) , b u +1 ,α,β (0 , x ) = , α ∈ ( −∞ , , − β − ln((1 − x ) / , α = 0 , − − β − α − (1 − x ) − α [1 + O (1 − x )] , α ∈ (0 , ∞ ) , β ∈ R . (4.34) HE KREIN–VON NEUMANN EXTENSION REVISITED 19
Combining the fact (4.32) with Theorem , the minimal operator T min, ,α,β cor-responding to τ α,β is essentially self-adjoint in L (( − , r α,β dx ) if and only if α, β ∈ R \ ( − , . Thus, boundary values for the maximal operator T max,α,β associ-ated with τ α,β at − exist if and only if α ∈ R , β ∈ ( − , , and similarly, boundaryvalues for T max,α,β at +1 exist if and only if α ∈ ( − , , β ∈ R .Employing the principal and nonprincipal solutions (4.33) , (4.34) at ± , accord-ing to (2.27) , (2.28) , generalized boundary values for g ∈ dom( T max,α,β ) are of theform e g ( −
1) = g ( − , β ∈ ( − , , − α +1 lim x ↓− g ( x ) / ln((1 + x ) / , β = 0 ,β α +1 lim x ↓− (1 + x ) β g ( x ) , β ∈ (0 , , e g ′ ( −
1) = g [1] ( − , β ∈ ( − , , lim x ↓− (cid:2) g ( x ) + e g ( − − α − ln((1 + x ) / (cid:3) , β = 0 , lim x ↓− (cid:2) g ( x ) − e g ( − − α − β − (1 + x ) − β (cid:3) , β ∈ (0 , , α ∈ R , (4.35) e g (1) = g (1) , α ∈ ( − , , β +1 lim x ↑ g ( x ) / ln((1 − x ) / , α = 0 , − α β +1 lim x ↑ (1 − x ) α g ( x ) , α ∈ (0 , , e g ′ (1) = g [1] (1) , α ∈ ( − , , lim x ↑ (cid:2) g ( x ) − e g (1)2 − β − ln((1 − x ) / (cid:3) , α = 0 , lim x ↑ (cid:2) g ( x ) + e g (1)2 − β − α − (1 − x ) − α (cid:3) , α ∈ (0 , , β ∈ R . (4.36) For a detailed treatment of solutions of the Jacobi differential equation and theassociated hypergeometric differential equations we refer to [31, Appendix A] .To shorten the presentation of this example and hence avoid case ( i ) in Theorem where τ α,β is in the limit point case at − or +1 and in the limit circle caseat the opposite endpoint ( i.e., the cases where n ± ( T min,α,β ) = 1) , we now assumethat τ α,β is in the limit circle case at ± i.e., n ± ( T min,α,β ) = 2 as in case ( ii ) ofTheorem . Thus, we assume that α, β ∈ ( − ,
1) (4.37) in the following. Next, we consider two linearly independent solutions of τ α,β y = 0 near x = − given by y ,α,β, − (0 , x ) = 1 ,y ,α,β, − (0 , x ) = (1 + x ) − β F (1 + α, − β ; 1 − β ; (1 + x ) / , β ∈ ( − , \{ } ,α ∈ ( − , , x ∈ ( − , . (4.38) Furthermore, using the connection formulas found in [1, Eq. 15.3.6, 15.3.10] yieldsthe behavior of y ,α,β, − (0 , x ) near x = 1 , y ,α,β, − (0 , x ) = (1 + x ) − β Γ(1 − β )Γ( − α )Γ( − α − β ) F (1 + α, − β ; 1 + α ; (1 − x ) / − (1 + x ) − β (1 − x ) − α α α − βF ( − α − β,
1; 1 − α ; (1 − x ) / ,α ∈ ( − , \{ } , − (1 + x ) − β β ∞ X n =0 ( − β ) n n ( n !) [ ψ ( n + 1) − ψ ( n − β ) − ln((1 − x ) / − x ) n , α = 0 ,β ∈ ( − , \{ } , x ∈ ( − , . (4.39) Here F ( · , · , · ; · ) denotes the hypergeometric function ( see, e.g., [1, Ch. 15]) , ψ ( · ) = Γ ′ ( · ) / Γ( · ) the Digamma function, γ E = − ψ (1) = 0 . . . . representsEuler’s constant, and ( ζ ) = 1 , ( ζ ) n = Γ( ζ + n ) / Γ( ζ ) , n ∈ N , ζ ∈ C \ ( − N ) , (4.40) abbreviates Pochhammer’s symbol ( see, e.g., [1, Ch. 6]) .Similarly, we consider linearly independent solutions of τ α,β y = 0 near x = 1 , y ,α,β, (0 , x ) = 1 ,y ,α,β, (0 , x ) = (1 − x ) − α F (1 + β, − α ; 1 − α ; (1 − x ) / , α ∈ ( − , \{ } ,β ∈ ( − , , x ∈ ( − , , (4.41) noting one can show that y ,α,β, (0 , x ) = (1 − x ) − α Γ(1 − α )Γ( − β )Γ( − α − β ) F (1 + β, − α ; 1 + β ; (1 + x ) / − (1 − x ) − α (1 + x ) − β β β − αF ( − α − β,
1; 1 − β ; (1 + x ) / ,β ∈ ( − , \{ } , − (1 − x ) − α α ∞ X n =0 ( − α ) n n ( n !) [ ψ ( n + 1) − ψ ( n − α ) − ln((1 + x ) / x ) n , β = 0 ,α ∈ ( − , \{ } , x ∈ ( − , . (4.42) For α, β ∈ ( − , , the following five cases are associated with a strictly positiveminimal operator T min,α,β ( see, [31]) and we now provide the corresponding choices u , u in (3.26) – (3.28) that yield R K,α,β and the Krein–von Neumann extension T ,R K ,α,β of T min,α,β , T ,R K ,α,β f = τ α,β f, (4.43) f ∈ dom( T ,R K ,α,β ) = (cid:26) g ∈ dom( T max,α,β ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) e g (1) e g ′ (1) (cid:19) = R K,α,β (cid:18) e g ( − e g ′ ( − (cid:19) (cid:27) . (I) The regular case α, β ∈ ( − , Choosing u (0 , x ) = 2 β Γ( − α − β )Γ( − α )Γ(1 − β ) y ,α,β, − (0 , x ) , u (0 , x ) = 1 − u (0 , x ) , (4.44) HE KREIN–VON NEUMANN EXTENSION REVISITED 21 yields R K,α,β = − α − β − Γ( − α )Γ( − β )Γ( − α − β )0 1 , α, β ∈ ( − , . (4.45) (II) The case α ∈ ( − , , β ∈ (0 , Choosing u (0 , x ) = 1 , u (0 , x ) = β − − α − y ,α,β, − (0 , x ) + 2 − α − β − Γ( − α )Γ( − β )Γ( − α − β ) , (4.46) yields R K,α,β = − − α − β − Γ( − α )Γ( − β )Γ( − α − β ) 1 − , α ∈ ( − , , β ∈ (0 , . (4.47) (III) The case α ∈ (0 , , β ∈ ( − , Choosing u (0 , x ) = β − − α − y ,α,β, − (0 , x ) , u (0 , x ) = 1 , (4.48) yields R K,α,β = −
11 2 − α − β − Γ( − α )Γ( − β )Γ( − α − β ) , α ∈ (0 , , β ∈ ( − , . (4.49)( In cases ( II ) and ( III ) we interpret / Γ(0) = 0 . ) (IV) The case α = 0 , β ∈ ( − , Choosing u (0 , x ) = β − − y , ,β, − (0 , x ) , u (0 , x ) = 1 , (4.50) yields R K, ,β = (cid:18) − − − β − [ γ E + ψ ( − β )] (cid:19) , α = 0 , β ∈ ( − , . (4.51) (V) The case α ∈ ( − , , β = 0: Choosing u (0 , x ) = 1 , u (0 , x ) = − α − − y ,α, , (0 , x ) , (4.52) yields R K,α, = (cid:18) − α − [ γ E + ψ ( − α )] 1 − (cid:19) , α ∈ ( − , , β = 0 . (4.53) Obviously, det ( R K,α,β ) = 1 in all five cases.Remark . In the remaining four cases in Example 4.3, given by all combinationsof α = 0 , β = 0 , α ∈ (0 , β ∈ (0 , T min,α,β , is not strictly positive. Thisis easily seen by considering the Jacobi polynomials and the boundary conditionsthey satisfy. The n th Jacobi polynomial is defined as (see [53, Eq. 18.5.7]) P α,βn ( x ) := ( α + 1) n n ! F ( − n, n + α + β + 1; α + 1; (1 − x ) / ,n ∈ N , − α / ∈ N , − n − α − β − / ∈ N , (4.54) and can be defined by continuity for all parameters α, β ∈ R . We note that P α,βn ( x )is a polynomial of degree at most n , and has strictly smaller degree if and only if − n − α − β ∈ { , . . . , n } (cf. [66, p. 64]). It satisfies the differential equation τ α,β P α,βn ( x ) = λ α,βn P α,βn ( x ) , (4.55)where λ α,βn = n ( n + 1 + α + β ) , n ∈ N . (4.56)One verifies that the Jacobi polynomials are solutions of the Jacobi operator eigen-value equation τ α,β y = λ α,βn y with Neumann boundary conditions in the regularcase where α, β ∈ ( − , α, β ∈ [0 , ∈ σ ( T F,α,β ), α, β ∈ [0 , T F,α,β denotesthe Friedrichs extension of T min,α,β , and hence T min,α,β > α, β ∈ [0 , ⋄ Acknowledgments.
We gratefully acknowledge discussions with Jussi Behrndt.We are indebted to Boris Belinskiy for kindly organizing the special session, “Mod-ern Applied Analysis” at the AMS Sectional Meeting at the University of Tennesseeat Chattanooga, October 10–11, 2020, and for organizing the associated special is-sue in Applicable Analysis.
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