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Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators Kaori Nagatou, Michael Plum and Mitsuhiro T. Nakao Proc. R. Soc. A 2012 468, doi: 10.1098/rspa.2011.0159 first published online 13 October 2011
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Proc. R. Soc. A (2012) 468, 545–562 doi:10.1098/rspa.2011.0159 Published online 13 October 2011
Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators BY KAORI NAGATOU1 , MICHAEL PLUM2, *
AND
MITSUHIRO T. NAKAO3
1 Institute
of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan 2 Institute for Analysis, Karlsruhe Institute of Technology, Karlsruhe 76133, Germany 3 Sasebo National College of Technology, Sasebo City, Nagasaki Prefecture 857-1193, Japan Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein’s bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account. Keywords: eigenvalue excluding; Schrödinger operator; error estimates
1. Introduction This article is concerned with one-dimensional Schrödinger equations of the form lu := −u + qu + su = lu
on R,
(1.1)
with q denoting a bounded, piece-wise continuous, r-periodic potential, and s ∈ L∞ (R) a perturbation potential satisfying s(x) → 0 as |x| → ∞. The canonical self-adjoint realization of l in L2 (R) is L : D(L) = H 2 (R) ⊂ L2 (R) −→ L2 (R), Lu = lu. We note that (at least for the more theoretical parts of our paper) the assumptions on q and s could be weakened; e.g. Teschl (2009, section 9.7). Our assumptions however are better suited for a direct use of the classical reference Eastham (1973), and furthermore for our detailed estimates in §6. It is well known that the unperturbed problem (s ≡ 0) has purely continuous spectrum sess , which is arranged in spectral bands (i.e. it is the union of countably many compact real intervals); see theorem 2.1 below for details. The perturbation *Author for correspondence (
[email protected]). Received 9 March 2011 Accepted 31 August 2011
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s, due to its decay at ±∞, does not change the continuous (or essential) spectrum sess , but it may or may not generate additional eigenvalues below the lowest spectral band or in the spectral gaps between the spectral bands. The actual knowledge about the presence or absence of such additional eigenvalues is of high importance in many physical applications, where s models a ‘doping’ of a periodic structure modelled by q. Below the lowest spectral band, the Rayleigh–Ritz method (based on Poincaré’s min–max principle) gives an easy access to numerically computing very accurate approximations to eigenvalues (if there are any), and also to computing at least upper bounds to them (which are ‘safe’ in a strict mathematical sense). For the latter purpose, assuming that we have numerically computed approximate eigenpairs (l˜ k , u˜ k ) (k = 1, . . . , N ), u˜ 1 , . . . , u˜ N ∈ H 1 (R) linearly independent, l˜ 1 ≤ l˜ 2 ≤ · · · ≤ l˜ N < inf sess , we can simply estimate min
U ⊂H 1 (R)subspace
max
u∈U \{0}
dim U =k
Lu, u
Lu, u
≤ max u, u L2 (R) u∈span{u˜ 1 ,...,u˜ k } u, u L2 (R)
(1.2)
u ≡0
(with Lu, u understood in the quadratic form sense). The computation of the right-hand side of (1.2) amounts to a finite-dimensional matrix eigenvalue problem, the eigenvalues of which can easily be safely enclosed by means of verifying numerical linear algebra; e.g. Rump (1999). So if (for k = N ) the right-hand side of (1.2) turns out to be less than inf sess by these verified matrix eigenvalue computations, then we can conclude by Poincaré’s min–max principle that there are at least N eigenvalues of problem (1.1) below inf sess , and that upper bounds to the N smallest of them are given by the matrix eigenvalues. Note however that this approach does not give lower eigenvalue bounds, and that, in particular, we cannot safely identify regions which do not contain eigenvalues. Lower eigenvalue bounds can be obtained by the Lehmann–Goerisch method (Behnke & Goerisch 1994) if some spectral pre-information is available, but we do not further address this method here since anyway we are mainly interested in the question of presence or absence of eigenvalues in spectral gaps. In gaps, the min–max principle is no longer available, and thus the above approach breaks down. A useful generalization of Poincaré’s ‘classical’ min–max principle to a min–max principle in a spectral gap (a, b) has been proposed in Dolbeault et al. (2000) and in Griesemer & Siedentop (1999). Under suitable assumptions, its application to our situation states that, if the left gap-endpoint a is not an accumulation point of gap eigenvalues, then the nth of these (counting from left to right, and regarding multiplicity) is given by inf
U ⊂L+ (H 2 (R)) subspace u∈U dim U =n
sup
L− (H 2 (R)) u ≡0
Lu, u L2 (R) u, u L2 (R)
,
(1.3)
with L− and L+ denoting the spectral projections of the unperturbed operator (with s ≡ 0) for the intervals (−∞, a] and (a, ∞), respectively. Actually, the min–max principle proposed in Dolbeault et al. (2000) and in Proc. R. Soc. A (2012)
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Griesemer & Siedentop (1999) is much more general and has been proved to be very useful, e.g. for Dirac operators with a potential, but it cannot be used to establish a Rayleigh–Ritz-type procedure for computing upper eigenvalue bounds (like the one described above for eigenvalues below inf sess ). The main reason is that the space U ⊕ L− (H 2 (R)) over which the supremum in (1.3) is taken is infinite-dimensional, and hence the upper bound (extracted from (1.3)) sup
u∈U
L− (H 2 (R)) u ≡0
Lu, u L2 (R) u, u L2 (R)
for the nth gap eigenvalue, with U ⊂ L+ (H 2 (R)) now denoting some specific n-dimensional subspace (e.g. using numerical eigenfunction approximations), amounts to an eigenvalue problem which is still infinite-dimensional and thus refuses a verifying numerical access. Furthermore, choosing such a specific subspace U is problematic in practice since it needs to lie in L+ (H 2 (R)). Obtaining safe lower eigenvalue bounds from (1.3) amounts to the same difficulty since also L+ (H 2 (R)) is infinite-dimensional. And again no regions containing no eigenvalue can be safely identified. Another variational characterization of eigenvalues in a spectral gap (a, b) has been proposed in Zimmermann & Mertins (1995). Starting from some r ∈ [a, b] which is not an eigenvalue, and introducing a local ordering − ≤ · · · ≤ l1− < r < l1+ ≤ l2+ ≤ · · · ≤ ln+ ≤ · · · < b a < · · · ≤ ln− ≤ ln−1
of gap eigenvalues, variational characterizations for lk− and for lk+ are given, which lead to lower bounds for lk− and upper bounds for lk+ , again by computing matrix eigenvalues like in the Rayleigh–Ritz approach. But no respective reverse bounds are available. Repeating this process with different choices of r, one gets a ‘good evidence’ of the location of gap eigenvalues, but no safe proof of eigenvalue enclosures unless the number of eigenvalues in (a, b) is known a priori which is usually not the case. In particular, again no regions which are free of eigenvalues can safely be identified. A variety of articles is concerned with asymptotics of gap eigenvalues, e.g. with the asymptotic number of eigenvalues in a given subinterval of the gap as the coupling constant c (when the perturbation s has the form s(x) = cs0 (x)) tends to infinity (Deift & Hempel 1986; Hempel 1989; Birman 1990, 1991; Sobolev 1991) or as an endpoint of the subinterval tends to an endpoint of the gap, considering in particular the question of accumulation of eigenvalues at this gap edge (Schmidt 2000; Krüger & Teschl 2008, 2009). For fast-decaying perturbations s, it is shown by Gesztesy & Simon (1993) that sufficiently high gaps contain at most two or precisely one gap eigenvalue. However, by their nature, these kinds of asymptotic results do not give information about the presence or absence of eigenvalues in a specific subregion of the gap for a specific perturbation potential. In view of these questions left open by variational gap eigenvalue characterizations and by asymptotic results, it is comparatively easy to obtain safe eigenvalue enclosures if one is not interested in indexing and multiplicity Proc. R. Soc. A (2012)
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˜ u), questions: starting from an approximate eigenpair (l, ˜ with u˜ ∈ H 2 (R)\{0}, D. Weinstein’s bound states that the interval [l˜ − d, l˜ + d],
with d :=
Lu˜ − l˜ u ˜ L2 (R)
u ˜ L2 (R)
,
(1.4)
contains at least one spectral point; the proof—by spectral decomposition—is very simple. So if we can make sure that this interval does not intersect with the essential spectrum, we obtain an eigenvalue enclosure. But of course no eigenvalue-free regions can be obtained by this approach as well. So there seems to be no method available which is suitable for computing regions in spectral gaps which can be guaranteed to contain no eigenvalue. But under many aspects, the knowledge of such regions is certainly desirable. In this article, we are proposing such an eigenvalue excluding method, which heavily relies on computer assistance. Nevertheless, it is completely rigorous in the strict mathematical sense, because all possible numerical errors are bounded in an appropriate way (e.g. rounding errors are captured by interval arithmetic). Our method uses a fixed-point formulation for u in (1.1), with some l or some ‘narrow’ interval L given, which is equivalent to the given problem (1.1), and by some kind of contraction argument we can show that this fixed-point problem has only the trivial solution u ≡ 0, whence l is no eigenvalue, or L contains no eigenvalue, respectively. Larger eigenvalue-free regions can then be computed by subdivision into ‘narrow’ intervals. The contraction argument needs a balance between a finite-dimensional projection of L − l and a corresponding projection error bound. Very roughly speaking, the product between the projection error bound and the inverse of this finite-dimensional projection (which can be computed by means of verifying numerical linear algebra) needs to be small enough. We wish to emphasize the following fact: if the given l in the gap (a, b) is no eigenvalue (or the given closed narrow interval L ⊂ (a, b) contains no eigenvalue, respectively), then our method will in principle always be able to prove this fact if the numerical effort is taken large enough (by refining the projection), because then we can (in principle) make the projection error bound as small as we like, while the inverse of the finite-dimensional projection stays stable under refinement. Of course, there are practical limitations to this statement, paying regard to limited computer power. One may ask if there is a generalization of our approach to higher dimensional Schrödinger equations. So far we are only able to state that the principles of our method can indeed be generalized, but we need rather detailed bounds for the inverse operator (L0 − l)−1 , with L0 denoting the unperturbed periodic operator, and l in a spectral gap. For example, we need (computable!) bounds for (L0 − l)−1 (su) L2 (Rn \UM ) which are small if the bounded ‘computational’ subdomain UM is chosen large. In the one-dimensional situation treated in this paper, we can obtain such bounds via the Green’s function which is composed of two fundamental solutions, and these can be enclosed by the method described in Nagatou (2008). This approach is clearly restricted to the one-dimensional case, and would have to be replaced by ‘something else’ in the higher dimensional case, which is not at hand so far. Proc. R. Soc. A (2012)
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2. Preliminaries We recall the structure of the essential spectrum of L. At first, the essential spectrum of the operator L0 u := −u + q(x)u,
D(L0 ) = H 2 (R)
is obtained using the result by Eastham (1973, ch. 2): Theorem 2.1. Consider the following two eigenvalue problems: I. Periodic eigenvalue problem: −u + q(x)u = lu, u(0) = u(r),
u (0) = u (r)
II. Anti-periodic eigenvalue problem: −u + q(x)u = mu, u(0) = −u(r),
u (0) = −u (r).
Then for the eigenvalues {ln }, {mn }, we have l0 < m0 ≤ m1 < l1 ≤ l2 < m2 ≤ m3 ≤ · · · , and the essential spectrum of L0 is obtained as the union of the intervals [l2m , m2m ], [m2m+1 , l2m+1 ]
m = 0, 1, 2, . . . .
Moreover, L is a relatively compact perturbation of L0 : given any sequence (un ) in H 2 (R) s.t. both (un ) and (L0 un ) are bounded in L2 (R), we find that (un ) is bounded in H 2 (R). Inductively, for each bounded subinterval [−k, k] (k ∈ N), we can choose a subsequence that converges in L2 (−k, k), by the Sobolev– Kondratiev–Rellich embedding theorem. A diagonal subsequence (unj ) converges in L2 (−k, k) for every k ∈ N. Using s → 0 as |x| → ∞, we find that (sunj ) is convergent in L2 (R). Therefore, the essential spectra of L and L0 coincide (cf. Kato 1995, theorem 5.35 in ch. IV).
3. Fixed-point formulation For a real number l ∈ / sess (L0 ) = sess (L), consider the linear equation u ∈ H 2 (R), (L − l)u = 0,
on R.
(3.1)
Our intention is to derive explicit conditions, by the method described in the following, implying that (3.1) has the unique solution u ≡ 0, which then obviously proves that l is not an eigenvalue of L. Using the same method, and regarding that we have to use interval arithmetic anyway for our verified computations, we can replace the real number l in (3.1) by a (narrow) real interval L not intersecting sess (L0 ), and thus prove that L does not contain any eigenvalue. Larger intervals can then be shown to be free of eigenvalues by subdivision into such narrow intervals L. Proc. R. Soc. A (2012)
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Since the inverse of L0 − l exists if l ∈ / sess (L0 ), we have (L − l)u = 0 ⇔ (L0 − l)u + su = 0 ⇔ u = (l − L0 )−1 (su).
(3.2)
This way of characterizing eigenvalues of L is in fact a variant of the well-known Birman–Schwinger principle (Reed & Simon 1978). To get access to the resolvent operator (l − L0 )−1 , we first express it using the Green’s function: By Floquet theory there exist, for l ∈ / sess (L0 ), fundamental solutions j1 (x), j2 (x) of (L0 − l)j = 0 s.t. j1 (x) = emx p1 (x)
and
j2 (x) = e−mx p2 (x),
(3.3)
where the characteristic exponent m has non-zero real part and p1 , p2 are rperiodic functions. See Eastham (1973, sections 1.1 and 2.3). Without loss of generality, we can assume that t := Re(m) is positive. (Concerning how to compute verified enclosures for the fundamental solutions j1 and j2 for (L0 − l)j = 0, see Nagatou 2008.) Using these fundamental solutions, the Green’s function G(x, y, l) is defined for −∞ < x, y < ∞ by ⎧ j1 (x)j2 (y) ⎪ ⎪ ⎨ (x ≤ y) W G(x, y, l) = (3.4) ⎪ j (x)j1 (y) ⎪ ⎩ 2 (x ≥ y), W where W := j1 (x)j2 (x) − j1 (x)j2 (x) stands for the Wronskian, which by simple calculations can be seen to be constant. See also Eastham (1973, section 5.3). Using this Green’s function, we have the usual representation for the resolvent operator (l − L0 )−1 : L2 (R) −→ D(L0 ) = H 2 (R) (see Eastham 1973): −1 (l − L0 ) f (x) = G(x, y, l)f (y) dy. R
Defining the operator Fl : L (R) −→ H 2 (R) ⊂ L2 (R) by Fl (u)(x) := G(x, y, l)s(y)u(y) dy (x ∈ R, u ∈ L2 (R)), 2
R
(3.5)
and using (3.2), we therefore find that (3.1) is equivalent to the fixed-point equation (3.6) u ∈ L2 (R), u = Fl (u). For the desired eigenvalue excluding, we are left to show that this fixed-point equation has the unique solution u ≡ 0. Remark 3.1. (a) Using the Rellich–Kondratiev embedding theorem on compact subintervals of R, and the decay of s, one can prove that Fl : L2 (R) −→ L2 (R) is compact, which however is not needed for our arguments. (b) Of course (3.6) has the unique solution u ≡ 0 if Fl < 1. In fact, we could try to prove this, using the explicit Green’s function bounds derived in §6, if we were willing to impose a smallness condition on the perturbation function s. Proc. R. Soc. A (2012)
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This is however not what we want to do, since we are aiming at a method that ‘always’ works (in principle) if l is in fact no eigenvalue and if the numerical effort is taken large enough. (See the corresponding remark in the introduction, and also remark 6.2a after theorem 6.1.) (c) By Eastham (1973) (section 1.2, case C and section 2.3, theorem 2.3.1), Re(m) = 0 holds if l ∈ sess (L0 ). This implies that by enclosing the characteristic exponent m in a (complex) set guaranteeing that Re(m) = 0, which is performed in the process of enclosing the fundamental solutions j1 and j2 by the method described in Nagatou (2008), the condition l ∈ / sess (L0 ) is proved at this stage and thus need not be checked a priori. Note that we will need these enclosures for j1 and j2 anyway to obtain the required Green’s function bounds discussed in §6.
4. Projection and interpolation For some large M > 0, we split the real line into the bounded interval UM := ˜ M := R\UM . [−M , M ], the ‘computational domain’, and the rest U Defining the projections PM : H 1 (R) −→ H 1 (UM ) and ˜ M) P˜ M : H 1 (R) −→ H 1 (U by PM (v) = v|UM and P˜ M (v) = v|U˜ M , we decompose (3.6) into the bounded interval part and the remainder: PM u = PM Fl (u) (4.1) and P˜ M u = P˜ M Fl (u). For treating the first equation in (4.1) numerically, we need to further split it into a finite- and an infinite-dimensional part: Let Sh (UM ) denote the set of continuous and piece-wise linear functions on UM with respect to the uniform mesh −M = x0 < x1 < · · · < xN = M and mesh size h. Let P : H 1 (UM ) −→ Sh (UM ) be the associated piece-wise linear interpolation operator. So we get the following equivalent splitting of the first equation in (4.1): PPM u = PPM Fl (u) (4.2) and (I − P)PM u = (I − P)PM Fl (u). The following error estimation holds for P which will help us to control the second equation in (4.2): Lemma 4.1. For f ∈ W 2,∞ (UM ),
f − Pf L∞ (UM ) ≤ 18 h 2 f L∞ (UM ) . For the proof in the classical case, i.e. f ∈ C 2 (UM ), see Schultz (1973). The proof generalizes in a straightforward way to f ∈ W 2,∞ (UM ). Proc. R. Soc. A (2012)
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5. Newton-like method and verification condition The system to be further considered consists of the two equations in (4.2) and the second equation in (4.1). Later, we will treat the second equations in (4.1) and (4.2), respectively, by suitable remainder term bounds. The first equation in (4.2) forms the numerical core and is captured by the Newton-like operator ˜ M ) −→ Sh (UM ), Nl : H 1 (UM ) × H 1 (U
u Nl M := PuM − [I − PPM Fl E0 ]−1 M P(uM − PM Fl (u)), u˜ M
where
u u := E M u˜ M
uM := u˜ M
on UM 2 ˜ M ∈ L (R) on U
(5.1)
˜ M ) −→ L2 (R), and E0 v := E( v0 ). Here, we have assumed with E : H 1 (UM ) × H 1 (U that the operator [I − PPM Fl E0 ]|Sh (UM ) : Sh (UM ) −→ Sh (UM ) has an inverse [I − PPM Fl E0 ]−1 M . The validity of this assumption is checked in the actual verified numerical computations, as explained later. ˜ M ) −→ H 1 (UM ) × H 1 (U ˜ M ) by We next define the operator Tl : H 1 (UM ) × H 1 (U ⎛
⎞
uM N + (I − P)PM Fl (u)⎠ u Tl M := ⎝ l u˜ M , (5.2) u˜ M ˜ PM Fl (u) where again u is defined by (5.1). Then we have the following equivalence relation:
u uM (5.3) = Tl M ⇐⇒ u = Fl (u), u˜ M u˜ M u
P u
where u and (uM , u˜ M ) are related via (5.1) or ( u˜ MM ) = ( P˜ M u ), respectively. M Our purpose is to show that ( 00 ) is the only fixed point of Tl in H 1 (UM ) × ˜ M ), and therefore, by the equivalence of (3.1) and (3.6), l is no eigenvalue H 1 (U of L. Given positive real numbers a, b− and b+ , a candidate set U ⊂ H 1 (UM ) × 1 ˜ H (UM ) is chosen as
UM + [a] U := , (5.4) [b] where UM := {vh ∈ Sh (UM )| vh L∞ (UM ) ≤ 1},
(5.5)
M M ∈ (I − P)(H 1 (UM ))| v⊥
L∞ (UM ) ≤ a}, [a] := {v⊥
(5.6)
˜ M )| v [b] := {v˜ ∈ H 1 (U ˜ L2 (−∞,−M ) ≤ b− , v ˜ L2 (M ,∞) ≤ b+ }.
(5.7)
˜ M ), but this fact does not affect Note that U is not bounded in H 1 (UM ) × H 1 (U our following argument. The constant 1 in (5.5) is just a normalization and could be replaced by any other fixed positive constant. Proc. R. Soc. A (2012)
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Theorem 5.1. Let a, b− , b+ > 0 be given. Suppose that for all ( u˜ MM ) ∈ U , and u defined by (5.1), Nl uM < 1, (5.8) u˜ M L∞ (U ) M
(I − P)PM Fl (u) L∞ (UM ) < a,
(5.9)
P˜ M Fl (u) L2 (−∞,−M ) < b− ,
(5.10)
P˜ M Fl (u) L2 (M ,∞) < b+
(5.11)
u
hold. Then ( u˜ MM ) = ( 00 ) is the unique fixed point of Tl . u
Proof. Assume that ( u˜ MM ) is a non-trivial fixed point of Tl . Then at least one of the four functions PuM , (I − P)uM , u˜ M |(−∞,−M ) , u˜ M |(M ,∞) is not identically zero, whence there exists a maximal c > 0 with the properties 1
PuM L∞ (UM ) ≤ , c
(I − P)uM L∞ (UM ) ≤
a c
and
u˜ M L2 (−∞,−M ) ≤
b− , c
u˜ M L2 (M ,∞) ≤
b+ . c
Define wM = cuM , w˜ M = c u˜ M and w := E(wM , w˜ M ). Therefore,
PwM L∞ (UM ) ≤ 1,
w˜ M L2 (−∞,−M ) ≤ b− ,
and
(I − P)wM L∞ (UM ) ≤ a
(5.12)
w˜ M L2 (M ,∞) ≤ b+ , w
with equality in at least one of these four inequalities. In particular, ( w˜ MM ) ∈ U . w Since Tl is linear, ( w˜ MM ) is a fixed point of Tl , too. Hence, by (5.2) and (5.8)–(5.11),
w M
PwM L∞ (UM ) = P Nl w˜ M + (I − P)PM Fl (w) ∞ L (UM ) wM = < 1, Nl w˜ M ∞ L (UM )
wM ∞ + (I − P)PM Fl (w)
(I − P)wM L (UM ) = (I − P) Nl ∞ w˜ M L (U ) M
= (I − P)PM Fl (w) L∞ (UM ) < a,
w˜ M L2 (−∞,−M ) = P˜ M Fl (w) L2 (−∞,−M ) < b− and
w˜ M L2 (M ,∞) = P˜ M Fl (w) L2 (M ,∞) < b+ ,
hold. This contradicts the equality in at least one of four inequalities in (5.12). Proc. R. Soc. A (2012)
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6. How to find a, b− and b+ Our aim is to find positive real numbers a, b− and b+ which satisfy the conditions (5.8)–(5.11), with U given by (5.4)–(5.7), respectively, to find a sufficient condition for their existence. In order to bound the left-hand sides of (5.8)–(5.11), we first use (3.5) and ˜ M ) and v := E(vM , v˜M ), (3.4) to obtain, for every (vM , v˜M ) ∈ H 1 (UM ) × H 1 (U j2 (x) x j1 (x) ∞ j1 (y)s(y)v(y) dy + j2 (y)s(y)v(y) dy, (6.1) Fl (v)(x) = W −∞ W x whence for x ∈ UM |Fl (v)(x)| ≤ Z1 (x) vM L∞ (UM ) + Z2 (x) v˜M L2 (−∞,−M ) + Z3 (x) v˜M L2 (M ,∞) (6.2) with |j2 (x)| Z1 (x) := |W | |j2 (x)| Z2 (x) := |W | and
|j1 (x)| Z3 (x) := |W |
x −M
|j1 (x)| |j1 (y)||s(y)| dy + |W |
−M
−∞ ∞
M |j2 (y)||s(y)| dy, x
|j1 (y)|2 s(y)2 dy
|j2 (y)|2 s(y)2 dy.
M
Furthermore, (6.1) gives
P˜ M Fl (v) L2 (−∞,−M ) ≤ G1 (v) L2 (−∞,−M ) + G2 (v) L2 (−∞,−M ) ,
(6.3)
P˜ M Fl (v) L2 (M ,∞) ≤ G1 (v) L2 (M ,∞) + G2 (v) L2 (M ,∞) ,
(6.4)
and
with G1 (v) and G2 (v) denoting the first and the second summand in (6.1), respectively. By the Green’s function estimates in Eastham (1973, section 5.3), we obtain, using t = Re(m), cp1 := p1 L∞ (R) , cp2 := p2 L∞ (R) (see (3.3)), and K := cp1 cp2 /(t|W |):
G1 (v) L2 (−∞,−M ) ≤ K s L∞ (−∞,−M ) v˜M L2 (−∞,−M ) , (6.5) and
G2 (v) L2 (M ,∞) ≤ K s L∞ (M ,∞) v˜M L2 (M ,∞) , as well as
1
s L∞ (M ,∞) v˜M L2 (M ,∞) 2 1 t
s 2L∞ (M ,∞) v˜M 2L2 (M ,∞) + |G1 (v)(M )|2 , + 4 2
G1 (v) L2 (M ,∞) ≤ K
Proc. R. Soc. A (2012)
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and
G2 (v) L2 (−∞,−M ) ≤ K +
1
s L∞ (−∞,−M ) v˜M L2 (−∞,−M ) 2
t 1 2 2 2
s L∞ (−∞,−M ) v˜M L2 (−∞,−M ) + |G2 (v)(−M )| . 2 4
Finally, |G1 (v)(M )| ≤ Z1 (M ) vM L∞ (−M ,M ) + Z2 (M ) v˜M L2 (−∞,−M ) and
(6.6)
|G2 (v)(−M )| ≤ Z1 (−M ) vM L∞ (−M ,M ) + Z3 (−M ) v˜M L2 (M ,∞) .
(6.7)
For example, in the special case s(x) = ce−x (with a coupling constant c ∈ R) treated in our numerical example, we obtain by direct calculation: 2 √ 2 (6.8) |Z1 (x)| ≤ |c|K min(1, te−t|x| et /4 p) + e−x 1 − e−t(M −|x|) , t −M 2 −t(M +x) |Z2 (x)| ≤ |c|K (6.9) e 2 t −M 2 −t(M −x) and |Z3 (x)| ≤ |c|K , (6.10) e 2 2
which makes the bounds (6.2)–(6.4) for Fl (v) completely explicit. Note that an upper bound for K and bounds for t can be computed using the enclosures for j1 , j2 obtained by the method described in Nagatou (2008).
(a) Sufficient condition for (5.10) and (5.11) The inequalities (6.3) and (6.5)–(6.7) imply, for v ∈ U (noting that
vM L∞ (−M ,M ) ≤ PvM L∞ (−M ,M ) + (I − P)vM L∞ (−M ,M ) ≤ 1 + a), 1
P˜ M Fl (v) L2 (−∞,−M ) ≤ K s L∞ (−∞,−M ) b− + s L∞ (−∞,−M ) b− 2 1 t 2
s 2L∞ (−∞,−M ) b− + + (Z1 (−M )(1 + a) + Z3 (−M )b+ )2 4 2 which gives the following sufficient condition for (5.10): 2 1 (1 − K s L∞ (−∞,−M ) )(1 − 2K s L∞ (−∞,−M ) ) · b− Z1 (−M ) < tK − Z3 (−M )b+ − Z1 (−M )a. (6.11) Similarly, a sufficient condition for (5.11) is obtained as 2 1 Z1 (M ) < (1 − K s L∞ (M ,∞) )(1 − 2K s L∞ (M ,∞) ) · b+ tK − Z2 (M )b− − Z1 (M )a. Proc. R. Soc. A (2012)
(6.12)
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(b) Sufficient condition for (5.9) For every
u ( u˜ MM
u
) ∈ U , and u = E( u˜ MM ), using lemma 4.1, we have 2 d h2
(I − P)PM Fl (u) L∞ (UM ) ≤ P F (u) M l ∞ 8 dx 2 L (UM ) 2 h2 PM d Fl (u) = . ∞ 8 dx 2 L (UM )
(6.13)
Setting f := Fl (u) = −(L0 − l)−1 (su), we have (L0 − l)f = −su, i.e. f = (q − l)f + su. Therefore, we obtain, using (5.5) and (5.6), 2 PM d Fl (u) = PM (q − l)Fl (u) + PM (su) L∞ (UM ) ∞ 2 dx L (UM ) ≤ q − l L∞ (UM ) PM Fl (u) L∞ (UM ) + s L∞ (UM ) uM L∞ (UM ) ≤ q − l L∞ (UM ) PM Fl (u) L∞ (UM ) + s L∞ (UM ) (1 + a).
(6.14)
Here, we can estimate PM Fl (u) L∞ (UM ) using (6.2), and again (5.5)–(5.7):
PM Fl (u) L∞ (UM ) ≤ Z1 L∞ (UM ) uM L∞ (UM ) + Z2 L∞ (UM ) u˜ M L2 (−∞,−M ) + Z3 L∞ (UM ) u˜ M L2 (M ,∞) ≤ Z1 L∞ (UM ) (1 + a) + Z2 L∞ (UM ) b− + Z3 L∞ (UM ) b+ .
(6.15)
Combining (6.13)–(6.15), we obtain the following sufficient condition for (5.9). h2 [ q − l L∞ (UM ) Z1 L∞ (UM ) + s L∞ (UM ) ] 8
h2 h2 < 1 − q − l L∞ (UM ) Z1 L∞ (UM ) − s L∞ (UM ) a 8 8 −
h2
q − l L∞ (UM ) { Z2 L∞ (UM ) b− + Z3 L∞ (UM ) b+ }. 8
(6.16)
Using explicit upper bounds for Z1 , Z2 , Z3 (e.g. (6.8)–(6.10) in our example 2 case s(x) = ce−x ), conditions (6.11), (6.12), (6.16) (which imply (5.9)–(5.11)) can together obviously be written in the form b− (component-wise inequality) (6.17) b < A b+ a with a matrix A ∈ R3×3 which is invertible (and has a non-negative inverse A−1 ) if M is sufficiently large and h is sufficiently small, and a positive vector b ∈ R3 Proc. R. Soc. A (2012)
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which is ‘small’ under the same assumptions on M and h. We can satisfy condition (6.17) choosing b− b+ := (1 + 3)A−1 b (6.18) a with a ‘theoretical’ sufficiently small 3 > 0 specified later, provided that A−1 b
is component-wise positive,
which can easily be checked by solving a 3 × 3 system using verified numerics. The remaining condition to be satisfied is (5.8), which we will now analyse using a, b− , b+ from (6.18).
(c) Sufficient condition for (5.8) u
Simple calculations show that, for every ( u˜ MM ) ∈ U ,
uM uM − PuM −1 = [I − PPM Fl E0 ]M PPM Fl E , Nl u˜ M u˜ M and thus, with {4j }N j=0 denoting the usual hat function basis of Sh (UM ), f := Nl
uM u˜ M
=
N
fj 4 j
(6.19)
j=0
solves the problem
uM − PuM , P[I − PM Fl E0 ]f = PPM Fl E u˜ M which amounts to the linear algebraic system Df = r
(6.20)
N for f = (fj )N j=0 , with D = (Dij )0≤i,j≤N and the vector r := (rj )j=0 given by
xi
j2 (xi )j1 (y) Dij := dij − s(y)4j (y) dy − W −M
u − PuM and ri := Fl E M (xi ). u˜ M
M xi
⎫ ⎪ j1 (xi )j2 (y) ⎪ s(y)4j (y) dy ⎪ ⎪ ⎬ W ⎪ ⎪ ⎪ ⎪ ⎭ (6.21)
Below, we will derive bounds for r0 , . . . , rN of the form |ri | ≤ (1 + 3)Ri Proc. R. Soc. A (2012)
(i = 0, . . . , N )
(6.22)
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with 3 coming from (6.18). So the solution f of (6.20) is given by f = (1 + 3)f(0) , where f(0) can be enclosed by solving the interval linear system Df(0) = [R]
(6.23)
with [R] = ([−Ri , Ri ])N i=0 . The system (6.23) is solved by a verifying linear algebraic solver (contained in the interval MATLAB toolbox INTLAB; Rump 1999), which encloses the solution vector f(0) and, as a by-product, proves the invertibility of the system matrix D. Since this invertibility is equivalent to the invertibility of the operator [I − PPM Fl E0 ]|Sh (UM ) , the validity of the corresponding assumption made after (5.1) is confirmed at this stage. By using (6.19) and the property of the hat functions 4j , we have (0) Nl uM = max |f | = (1 + 3) max |fj | u˜ M L∞ (U ) 0≤j≤N j 0≤j≤N M
(0)
whence a sufficient condition for (5.8) is (1 + 3) max0≤j≤N |fj | < 1, i.e. when 3 > 0 is chosen small enough, (0) max |fj | < 1, 0≤j≤N
which can directly be checked using the computed enclosure for f(0) . u Now we show how to get R0 , . . . , RN in (6.22). For fixed ( u˜ MM ) ∈ U and i = 0, . . . , N , we have by (6.2) and (5.6), (5.7)
u − Pu M M (xi ) |ri | = Fl E u˜ M ≤ Z1 (xi ) uM − PuM L∞ (UM ) + Z2 (xi ) u˜ M L2 (−∞,−M ) + Z3 (xi ) u˜ M L2 (M ,∞) ≤ Z1 (xi ) · a + Z2 (xi ) · b− + Z3 (xi ) · b+ , i.e. using (6.18) and
⎛
⎞ Z2 (x0 ) Z3 (x0 ) Z1 (x0 ) .. .. ⎠ ∈ R(N +1)×3 , Z := ⎝ ... . . Z2 (xN ) Z3 (xN ) Z1 (xN )
(6.24)
we obtain the desired bounds (6.22): ⎞ ⎛ ⎞ ⎛ R0 |r0 | ⎝ ... ⎠ ≤ (1 + 3)ZA−1 b =: (1 + 3) ⎝ ... ⎠. |rN | RN For the practical computation of R0 , . . . , RN , the components of Z are of course replaced by their known upper bounds (e.g. (6.8)–(6.10) in our example case). So we have arrived at our main theorem: Theorem 6.1. Let l ∈ / sess (L0 ) (or let L be some ‘narrow’ closed interval with empty intersection with sess (L0 ), respectively). Choose some ‘large’ M > 0 and Proc. R. Soc. A (2012)
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some ‘small’ h > 0. Depending on l (respectively L), M and h, let b ∈ R3 and A ∈ R3×3 denote the vector and the matrix generated by the left-hand sides and the right-hand sides, respectively, of (6.11), (6.12), (6.16), and suppose that A−1 b is (component-wise) positive. Let Z be defined by (6.24), (R0 , . . . , RN )t := ZA−1 b, and [R] = [−Ri , Ri ]N i=0 . Suppose that a verified algorithm for enclosing the solution of the interval linear system Df (0) = [R] can be carried out successfully, and that the solution enclosure guarantees (0)
max |fj | < 1.
0≤j≤N
(6.25)
Then l is no eigenvalue of (3.1) (or L does not contain any eigenvalue of (3.1), respectively). Remark 6.2. (a) If M is chosen sufficiently large and h sufficiently small, the numbers R0 , . . . , RN are also ‘small’. So if the matrix D is not too close to being singular, we will be able to verify the crucial condition (6.25). Since the matrix D is a projection of I − Fl (see (6.21)) and thus ‘close to’ I − Fl if M is large and h is small, it will be not too close to being singular, unless l is an eigenvalue of L (or L contains an eigenvalue of L, respectively). So if some (narrow) closed interval L is free of eigenvalues, our method will in principle always be able to prove this, if the numerical effort is taken high enough (by choosing M large enough and h small enough, and possibly the computer arithmetic accurate enough). (b) As mentioned earlier already, larger intervals L can be treated by subdivision into narrow subintervals. Thus, in the above sense, our method will in principle again always be successful. (c) If l (respectively L) is too close to a true eigenvalue of L, or to the essential spectrum, our method will however fail in practice, since then D is too close to a singular matrix, and hence the required conditions cannot be verified within the practically available numerical framework. In particular, we can never go up to the gap-endpoints with our excluding technique. (d) By making use of INTLAB (Rump 1999), all rounding errors are taken into account, i.e. all computations until checking the final condition (6.25) are rigorous and guarantee the correctness of the result in the strict mathematical sense. Remark 6.3. In order to get an idea in which parts of the spectral gap there is a chance to prove absence of eigenvalues by our method, i.e. which subintervals L should be checked, usually some approximate computations are needed first which localize possible eigenvalues (in the sense of ‘numerical evidence’, i.e. without proof). These approximate computations need some care, in order to avoid the problem of ‘Spectral Pollution’. (See Davies & Plum 2004; Boulton & Levitin 2007; Teschl 2008.)
7. Numerical examples We exclude eigenvalues of L in spectral gaps by the method proposed in this paper for the case q(x) = a · cos(2px), a ∈ R. Then we obtain approximate {mi } Proc. R. Soc. A (2012)
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K. Nagatou et al. Table 1. Eigenvalue-free intervals.
a and c
L
the first spectral gap (approximations)
a = 3, c = 1 a = 3, c = 2 a = 5, c = 1
[8.822, 10.585] [9.0, 9.35] [7.55, 11.335]
[8.341644,11.340563] [8.341644,11.340563] [7.292924,12.287917]
Table 2. Detailed exclusion results for the case a = 5, c = 1. (0)
N +1
the width of Lk
Lk
max0≤j≤N |fj |
400
0.01
[7.54,7.55] [7.55,7.56] [7.56,7.57] [7.57,7.58] [7.58,7.59] ··· [7.74,7.75]
1.580639 > 1 0.932369 0.651513 0.495527 0.396753 ··· 0.823996
400
0.02
[7.75,7.77] [7.77,7.79] [7.79,7.81] [7.81,7.83] [7.83,7.85] [7.85,7.87] ··· [8.83,8.85] [8.85,8.87] [8.87,8.89] ··· [10.19,10.21] [10.21,10.23] [10.23,10.25]
0.400467 0.303159 0.243087 0.202494 0.173345 0.151479 ··· 0.036461 0.036511 0.036582 ··· 0.517442 0.735068 1.265683 > 1
400
0.01
[10.23,10.24] [10.24,10.25] ··· [11.08,11.09] [11.09,11.10]
0.056358 0.057623 ··· 0.937856 1.686884 > 1
500
0.005
[11.09,11.095] [11.095,11.10] ··· [11.21,11.215] [11.215,11.22]
0.131111 0.136201 ··· 0.915229 1.195596 > 1
550
0.05
[11.215,11.22] [11.22,11.225] ··· [11.325,11.33] [11.33,11.335]
0.132834 0.138593966 ··· 0.993590 1.369085 > 1
Proc. R. Soc. A (2012)
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and {li } (see theorem 2.1) for a = 3.0 as follows: l0 (−0.226751) < m0 (8.341644) < m1 (11.340563) < l1 (39.45942034) < l2 (39.686164) < m2 (88.84015) < · · · . In case of a = 5.0, {mi } and {li } are obtained approximately as follows: l0 (−0.624017) < m0 (7.292924) < m1 (12.287917) < l1 (39.425660) < l2 (40.049607) < m2 (88.863540) < · · · . As we have seen at the end of §2, the spectral gaps are (m2m , m2m+1 ), (l2m+1 , l2m+2 ),
m = 0, 1, 2, . . . .
Although we computed only approximations to li and mi , the eigenvalue exclusions described in the following and listed in tables 1 and 2 are completely rigorous, including a proof that the exclusion intervals L do not intersect the essential spectrum as well; see remark 3.1c). Our target is to exclude eigenvalues in a subinterval of the first spectral gap. 2 We consider the case q(x) = a cos 2px and s(x) = ce−x for two coupling constants c ∈ R. The interval L was subdivided into narrow subintervals Lk whose widths are between 0.001 and 0.01. We chose M = 3 and N + 1 between 400 and 550. All computations were carried out on the DELL Studio 540 (Intel Core2 Quad CPU 2.67 GHz) using MATLAB (R2010a) and INTLAB (Rump 1999). The eigenvalue-free intervals are given in tables 1 and 2. For example, in case of a = 5, c = 1, Lk = [9.01, 9.03], the real part t of the characteristic exponent m in (3.3) was enclosed in [0.398197, 0.410843] and cp1 = p1 L∞ (R) ≤ 1.638409, cp2 = (0)
p2 L∞ (R) ≤ 2.464511, W ∈ [2.017304, 2.468351]. The value of max0≤j≤N |fj | in (6.25) was ≤ 0.037637. The authors are grateful to the four anonymous referees for their useful comments. This work was supported by PRESTO, Japan Science and Technology Agency and a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (no. 20224001).
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Proc. R. Soc. A (2012)
