Interlaced dense point and absolutely continuous spectra for Hamiltonians with concentric-shell singular interactions
aa r X i v : . [ m a t h - ph ] J a n INTERLACED DENSE POINT AND ABSOLUTELYCONTINUOUS SPECTRA FOR HAMILTONIANSWITH CONCENTRIC-SHELL SINGULARINTERACTIONS
Pavel Exner and Martin Fraas
Nuclear Physics Institute, Czech Academy of Sciences, 25068 ˇReˇz near Prague,Doppler Institute, Czech Technical University, Bˇrehov´a 7, 11519 Prague, Czechiae-mail: [email protected], [email protected]
Abstract
We analyze the spectrum of the generalized Schr¨odinger operatorin L ( R ν ) , ν ≥
2, with a general local, rotationally invariant singularinteraction supported by an infinite family of concentric,equidistantly spaced spheres. It is shown that the essential spectrumconsists of interlaced segments of the dense point and absolutelycontinuous character, and that the relation of their lengths at highenergies depends on the choice of the interaction parameters;generically the p.p. component is asymptotically dominant. We alsoshow that for ν = 2 there is an infinite family of eigenvalues belowthe lowest band. PACS number : 03.65.Xp keywords : Schr¨odinger operators, singular interactions, absolutelycontinuous spectrum, dense pure point spectrum
Quantum systems with the spectrum consisting of components of a differentnature attract attention from different points of view. Probably the mostimportant among them concerns random potentials in higher dimensions —a demonstration of existence of a mobility edge is one of the hardest questionsof the present mathematical physics. At the same time, a study of specificnon-random systems can reveal various types of spectral behaviour whichdiffer from the generic type.An interesting example among these refers to the situation where thespectrum is composed of interlacing intervals of the dense point and abso-lutely continuous character. A way to construct such models using radially1eriodic potentials was proposed in [1]: since at large distances in such asystem the radial and angular variables “almost decompose” locally and theradial part behaves thus essentially as one-dimensional there are spectral in-tervals where the particle can propagate, with the gaps between them filleddensely by localized states.To be specific consider, e.g., the operator t = − d / d x + q ( x ) on L ( R )with q bounded and periodic. By the standard Floquet analysis the spec-trum of t is purely absolutely continuous consisting of a family of bands, σ ( t ) = S Nk =0 [ E k , E k +1 ], corresponding to a strictly increasing, generically in-finite sequence { E k } Nk =0 . Suppose now that the potential is mirror-symmetric, q ( x ) = q ( − x ), and consider the operator T = −△ + q ( | x | )on L ( R ν ) , ν ≥
2. It was shown in [1] that the essential spectrum of T coversthe half-line [ E , ∞ ), being absolutely continuous in the spectral bands of t and dense pure point in the gaps ( E k − , E k ) , k = 1 , . . . N .The well-known properties of one-dimensional Schr¨odinger operators tellus that the dense point segments in this example shrink with increasingenergy at a rate determined by the regularity of the potential. If we replacethe bounded q by a family of δ interactions, the segment lengths tend insteadto a positive constant [2] , nevertheless, the absolutely continuous componentstill dominates the spectrum at high energies.The aim of this paper is to investigate a similar model in which a familyof concentric, equally spaced spheres supports generalized point interactionswith identical parameters. We will demonstrate that the interlaced spectralcharacter persists and, depending on the choice of the parameters, each ofthe components may dominate in the high-energy limit, or neither of them.Specifically, the ratio of the adjacent pp and ac spectral segments, ( E k − E k − ) / ( E k +1 − E k ), has three possible types of behaviour, namely like O ( k µ ) with µ = 0 , ±
1. What is more, in the generic case we have µ = 1so the dense point part dominates, which is a picture very different fromthe mobility-edge situation mentioned in the opening. Apart of this mainresult, we are going to show that the interesting result about existence of theso-called “Welsh eigenvalues” in the two-dimensional case[3, 4] also extendsto the case of generalized point interactions.2 The model
As we have said, we are going to investigate generalized Schr¨odinger operatorsin R ν , ν ≥
2, with spherically symmetric singular interaction on concentricspheres, the radii of which are supposed to be R n = nd + d/ , n ∈ N .It is important that the system is radially periodic, hence the interactionson all the spheres are assumed to be the same. In view of the sphericalsymmetry we may employ the partial-wave decomposition: the isometry U : L ((0 , ∞ ) , r ν − dr ) → L (0 , ∞ ) defined by U f ( r ) = r ν − f ( r ) allows us towrite L ( R ν ) = L l ∈ N U − L (0 , ∞ ) ⊗ S l , where S l is the l -th eigenspaceof the Laplace-Bertrami operator on the unit sphere. The operator we areinterested in can be then written as H Λ := M l U − H Λ ,l U ⊗ I l , (2.1)where I l is the identity operator on S l and the l -th partial wave operators H Λ ,l := − d d r + 1 r (cid:20) ( ν − ν − l ( l + ν − (cid:21) (2.2)are determined by the boundary conditions at the singular points R n , (cid:18) f ( R n +) f ′ ( R n +) (cid:19) = e iχ (cid:18) γ βα δ (cid:19) (cid:18) f ( R n − ) f ′ ( R n − ) (cid:19) ; (2.3)in the transfer matrix Λ := e iχ (cid:0) γ βα δ (cid:1) the parameters α, β, γ, δ are real andsatisfy the condition αβ − γδ = −
1. In other words, the domain of theselfadjoint operator H Λ ,l is D ( H Λ ,l ) = n f ∈ L (0 , ∞ ) : f, f ′ ∈ AC loc (cid:0) (0 , ∞ ) \ ∪ n { R n } (cid:1) ; − f ′′ + r h ( ν − ν − + l ( l + ν − i f ∈ L (0 , ∞ ); F ( R n +) = Λ F ( R n − ) o , (2.4)where the last equation is a shorthand for the boundary conditions (2.3). Ifthe dimension ν ≤ f ∈ D ( H l )at the origin: for ν = 2 , l = 0 we assume that lim r → [ √ r ln r ] − f ( r ) = 0 , and for ν = 3 , l = 0 we replace it by f (0+) = 0. Since the generalized pointinteraction is kept fixed, we will mostly drop the symbol Λ in the following. For relations of these conditions to the other standard parametrization of the gener-alized point interaction, ( U − I ) F ( R n ) + i ( U + I ) F ′ ( R n ) = 0, see [ ? ] Generalized Kronig-Penney model
As in the regular case the structure of the spectrum is determined by theunderlying one-dimensional Kronig-Penney model. We need its generalizedform where the Hamiltonian acts as the one dimensional Laplacian exceptat the interaction sites, x n := nd + d/ , n ∈ Z , where the wave functionssatisfy boundary conditions analogous to (2.3). To be explicit we considerthe four-parameter family of self-adjoint operators h Λ f := − f ′′ , D ( h Λ ) = n f ∈ H , (cid:0) R \ ∪ n { x n } (cid:1) : F ( x n +) = Λ F ( x n − ) o (3.1)with the matrix Λ same as above (without loss of generality we may assume χ = 0 because it is easy to see that operators differing by the value of χ are isospectral). Spectral properties of this model were investigated in [6, 7]where it was shown that the following three possibilities occur:(i) the δ -type: β = 0 and γ = δ = 1. In this case the gap width isasymptotically constant; it behaves like 2 | α | d − + O ( n − ) as the bandindex n → ∞ . This is the standard Kronig-Penney model.(ii) the intermediate type: β = 0 and | γ + δ | >
2. Now the quotient ofthe band width to the adjacent gap width is asymptotically constantbehaving as arcsin(2 | δ + γ | − ) / arccos(2 | δ + γ | − ) + O ( n − ).(iii) the δ ′ -type: the generic case, β = 0. In this case the band width isasymptotically constant; it behaves like 8 | βd | − + O ( n − ) as n → ∞ .Recall that these types of spectral behaviour correspond to high-energy prop-erties of a single generalized point interaction as manifested through scatter-ing, resonances [5], etc.There is one more difference from standard Floquet theory which we wantto emphasize. It is well known [8] that in the regular case the spectral edge E corresponds to a symmetric eigenfunction. In the singular case this is nolonger true; one can check easily the following claim. Proposition 3.1
Let u be an α -periodic solution of the equation − u ′′ = E u on ( − d/ , d/ with U ( x n +) = Λ U ( x n − ) , where E := inf σ ( h Λ ) . Then u isperiodic for β ≥ and antiperiodic for β < . To finish the discussion of the one-dimensional comparison operator, letus state three auxiliary results which will be needed in the next section.4 emma 3.2
There is a constant
C > such that for every function u in thedomain of the operator h Λ it holds that k u ′ k ≤ C ( k h Λ u k + k u k ) . (3.2) Proof:
We employ Redheffer’s inequality [9] which states that b Z a | u ′ ( x ) | d x ≤ C ′ b Z a | u ′′ ( x ) | d x + b Z a | u ( x ) | d x holds for any u twice differentiable in an interval [ a, b ] and some C ′ > a = x n , b = x n +1 , and the sought resultwith C = 2 C ′ follows easily. (cid:4) Lemma 3.3
The set of functions from D ( h Λ ) with a compact support is acore of the operator h Λ . Proof:
To a given u ∈ D ( h Λ ) and ε > u ε ∈ D ( h Λ ) which is compactly supported to the right, i.e. it satisfiessup supp u ε < ∞ , and Z R (cid:0) | u − u ε | + | u ′′ − u ′′ ε | (cid:1) ( t ) d t ≤ ε . Given x ∈ R and d > v ∈ H , ( x, x + d ) theSobolev embedding, | v ( x ) | + | v ′ ( x ) | ≤ sup t ∈ [ x, x + d ] | v ( t ) | + | v ′ ( t ) | ≤ C Z x + dx ( | v | + | v ′′ | )( t ) d t with a constant C which depends on d but not on x . Let us take next a pairof functions, φ i ∈ C ∞ (0 , d ) , i = 1 ,
2, such that they satisfy φ (0) = φ ′ (0) = 1and φ ′ (0) = φ (0) = φ i ( d ) = φ ′ i ( d ) = 0. Denote by M i the maximum of | φ i ( t ) | + | φ ′′ i ( t ) | and put M := max { M , M } ; then it holds Z d ( | aφ + bφ | + | aφ ′′ + bφ ′′ | )( t ) d t ≤ M d ( a + b ) .
5n view of the assumption made about the function u we can find n such that R ∞ x n ( | u | + | u ′′ | )( t ) d t ≤ ˜ ε := ε/ (2 + 8 M dC ) and define u ε ( x ) := u ( x ) if x ≤ x n u ( x n +) φ ( x ) + u ′ ( x n +) φ ( x ) if x ∈ ( x n , x n + d )0 if x ≥ x n + d Then u ε belongs to D ( h Λ ) being compactly supported to the right and Z R ( | u − u ε | + | u ′′ − u ′′ ε | )( t ) d t ≤ Z ∞ x n ( | u | + | u ′′ | + | u ε | + | u ′′ ε | )( t ) d t ≤ Z ∞ x n ( | u | + | u ′′ | )( t ) d t + 8 M d ( | u ( x n ) | + | u ′ ( x n ) | ) ≤ (2 + 8 M dC ) Z ∞ x n ( | u | + | u ′′ | )( t ) d t ≤ (2 + 8 M dC )˜ ε = ε . Furthermore, one can take this function u ε and perform on it the analogousconstruction to get the support compact on the left, arriving in this way ata compactly supported ˜ u ε such that Z R (cid:0) | u − ˜ u ε | + | u ′′ − ˜ u ′′ ε | (cid:1) ( t ) d t ≤ ε , and since ε was arbitrary by assumption the lemma is proved. (cid:4) The last one is a simple observation, which is however the main tool forconversion of the proofs in the regular case to their singular counterparts.
Lemma 3.4
Let u, v ∈ D ( h Λ ) , then the Wronskian W [¯ u, v ]( x ) := ¯ u ( x ) v ′ ( x ) − ¯ u ′ ( x ) v ( x ) (3.3) is a continuous function of x on the whole real axis. Proof:
The condition αβ − γδ = − ∗ σ Λ = σ , where σ is the second Pauli matrix[6]. Then we have W [¯ u, v ]( x n +) = iU ∗ ( x n +) σ V ( x n +) = i (Λ U ( x n − )) ∗ σ Λ V ( x n − )= iU ∗ ( x n − ) σ V ( x n − ) = W [¯ u, v ]( x n − ) , (cid:4) The way in which we are going to employ this result is the following. Supposewe have real-valued functions u , v , u which are H , away from the points x n and satisfy the boundary conditions (2.3) at them. Let, in addition, W [ u , v ] be nonzero – in the applications below this will be true as u , v will be linearly independent generalized eigenfunctions of h Λ – then by thelemma the vector function y = (cid:20) u v u ′ v ′ (cid:21) − (cid:18) uu ′ (cid:19) = W [ u , v ] − (cid:18) v ′ u − v u ′ − u ′ u + u u ′ (cid:19) (3.4)is continuous everywhere including the points x n . Now we are going to demonstrate the spectral properties of H Λ announcedin the introduction. We follow the ideology used in the regular case [10,1], localizing first the essential spectrum and finding afterwards the subsetswhere it is absolutely continuous. In view of the partial wave decomposition(2.1) it is natural to start with the partial wave operators H l .The essential spectrum is stable under a rank one perturbation, henceadding the Dirichlet boundary condition at a point a > H l , h Λ we do not change their essential spectrum. Moreover, mul-tiplication by Cx − is a relatively compact operator on L ( a, ∞ ), thus theessential spectra of the said operators coincide, σ ess ( H l ) = σ ess ( h Λ ) . (4.1)With this prerequisite we can pass to our first main result. Theorem 4.1
The essential spectrum of the operator (2.1) is equal to σ ess ( H Λ ) = [inf σ ess ( h Λ ) , ∞ ) . (4.2)The idea of the proof is the same as in [10]: first we check that inf σ ess ( H Λ )cannot be smaller then inf σ ess ( h Λ ), after that we show that σ ess ( H Λ ) containsthe whole interval [inf σ ess ( h Λ ) , ∞ ). 7 roposition 4.2 Under the assumptions stated we have inf σ ess ( H Λ ) ≥ inf σ ess ( h Λ ) . (4.3) Proof: If ν > σ ess ( H Λ ) ≥ inf l inf σ ( H l ) = inf σ ess ( h Λ ) ;notice that with the exception of the case ν = 2 , l = 0 the centrifugal termin the partial waves operators (2.2) is strictly positive, and consequently, themini-max principle impliesinf σ ( H l ) ≥ inf σ ( h Λ ) = inf σ ess ( H l ) ≥ inf σ ( H l ) . (4.4)For ν = 2 the argument works again, we have just to be a little more cautiousand consider in the first partial wave the infimum over the essential spectrumonly. (cid:4) Proposition 4.3 σ ess ( H Λ ) ⊃ [inf σ ess ( h Λ ) , ∞ ] . Proof:
The idea is to employ Weyl criterion [11]. Let λ ∈ σ ess ( h Λ ) and λ >
0, then we have to show that for every ε > φ ∈ D ( H Λ ) satisfying k ( H Λ − λ − λ ) φ k ≤ ε k φ k . Basic properties of the essential spectrum together with Lemma 3.3 provideus with a compactly supported u ∈ D ( h Λ ) such that k u ′′ − λ u k ≤ ε . inview of the periodicity we may suppose that supp u ⊂ (0 , L ). Next we aregoing to estimate λ by the repulsive centrifugal potential in a suitably chosenpartial wave. Putting l R := [ √ λR ] we have1 r (cid:20) ( ν − ν − l R ( l R + ν − (cid:21) = λ + O ( R − ) for r ∈ [ R, R + L ]as R → ∞ , hence choosing R large enough one can achieve thatsup r ∈ [ R, R + L ] (cid:12)(cid:12)(cid:12)(cid:12) r (cid:20) ( ν − ν − l R ( l R + ν − (cid:21) − λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . Y ∈ S l nε and putting φ ( x ) := U − u ( | x | − R ) Y ( x/ | x | ). It holds obviously φ ∈ D ( H Λ ) , k φ k = k u ( · − R ) k , and k H Λ φ − ( λ + λ ) φ k = k H l R u ( r − R ) − ( λ + λ ) u ( r − R ) k≤ k u ′′ ( r − R ) − λ u ( r − R ) k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) r (cid:20) ( ν − ν − l R ( l R + ν − (cid:21) − λ (cid:19) u ( r − R ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε k φ k , which concludes the proof. (cid:4) Once the essential spectrum is localized, we can turn to its continuouscomponent. In view of the decomposition (2.1) we have to describe the con-tinuous spectrum in each partial wave and the results for H Λ will immediatelyfollow; recall that the essential spectrum of H l consists of the bands of theunderlying one-dimensional operator h Λ . Our strategy is to prove that thetransfer matrix — defined in the appendix, Sec. ?? below — is boundedinside the bands, which implies that the spectrum remains absolutely con-tinuous [12, 13]. The following claim is a simple adaptation of the Lemma 2from [1] to the singular case. Lemma 4.4
Let ( a, b ) be the interior of a band of the operator h Λ in L ( R ) .Let further K ⊂ ( a, b ) be a compact subinterval, c ∈ R , and x > . Thenthere is a number C > such that for every λ ∈ K any solution u of − u ′′ + cr u = λu, u ∈ D ( h Λ ) (4.5) with the normalization | u ( x ) | + | u ′ ( x ) | = 1 (4.6) satisfies in ( x , ∞ ) the inequality | u ( x ) | + | u ′ ( x ) | ≤ C . (4.7) For simplicity we allow ourselves the licence to write k f k ≡ k f ( · ) k = k f ( r ) k in thefollowing formula. roof: For a fixed λ ∈ K the equation h Λ w = λw has two real-valued,linearly independent solutions, u = u ( · , λ ) and v = v ( · , λ ), such that u , v ∈ D ( h Λ ) and the functions | u | , | u ′ | , | v | , | v ′ | are periodic, bounded,and continuous with respect to λ , cf. [8]. Without loss of generality we mayassume that the determinant of the matrix Y = (cid:20) u v u ′ v ′ (cid:21) equals one; note that u , v are real-valued and hence det Y is continuous atthe singular points in view of to the Lemma 3.4. It is also nonzero, hence toany solution u of (4.5) we can define the function y := Y − (cid:20) uu ′ (cid:21) which satisfies y ′ = Ay on every interval ( na, ( n + 1) a ) , (4.8)where the the matrix A is given by A := − cx (cid:20) u v v − u − u v (cid:21) , being integrable away of zero. By a straightforward calculation we get y = (cid:20) v ′ u − v u ′ − u ′ u + u u ′ (cid:21) and using Lemma 3.4 again we infer that y is continuous at the singularpoints. Consequently, y ( x ) = exp (cid:26) x Z x A ( t ) d t (cid:27) y ( x )is a solution of (4.8) and following [1] we arrive at the estimates12 ( | y | ) ′ ≤ | ( y, y ′ ) | ≤ k A k| y | | y ( x ) | ≤ | y ( x ) | exp (cid:26) x Z x k A k ( t ) d t (cid:27) ≤ | Y − ( x ) | exp (cid:26) ∞ Z x k A k ( t ) d t (cid:27) for x ≥ x and every solution of (4.5) with the normalization (4.6). From (cid:20) u ( x ) u ′ ( x ) (cid:21) = Y ( x ) Y − ( x ) (cid:20) u ( x ) u ′ ( x ) (cid:21) + x Z x Y ( x ) A ( t ) y ( t ) d t we then infer that the function | u ( · ) | + | u ′ ( · ) | is bounded in the interval( x , ∞ ) which we set out to prove. (cid:4) Now we are ready to describe the essential spectrum of H Λ . Theorem 4.5
For H Λ defined by (2.1) the following is true:(i) For any gap ( E k − , E k ) in the essential spectrum of h Λ ,(a) H Λ has no continuous spectrum in ( E k − , E k ) , and(b) the point spectrum of H Λ is dense in ( E k − , E k ) .(ii) On any compact K contained in the interior of a band of h Λ the spec-trum of H Λ is purely absolutely continuous. Proof: (i) By (4.1), none of the operators H l , l = 0 , , , . . . has a continu-ous spectrum in ( E k − , E k ), hence H Λ has no continuous spectrum in thisinterval either. On the other hand, the entire interval ( E k − , E k ) is con-tained in the essential spectrum of H Λ ; it follows that the spectrum of H Λ in ( E k − , E k ) consists solely of eigenvalues which are necessarily dense inthat interval.(ii) The claim follows from the previous lemma and [12, 13]. To make thearticle self-contained we prove in the appendix A a weaker result which stillguarantees the absolute continuity of the spectrum in the bands in our sin-gular case. (cid:4) The discrete spectrum
Recall that with the exception of the case ν = 2 , l = 0 the centrifugal termin the partial waves operators (2.2) is strictly positive, hence by the mini-max principle there is no discrete spectrum below E . On the other hand, inthe two-dimensional case Brown et al. noticed that regular radially periodicpotentials give rise to bound states [3] which they named in a nationalistspirit. Subsequently Schmidt [4] proved that there are infinitely many sucheigenvalues of the operator H below inf σ ess ( H Λ ). Our aim is to show thatthis result persists for singular sphere interactions considered here. Theorem 5.1
Let ν = 2 , then except of the free case the operator H Λ hasinfinitely many eigenvalues in ( −∞ , E ) , where E := inf σ ess ( H Λ ) . Proof:
The argument is again similar to that of the regular case [4], hencewe limit ourselves to just sketching it. First of all, it is clear that we have toinvestigate the spectrum of H Λ , .Let u, v be linearly independent real-valued solutions of the equation h Λ z = E z , where u is (anti)periodic — cf. Proposition 3.1. — satisfying W [ u, v ] = 1. We will search the solution of H y ≡ − y ′′ − r y = E y , we areinterested in, using a Pr¨ufer-type Ansatz, namely (cid:18) yy ′ (cid:19) = (cid:18) u vu ′ v ′ (cid:19) a (cid:18) sin γ − cos γ (cid:19) , where a is a positive function and γ is chosen continuous recalling Lemma 3.4and eq. (3.4). It is demonstrated in [4] that the function γ ( · ) and the standardPr¨ufer variable θ ( · ), appearing in (cid:18) yy ′ (cid:19) = ρ (cid:18) cos θ sin θ (cid:19) , are up to constant asymptotically equal to each other as r → ∞ . Accordingto Corollary B.3 there are then infinitely many eigenvalues below E if θ , andtherefore also γ , is unbounded from below.Now a straightforward computation yields γ ′ = − r ( u sin γ − v cos γ ) = −
14 cos γ u (cid:16) r tan γ − vr u (cid:17) . φ = ( r − tan γ − r − v/u ) satisfies γ ( r ) = φ ( r ) + O (1) as r → ∞ , and φ ′ = 1 r (cid:16) − sin φ cos φ − u sin φ − u cos φ (cid:17) = − r (cid:18) u + 14 u + sin 2 φ + (cid:16) u − u (cid:17) cos 2 φ (cid:19) (5.1)holds on R \ ∪ n { r n } with the discontinuitytan φ ( r n +) − tan φ ( r n − ) = − r n βu ( r n +) u ( r n − ) , (5.2)where β is the parameter appearing in (2.3). A direct analysis of the equation(5.1) shows that φ ′ ≤
0, and owing to (5.2) and Proposition 3.1 the corre-sponding discontinuity is strictly negative for β = 0. Hence φ is decreasingand there is a limit L = lim r →∞ φ ( r ). Suppose that L is finite. Then thecondition | R ∞ φ ′ ( t ) d t | < ∞ gives1 u ( r ) + 14 u ( r ) + sin 2 φ ( r ) + (cid:18) u ( r ) − u ( r ) (cid:19) cos 2 φ ( r ) → r → ∞ (5.3)and, as u is (anti)-periodic and φ tends to a constant, we infer that u isconstant also, not only asymptotically but everywhere. With the excep-tion of the free case this may happen only for pure repulsive δ ′ interaction, β > , α = 0 , γ = δ = 1. To finish the proof we employ eq. (5.3) again andobserve that L = π/ π ) holds necessarily. We thus find a monotonoussequence of points r n such that φ ( r n − ) < π (cid:0) (cid:2) Lπ (cid:3)(cid:1) , where [ · ] is the in-teger part. Since φ is monotonous we have φ ( r n ± ) ≥ L , hence all thesepoints belong to the same branch of the tan function. Summing then thediscontinuities (5.2) we gettan φ ( r N +) − tan φ ( r n − ) ≤ N X i = n tan φ ( r i +) − tan φ ( r i − )= − N X i = n r i βu ( r i +) u ( r i − ) , where the right-hand side diverges as N → ∞ for any β >
0, while theleft-hand side tends to a finite number tan( L ) − tan φ ( r n − ). Hence L can befinite for the free Hamiltonian only, which was to be demonstrated. (cid:4) Continuous spectra for one dimensionalSchr¨odinger operators with singular inter-actions
In this appendix we consider Schr¨odinger operators on a halfline,( H u )( x ) = − u ′′ ( x ) + V ( x ) u ( x ) (1.1) u (0) = 0 , U ( x n +) = Λ U ( x n − ) , (1.2)where we suppose that the condition Z ∞ K | u ′ | ≤ β Z ∞ K ( | H u | + | u | ) , (1.3)holds for some β, K > u ∈ D ( H ). This is obviously the case ofoperators H λ, l , where in the dimension ν > K = 0, while for ν = 2 we have to choose K > u of H u = Eu we define the transfer matrix T ( E, x, y )at energy E by T ( E, x, y ) (cid:18) u ′ ( y ) u ( y ) (cid:19) = (cid:18) u ′ ( x ) u ( x ) (cid:19) . (1.4)Our purpose is to prove the following result. Theorem A.1
Let T ( E, x, y ) be bounded on S . Then for every interval ( E , E ) ⊂ S we have ρ ac (( E , E )) > and ρ sc (( E , E )) = 0 , where ρ denotes the spectral measure associated with the operator H . Following [13] we employ the theory of Weyl m-functions. For E ∈ C + = { z, Im z > } , there is a unique solution u + ( x, E ) of H u + ( x, E ) = Eu + ( x, E )with u + ∈ L at infinity, which is normalized by u + (0 , E ) = 1. We definethe m-function by m + ( E ) = u ′ + (0 , E ) ;the spectral measure ρ is then related to it by d ρ ( E ) = 1 π lim ε ↓ Im m + ( E + iε ) , where the imaginary part at the right-hand side can be expressed asIm m + ( E ) = Im E Z ∞ | u + ( x, E ) | d x. (1.5)14t is known, see [13] and references therein, thatsupp ρ sc = n E : lim ε ↓ Im m + ( E + iε ) = ∞ o , while d ρ ac ( E ) = π Im m + ( E + i d E . Theorem A.1 is then an immediateconsequence of the following result. Theorem A.2 If T ( E, x, y ) be bounded as above and E ∈ ( E , E ) , then lim inf Im m + ( E + i > m + ( E + i < ∞ . Proof:
For x = x n we have the relations dT ( E, x, y ) d x = (cid:18) V ( x ) − E (cid:19) T ( E, x, y ) , dd y (( T ( E , x, y ) T ( E , y, x )) = ( E − E ) T ( E , x, y ) (cid:18) (cid:19) T ( E , y, x ) . It is straightforward to verify that T ( E , x, y ) T ( E , y, x ) is continuous atsingular points with respect to y and hence1 − T ( E , x, T ( E , , x ) = Z x ( E − E ) T ( E , x, y ) (cid:18) (cid:19) T ( E , y, x ) d y. Now we put E = E, E = E + iε and multiply by T ( E + iε, x,
0) from theright to get the formula T ( E + iε, x,
0) = T ( E, x, − ( iε ) Z x T ( E, x, y ) (cid:18) (cid:19) T ( E + iε, y, d y. By assumption we have k T ( E, x, y ) k ≤ C , and therefore k T ( E + iε, x, k ≤ C + ε Z x C k T ( E + iε, y, k d y , so by iteration we get k T ( E + iε, x, k ≤ Ce εCx . T = 1 so k T k = k T − k . Putting now γ = (( E + 1) β + 1) − and using the condition (1.3) we get Z ∞ | u ( x ) | d x ≥ γ Z ∞ K ( | u ( x ) | + | u ′ ( x ) | ) d x ≥ C − γ (1 + | m + | ) Z ∞ K e − εCx d x , hence by (1.5) we infer thatIm m + ≥ C − γ (1 + | m + | ) . From here the first claim follows immediately, and since2 C γ − ≥ | m + | Im m + ≥ | m + | , we get also the remaining part. (cid:4) B Oscillation theory for singular potentials
In the case of point interactions the classical oscillation theory fails due todiscontinuity of the wave functions. Nevertheless, we can employ the continu-ity of the Wronskian and formulate the oscillation theory using the approachof relative oscillations [14]. The aim of this appendix is to present briefly thebasic theorems; since the claims are the same as in the regular case we followclosely the above mentioned article.We consider Schr¨odinger-type operators on L ( l − , l + ) with the singularinteractions at the points x n ∈ ( l − , l + ) , n ∈ M ⊂ N which act as T u ( x ) = − u ′′ ( x ) + q ( x ) u ( x ) , with a real-valued potential q ∈ L ( l − , l + ) and the domain D ( T ) = (cid:26) u, u ′ ∈ AC loc ( l − , l + ) \ [ n ∈ M { x n } : T u ∈ L ( l − , l + ) and U ( x n +) = Λ n U ( x n − ) (cid:27) . H an arbitrary self-adjoint extension of it satisfying either(a) T is limit point in at least one endpoint, or(b) H is defined by separated boundary conditions.By ψ ± ( E, x ) we denote real-valued solutions of the equation T ψ ± ( E, x ) = Eψ ± ( E, x ), which satisfy the boundary conditions defining H at the points l ± , respectively. Note that such solutions may not exist, the theorems givenbelow implicitly assume their existence. In particulary, their existence isassure for energies E outside the essential spectrum. And with respect toanalyticity in spectral parameter we may use the oscillation theory also atthe edge of the essential spectrum.The first theorem to follow provides the basic oscillation result, while thecorollary of the second one is the result used in Section 5. By W ( u , u )we denote the number of zeros of the Wronskian W [ u , u ]( x ) in the openinterval ( l − , l + ), and given E < E , we put N ( E , E ) = dim Ran P ( E , E ) ,where P is a spectral measure of the self-adjoint operator H . In particular,in case of the pure point spectrum N ( E , E ) simply denotes the number ofeigenvalues in the interval ( E , E ). Theorem B.1
Suppose that E < E and put u = ψ − ( E ) , u = ψ + ( E ) .Then W ( u , u ) = N ( E , E ) . Theorem B.2
Let E < E . Assume that either u = ψ + ( E ) or u = ψ − ( E ) holds, and similarly either u = ψ + ( E ) or u = ψ − ( E ) . Then W ( u , u ) ≤ N ( E , E ) . Next we introduce Pr¨ufer variables ρ i , θ i defined by (cid:18) u i ( x ) u ′ i ( x ) (cid:19) = ρ i ( x ) (cid:18) cos θ i ( x )sin θ i ( x ) (cid:19) , where ρ i is chosen positive and θ i is uniquely determined by its boundaryvalue and the requirement that θ i is continuous on ( l − , l + ) \ S n ∈ M { x n } whileits discontinuity at the sites x n of the point interactions satisfies | θ i ( x n +) − θ i ( x n − ) | = 0 (mod π ). Corollary B.3
Suppose that E is the edge of the essential spectrum, and u = ψ − ( E ) or u = ψ + ( E ) . Then H has infinitely many eigenvalues below E if θ ( · ) is unbounded. roof: In analogy with the regular case the function θ corresponding to u = ψ ± ( E ) is bounded for negative E large enough. This implies that | θ − θ | → ∞ and since W [ u , u ]( x ) = ρ ( x ) ρ ( x ) sin( θ ( x ) − θ ( x )) we get W ( u , u ) = ∞ . Hence Theorem B.2. completes the proof. (cid:4) Acknowledgment
The research was supported by the Czech Ministry of Education, Youth andSports within the project LC06002.
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