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Bounds on Perturbations of Self-Adjoint Operators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
201, 577]587 Ž1996.
0274
Bounds on Perturbations of Self-Adjoint Operators Edward L. Green Department of Mathematics, North Georgia College, Dahlonega, Georgia 30597 Submitted by Robert E. O’Malley, Jr. Received September 6, 1995
An improvement of a perturbation theory lemma by M. M. Skriganov which gives an upper bound to the shift of eigenvalues is presented along with other related theorems. These results are also compared with Temple’s inequality and the generalized Temple’s inequality. Applications to spectral theory of differential operators, inverse spectral theory, and quantum mechanics are included. In conjunction with the Rayleigh-Ritz method, a method for bracketing the eigenvalues is developed. Q 1996 Academic Press, Inc.
I. INTRODUCTION Suppose the spectrum of the self-adjoint operator T 0 is known while the spectrum of the operator T 0 q V is unknown. If T 0 q V is bounded from below, the Rayleigh-Ritz method may be used to obtain upper bounds of the eigenvalues; however, lower bounds are more difficult to obtain Žsee w1, 2x.. Temple’s inequality provides a method for bounding the perturbation of the lowest eigenvalue of such an operator even when the perturbation V is unbounded w3, 4x. Although Temple’s inequality is optimal it requires the input of a trial function. Typically this trial function is the eigenfunction of the unperturbed operator. In a similar way the generalized Temple’s inequality w4, 5x may be applied to bound the perturbation of other isolated, nondegenerate eigenvalues even when 5 V 5 o p s `. The weakness of this procedure is that the eigenvalue must be nondegenerate and, again, a trial function must be input. A method called the minimization of the variance method w6x is based on Temple’s inequality and the Rayleigh-Ritz method. Tight two-sided bounds may be obtained by this method, but only with a considerable amount of computation. 577 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
578
EDWARD L. GREEN
When the perturbation is nonnegative and T 0 is semibounded, methods have been developed for obtaining lower bounds for the bottom of the spectrum w7]13x. Along with the Rayleigh-Ritz upper bounds, a bracketing of these eigenvalues may be obtained. Some methods only apply to the case where the perturbation is of the form Ý i,N js1, i/ j Ž1r< x i y x j <. w12, 13x. Aronszajn’s method w9, 10x and the method of truncation which is a modification of Aronszajn’s method w7, 8x require that the spectrum of some intermediate operator be obtained which may be difficult. Although the bracketing method developed in this paper only applies to bounded perturbations, nonnegativity is not required and the numerical scheme is very simple. The results here also give insight into situations where the first few eigenvalues may be only slightly perturbed and yet the operator norm of V is large. When 5 V 5 o p - `, the spectrum of the operator T 0 q V can be expected to be shifted from that of T 0 by no more than 5 V 5 o p . A lemma by Skriganov w14x which proves that in certain situations the size of the perturbation is much less than 5 V 5 o p may be improved to yield even tighter bounds on the perturbation.
II. MAIN RESULTS On a Hilbert space H let T be a self-adjoint operator of the form T s T 0 q V, where T 0 and V are also self-adjoint and 5 V 5 o p - `. Let Ej 4 and Ej0 4 respectively denote the eigenvalues of T and T 0 . If P 0 denotes any spectral projection of T 0 , we define V s P 0 VP 0 q Ž I y P 0 . VP 0 q P 0 V Ž I y P 0 . q Ž I y P 0 . V Ž I y P 0 . ' V11 q V21 q V12 q V22 ,
Ž 1.
respectively. THEOREM 1. With the preceding notation suppose for j s J that P 0 VP 0 s 0 Ž i.e., V11 s 0., where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' Ž E J0 y r, E J0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p F a 5 V 5 o p where 0 - a F 1. Then if there exists a positi¨ e number q such that Ž2 q q .5 V 5 o p F r, then < E J y E J0 < F
a2 4
ž
Ž q q 2 . ln 1 q
ž
2 q
/
q
2 q
y 1 5V 5 op .
/
Proof. Using the methods of regular perturbation theory w1, 2, 15x, make a one-parameter family of self-adjoint operators by T Ž t . s T 0 q tV,
579
BOUNDS ON PERTURBATIONS
0 F t F 1. Let Ej Ž t .4 denote the eigenvalues of T Ž t ., with normalized eigenfunctions w j Ž t .. The functions Ej Ž t . are continuous piecewise differentiable functions and dEj dt
s Ž w j Ž t . , Vw j Ž t . . .
The Hilbert space may be decomposed using P 0 into H1 s P 0 H and H2 s Ž I y P 0 . H. Thus
w J Ž t . s P 0w J Ž t . q Ž I y P 0 . w J Ž t . ' F 1 Ž t . q F 2 Ž t . ,
Ž 2.
T 0 s P 0 T 0 P 0 q Ž I y P 0 . T 0 Ž I y P 0 . ' T1 q T2 .
Ž 3.
and
Also note that V s V12 q V21 q V22 , since P 0 VP 0 s 0. Applying P 0 on the left on both sides of the eigenvalue equation ŽT 0 q tV . f J Ž t . s E J Ž t . f J Ž t . and using Ž1. ] Ž3., one obtains T1F 1 Ž t . q tV12 F 2 Ž t . s E J Ž t . F 1 Ž t . . If one applies Ž I y P 0 . on the left on both sides of the eigenvalue equation, one obtains T2 F 2 Ž t . q tV21F 1 Ž t . q tV22 F 2 Ž t . s E J Ž t . F 2 Ž t . , and so tV21 F 1Ž t . s w E J Ž t . y T2 y tV22 xF 2 Ž t .. Solving for F 2 Ž t . gives F 2 Ž t . s yt T2 q tV22 y E J Ž t .
y1
V21F 1 Ž t . ' ytR Ž t . V21F 1 Ž t . , Ž 4 .
where RŽ t . is the resolvent of T2 q tV22 . Now T2 Ž H1 . s 0 and V22 Ž H1 . s 0, thus the spectrum of T2 has the property that s T 2 l Ž E J0 y r, E J0 q r . s B. Now turn on the perturbation V22 . Then the spectrum of T2 q tV22 is shifted less than t 5 V 5 o p . This implies that the spectrum of T2 q tV22 on H2 does not intersect the interval Ž E J0 y r q t 5 V 5 o p , E J0 q r y t 5 V 5 o p .. Since E J0 y t 5 V 5 o p F E J Ž t . F E J0 q t 5 V 5 o p this implies
Ž I y P 0 . RŽ t . Ž I y P 0 . F
1 r y 2 t 5V 5 op
-
1
Ž q q 2 y 2 t . 5V 5 op
. Ž 5.
580
EDWARD L. GREEN
Now we have dEJ Ž t .
s Ž F 1 q F 2 , Ž V12 q V21 q V22 . Ž F 1 q F 2 . .
dt
s Ž F 1 , V12 F 2 . q Ž F 2 , V21F 1 . q Ž F 2 , V22 F 2 . s yt Ž F 1 , V12 R Ž t . V21F 1 . y t Ž R Ž t . V21F 1 , V21F 1 . q t 2 Ž V22 R Ž t . V21F 1 , R Ž t . V21F 1 . .
Ž 6.
But because of the self-adjointness, the first and second terms in the last U expression are identical Žnote V21 s V12 ., and thus dEJ Ž t . dt
s y2 t Ž F 1 , V12 R Ž t . V21F 1 . q t 2 Ž F 1 , V12 R Ž t . V22 R Ž t . V21F 1 . .
Ž 7. Using < E J y
E J0 <
< E J y E J0 < F
F 1
ž
H0
H01 < dEJ Ž t .rdt < 2t
a 2 5 V 5 2o p a 2 5 V 5 3o p q t2 2 Ž q q 2 y 2 t . 5V 5 op Ž q q 2 y 2 t . 5 V 5 2o p
s a 2 5V 5 op s
a2 4
ž
dt and 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p gives
1
H0
ž
2t q q 2 y 2t
Ž q q 2 . ln 1 q
ž
2 q
/
t2
q q
Ž q q 2 y 2t. 2 q
y 2 5V 5 op .
/
2
/
/
dt
dt
Ž 8.
Note. Skriganov assumes that r ) 4 5 V 5 o p with a s 1 and proves that < E J y E J0 < F 4 5 V 5 2o prr. If q s 2 and a s 1, we may compare with Skriganov’s result, which is that the perturbation is bounded by 4 5 V 5 2o prr 5 V 5 o p . With the improvement one gets approximately 0.44315 5 V 5 o p which is a better result. If q ) 1 then f Ž q . s Ž1r4.Ž q q 2.lnŽ1 q 2rq . q 1r2 q y 1r2 is less than 1rq. Since lim q ª`Ž f Ž q . y 1rq . s 0, it may be more practical to use < E J y E J0 < F Ž a 2rq .5 V 5 o p for large q. We also have the following immediate result: THEOREM 2. Suppose for j s J that 5 P 0 VP 0 5 s e , where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' Ž E J0 y r, E J0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p where 0 - a F 1. Then if there exists a positi¨ e number q such that Ž2 q q .Ž5 V 5 o p q e . - r, then < E J y E J0 < F
a2 4
ž
Ž q q 2 . ln 1 q
ž
2 q
/
q
2 q
y2
/Ž
5 V 5 o p q e . q e . Ž 9.
581
BOUNDS ON PERTURBATIONS
Proof. Consider V˜ s V y P 0 VP 0 and note that 5 V˜ 5 o p F 5 V 5 o p q e . ˜ 0 s 0, V˜12 s V12 , V˜21 s V21 , and Thus Theorem 1 applies to V˜ since P 0 VP ˜ q is such that r ) Ž2 q q .5 V 5 o p . Thus < E˜J y E J0 < F F
a2 4
a2 4
ž ž
2
Ž q q 2 . ln 1 q
ž ž
q 2
Ž q q 2 . ln 1 q
q
/ /
q q
2 q 2 q
y 2 5 V˜ 5 o p y2
/ /Ž
5V 5 op q e . .
Since 5 P 0 VP 0 5 o p s e , then 5 E J y E˜J < F e and hence the result follows. THEOREM 3. Let T 0 be a self-adjoint operator which is bounded from below. For j G 1, let Ej 4 and Ej0 4 denote the eigen¨ alues of T 0 q V and T 0 , respecti¨ ely, in increasing order, counting multiplicities. Suppose for j F J that P 0 VP 0 s 0, where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' w E10 , Ej0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p where 0 - a F 1. If there exists a positi¨ e number q such that Ž2 q q .5 V 5 o p - r, then Ej0 y Ej F
a 2 5V 5 op qq1
.
Ž 10 .
Proof. Let T 0f j0 s Ej0f j0 and thus the eigenspace, H1 s P 0 H, associJ ated with I has eigenbasis f j0 4js1 . Since P 0 VP 0 s 0, then for all c g H1 , 0 0 Ž c , ŽT q V . c . s Ž c , T c .. Thus the eigenvalues of T1 s P 0 TP 0 are also J Ej0 4js1 . By the Rayleigh-Ritz technique, the eigenvalues Ej Ž t . of T 0 q tV satisfy Ej Ž t . F Ej0 , that is, the eigenvalues are not perturbed upward. Now follow the logic in Theorem 1 up to Ž5.. Since the spectrum of H2 does not intersect w E10 , Ej0 q r y t 5 V 5 o p ., then, in place of Ž5., we have
Ž I y P 0 . RŽ t . Ž I y P 0 . F
1 r y t 5V 5 op
s
1
Ž q q 2 y t . 5V 5 op
. Ž 11 .
Therefore Ej0 y Ej F
s
1
H0
ž
2t
a2 qq1
a 2 5 V 5 2o p a 2 5 V 5 3o p q t2 2 Ž q q 2 y t . 5V 5 op Ž q q 2 y t . 5 V 5 2o p 5V 5 op .
/
dt
582
EDWARD L. GREEN
III. DISCUSSION AND APPLICATIONS A. Comparison with Temple’s Inequality and Generalized Temple’s Inequality. Suppose the spectrum of s ŽT 0 . intersected with interval I s Ž E10 y r, E10 q r . is only E10 4 and that r ) 5 V 5 o p . Then if f 10 is used as a test function and Ž f 10 , Vf 10 . s 0, Temple’s inequality indicates that E10 y E1 F
Ž f 10 , V 2f 10 . Ž q q 1. 5 V 5 o p
.
Ž 12 .
The value of q is determined by the condition Ž2 q q .5 V 5 o p - r Žthus Ž q q 1.5 V 5 o p is a lower bound for < E10 y m 2 < where m 2 - E2 .. Generally we may bound Ž f 10 , V 2f 10 . since for any normalized f ,
Ž f , V 2f . F 5 V 5 2o p .
Ž 13 .
When additionally Ž13. is used, then Theorem 3 and Temple’s inequality give the same result with a s 1. One of the advantages of these results over Temple’s inequality is that they give similar bounds when E10 is degenerate. The other advantage is that if P 0 VP 0 s 0 or is small on a larger interval about E10 , then tighter bounds are possible. Now let J ) 1. If Ž f J0 , Vf J0 . s 0, f J0 is used as a test function, the interval I s Ž E J0 y r, E J0 q r . contains only the eigenvalue E J0 , a s 1, and r - 5 V 5 o p , then the generalized Temple’s inequality w4, 5x gives a better bound than Theorem 1, indicating when Ž13. is used that < E J y E J0 < F 5 V 5 o prŽ q q 1., where Ž q q 2.5 V 5 o p - r. Again, one advantage of Theorem 1 is that it applies when the eigenvalue is degenerate. If P 0 VP 0 s 0 or is small on an interval about E J0 which includes other eigenvalues, or if a - 1, these results give improved bounds on the size of the perturbation. B. Perturbation of a One-Dimensional Laplacian Operator. Let T 0 s yd 2rdx 2 on the interval w0, 1x with Dirichlet boundary conditions. Let V s V Ž x . s 100 cos 100p x so that 5 V 5 o p s 100. In applying Temple’s inequality, Ž12. cannot be used since r - 5 V 5 o p . Instead we use f 10 s '2 sin p x as a trial function in E10 y E1 F Ž f , V 2f .r< E10 y m 2 <. An upper bound for < E10 y m 2 < is 3p 2 q 100. Thus the bound from Temple’s inequality is greater than 5000rŽ3p 2 q 100. f 38.58. Thus with f 10 as a trial function. Temple’s inequality does not give a small bound on the perturbation. Let P 0 be the projection of T 0 onto the eigenspace associated with the open interval Ž0, 2500p 2 .. The condition P 0 VP 0 s 0 is easily verified since the eigenfunctions are sine functions. Thus Theorem 3 applies and since 5 V12 5 o p s 5 V21 5 o p s 50 we may choose a s 1r2. The condition r ) Ž2 q q .5 V 5 o p will also be satisfied for eigenvalues p 2 . . . 2401p 2 for different values of q. For example, apply Theorem 3 to E10 s p 2 . Then q
583
BOUNDS ON PERTURBATIONS
may be chosen to be 244.64 and hence E10 y E1 F Ž1r4.Ž0.004071.Ž100. s 0.1018. Using numerical methods Žshooting method with Runge-Kutta., E1 f 9.82 and so the actual shift is E10 y E1 f 0.05. Table I compares the theoretical bound of the shift with the approximate value found numerically by the shooting method. C. In¨ erse Spectral Theory. The inverse spectral problem for operators of the form y
d2 dx 2
c Ž x . q V Ž x . c Ž x . s lc Ž x .
with self-adjoint boundary conditions at x s 0 and x s 1, consists of determining the potential V Ž x . from the spectrum w16]18x. In general, due to isospectral sets V Ž x . cannot be constructed unless additional information is given. If the potential is symmetric with Dirichlet boundary conditions a unique solution for V Ž x . exists w14x. A practical version of this problem is the reconstructing of V Ž x . from finite spectral data w19x. The implications of Example B to this problem is this: 5 V 5 o p cannot be determined with finite spectral data. ŽThis fact is probably known, but Example B makes it clear.. D. One-Dimensional Harmonic Oscillator w20x. Let T 0 s ydrdx 2 q x 2 on L2 Ž R ., whose eigenvalues are given by l0n s 2 n q 1, n s 0, 1, . . . , ` 2 and whose eigenfunctions are fnŽ x . s eyŽ1 r2. x HnŽ x . where HnŽ x . is a Hermite polynomial. Let P 0 be the projection onto the eigenspace associ4 of T 0 , i.e., the interval Žy`, 103.. ated with eigenvalues l00 , l10 , . . . , l50 0 TABLE I
n 1 2 3 4 5 10 20 30 40 49 50 51 60
En f 9.82 39.43 88.78 157.86 246.69 986.91 3947.79 8882.59 15791.32 23695.80 24624.01 25672.0 35530.6
Shifts of eigenvalues En0 s n2p 2 Theoretical bound 9.87 39.48 88.83 157.91 246.74 986.96 3947.84 8882.64 15791.37 23696.92 24674.01 25670.84 35530.58
0.1018 0.1019 0.1021 0.1024 0.1028 0.1060 0.1212 0.1593 0.2847 2.850 50.00 2.788 0.2399
Actual shift y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y1.12 y50.00 1.16 0.02
584
EDWARD L. GREEN
Let 2
VŽ x. s
½
e x P101 Ž x . , 0,
if < x < F 1 if < x < ) 1,
where P101Ž x . is a Legendre polynomial w21x. The condition P 0 VP 0 is easily verified since products of polynomials of degree less than or equal to fifty can be written as a linear combination of P0 Ž x ., P1Ž x ., . . . , P100 Ž x .4 a n d h e n c e t h e i n t e g r a l 1 1 Hy1 Hn Ž x . V Ž x . Hm Ž x . eyx dx s Hy1 Ý100 is0 a i Pi Ž x . P101 Ž x . dx s 0. 2
In this example 5 V 5 o p s e. For the first eigenvalue l00 s 1 we can let q s 35.52. Applying Theorem 3 with a s 1, gives l00 y l 0 F 0.0745. E. Separable Potentials in R n . Let V Ž x 1 , . . . , xn . s
n
Ý ai cos 100p x i is1
and T s yD on w0, 1 with Dirichlet boundary conditions. Let P 0 be the projection of T 0 on the interval Žy`, 2500p 2 q Ž n y 1.p 2 .. Then one easily verifies the condition P 0 VP 0 s 0. In this case En0 s Ýnis1Ž ji . 2p 2 , where ji is a positive integer. This eigenvalue is often degenerate. Consider the perturbation of one of these eigenvalues En0 , where En0 - 2500p 2 . If this eigenvalue is p-fold degenerate, the perturbation will possibly split it into p nondegenerate eigenvalues. Let En be one of these perturbed eigenvalues. Then with a s 1r2 we may use Theorem 3 with 5 V 5 o p s Ýnis1 < a i <. For example, consider the perturbation of the lowest eigenvalue with n s 3 and a1 s a2 s a3 s 10. Then 5 V 5 o p s 30 and q f 830 so that 3p 2 y E1 - 0.00903. 0
xn
F. Radially Symmetric Potential in R 3. Consider the Schrodinger equa¨ tion ŽyD y 1rr . c s Ec on R 3. One can separate variables w20x taking c Ž r, u , f . s uŽ r .Yl mŽ u , f . resulting in yŽ1rr 2 .Ž drdr .Ž r 2 Ž durdr .. y Ž1rr . u q Ž l Ž l q 1.rr 2 . u s Eu. The eigenvalues do not depend on l and are given by En0 s y1r4n2 for n s 1, 2, . . . , and the unnormalized eigenlq1 Ž y1 ' functions are given by cn0 s r l L2nql n 2 r . eyr r '2 . The degree of l 2 lq1 Ž y1 ' r L nq l n 2 r . is n y 1. Consider now the perturbation y '2 '2 r U e P2001 Ž r . , V Ž x . s ke 0,
½
if r - 1 if r G 1,
U Ž r . is the shifted Legendre polynomial w21x of degree 2001. Let where P2001 0 P be the projection onto the eigenspace associated with En0 for n s
BOUNDS ON PERTURBATIONS
585
1, 2, . . . , 1000. The condition P 0 VP 0 s 0 is easily verified since the product of two functions in this eigenspace times the factors of ey '2 r and r 2 would yield a polynomial with degree less than or equal to 2000. Note that 5 V 5 o p s k. For E10 take r s 0.24999975 and assume that k 0.24999975r4. Thus q s 0.24999975rk y 2 is greater than or equal to 2. Under these conditions with a s 1, we have E10
y E1 F
k2 0.24999975 y k
.
For example, if k s 0.01 then E10 y E1 F 0.000417. This perturbation function oscillates rapidly with 2001 zeros between r s 0 and r s 1. Thus this shows that small fluctuations in the potential may have little effect on the bottom of the spectrum.
IV. METHOD FOR BRACKETING EIGENVALUES Let T 0 be a self-adjoint operator bounded from below and let V be a bounded, self-adjoint perturbation of T 0 . Let the bottom of the spectrum of T 0 consist of eigenvalues E10 F E20 F ??? F E J0 with corresponding eigenfunctions f 10 , f 20 , . . . , f J0 . Let P 0 be the spectral projection of the separable Hilbert space H onto the eigenspace of T 0 associated with w E10 , E J0 x. Step One. Find the spectrum of T˜ s T 0 q P 0 VP 0 on P 0H . This is a finite dimensional matrix problem. Denote this spectrum by E˜1 F E˜2 F ??? F E˜J . By the Rayleigh-Ritz theory, it is known that the eigenvalues of Ei of T 0 q V satisfy the relations Ei F E˜i provided there are i eigenvalues of T 0 q V. Step Two. Shift the spectrum by replacing V with Vk s V y k in such a way that the spectrum of Tk0 s T 0 q P 0 Vk P 0 does not intersect the 0 . interval w E Jq 1 , ` . The value of k may also be chosen in order to reduce 5 V y k 5 o p. Step Three. Compute 5 Vk y P 0 Vk P 0 5 o p , 5Ž Vk y P 0 Vk P 0 .12 5 o p , and a . For eigenvalue E˜i , compute r and q. Theorem 3 may be used to obtain bounds on the perturbation of Tk s T 0 q Vk s Tk0 q Ž Vk y P 0 Vk P 0 ., since the perturbation term satisfies P 0 Ž Vk y P 0 Vk P 0 . P 0 s 0. Thus a bound to the quantity Eik y Ž Ei y k . may be attained. Since Eik s E˜i y k we have a bound on the quantity E˜i y Ei . In Step 3, one may use the inequality 5 Vk y P 0 Vk P 0 5 o p F 5 Vk 5 o p q 5 P 0 Vk P 0 5 o p . The value of 5 P 0 Vk P 0 5 o p can be approximated numerically.
586
EDWARD L. GREEN
EXAMPLES OF METHOD. Let T 0 s yd 2rdx 2 on wy1, 1x with Dirichlet boundary conditions and let V s x 2 . Let P 0 be the projection onto the eigenspace associated with the first four eigenvalues of T 0 which is associated with the interval wp 2r4, 4p 2 x. Using the inverse power method the value of E˜1 is approximately 2.5969325. Applying the method with k s 0.5 yields 5 Vk y P 0 Vk P 0 5 o p - 0.88. For E˜1 , r s 59.58809, and thus q s 65.7137. Using a s 1 yields E˜1 y E1 - 0.0132. By the shooting method the actual shift is approximately 10y5 . For the same T 0 let V s < x < and let P 0 project onto the eigenspace associated with wp 2r4, 9p 2 x. Use k s 0.5. Then 5 Vk y P 0 Vk P 0 5 o p - 0.886. Using a s 1 yields the following bracketing: 2.75602086 F E1 F 2.76268727.
ACKNOWLEDGMENT It is a pleasure to thank my thesis advisor Evans M. Harrell, II for suggestions and discussions.
REFERENCES 1. M. Reed and B. Simon, ‘‘Methods of Modern Mathematical Physics,’’ Vol. IV, Academic Press, San Diego, 1978. 2. T. Kato, ‘‘Perturbation Theory of Eigenvalue Problems,’’ Gordon & Breach, New York, 1976. 3. G. Temple, The theory of Rayleigh’s principle as applied to continuous systems, Proc. Roy. Soc. Sect. A 119 Ž1928., 276]293. 4. T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4 Ž1949., 334]339. 5. E. M. Harrell, II, Generalizations of Temple’s inequality, J. Amer. Math. Soc. 69 Ž1978., 271]276. 6. H. Kleindienst and R. Emrich, Minimization of the variance: A method for two-sided bounds for eigenvalues of selfadjoint operators, in ‘‘International Series of Numerical Mathematics,’’ Vol. 96, pp. 131]142, Birkhauser, Basel, 1991. ¨ 7. N. Bazley and D. Fox, Lower bounds for eigenvalues of Schrodinger’s equation, Phys. ¨ Re¨ . 124 Ž1961., 483]492. 8. P.-O. Lowdin, Studies in perturbation theory. X. Lower bounds to energy eigenvalues in ¨ perturbation theory ground state, Phys. Re¨ . A 139 Ž1965., 357]372. 9. C. A. Beattie and W. M. Greenlee, Some remarks concerning closure rates for Aronszajn’s method, in ‘‘International Series of Numerical Mathematics,’’ Vol. 96, pp. 23]39, Birkhauser, Basel, 1991. ¨ 10. N. Aronszajn, Approximation methods for eigenvalues of completely continuous symmetric operators, in ‘‘Proceedings of the Symposium on Spectral Theory and Differential Problems, Stillwater, Oklahoma,’’ pp. 179]202. 11. W. Thirring, Vorlesungen ¨ uber Mathematische Physik, T7, Quantenmechanik, Univ. Wien Lecture Notes, Vienna.
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12. H. Grosse, P. Hertel, and W. Thirring, Lower bounds to the energy levels of atomic and molecular systems, Acta Phys. Austriaca 49 Ž1978., 89]112. 13. P. Hertel, E. Lieb, and W. Thirring, Lower bound to the energy of complex atoms, J. Chem. Phys. 62 Ž1975., 3355]3356. 14. M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, in ‘‘Proceedings of the Steklov Institute of Mathematics,’’ Vol. 171, Amer. Math. Soc., Providence, RI, 1987. 15. J. Weidmann, ‘‘Linear Operators in Hilbert Spaces,’’ Springer-Verlag, New York, 1980. 16. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung Durch die Eigenwerte, Acta Math. 78 Ž1946., 1]96. 17. J. Poschel, and E. Trubowitz, ‘‘Inverse Spectral Theory,’’ Academic Press, Orlando, 1987. ¨ 18. J. R. McLaughlin, Analytic methods for recovering coefficients in differential equations from spectral data, SIAM Re¨ . 28 Ž1986., 53]72. 19. B. D. Lowe, M. Pilant, and W. Rundell, The recovery of potentials from finite spectral data, SIAM J. Math. Anal. 23 Ž1991., 482]504. 20. L. I. Schiff, ‘‘Quantum Mechanics,’’ McGraw]Hill, New York, 1949. 21. M. Abramowitz and I. A. Stegun, ‘‘Handbook of Mathematical Functions,’’ National Bureau of Standards, Washington, 1964.
201, 577]587 Ž1996.
0274
Bounds on Perturbations of Self-Adjoint Operators Edward L. Green Department of Mathematics, North Georgia College, Dahlonega, Georgia 30597 Submitted by Robert E. O’Malley, Jr. Received September 6, 1995
An improvement of a perturbation theory lemma by M. M. Skriganov which gives an upper bound to the shift of eigenvalues is presented along with other related theorems. These results are also compared with Temple’s inequality and the generalized Temple’s inequality. Applications to spectral theory of differential operators, inverse spectral theory, and quantum mechanics are included. In conjunction with the Rayleigh-Ritz method, a method for bracketing the eigenvalues is developed. Q 1996 Academic Press, Inc.
I. INTRODUCTION Suppose the spectrum of the self-adjoint operator T 0 is known while the spectrum of the operator T 0 q V is unknown. If T 0 q V is bounded from below, the Rayleigh-Ritz method may be used to obtain upper bounds of the eigenvalues; however, lower bounds are more difficult to obtain Žsee w1, 2x.. Temple’s inequality provides a method for bounding the perturbation of the lowest eigenvalue of such an operator even when the perturbation V is unbounded w3, 4x. Although Temple’s inequality is optimal it requires the input of a trial function. Typically this trial function is the eigenfunction of the unperturbed operator. In a similar way the generalized Temple’s inequality w4, 5x may be applied to bound the perturbation of other isolated, nondegenerate eigenvalues even when 5 V 5 o p s `. The weakness of this procedure is that the eigenvalue must be nondegenerate and, again, a trial function must be input. A method called the minimization of the variance method w6x is based on Temple’s inequality and the Rayleigh-Ritz method. Tight two-sided bounds may be obtained by this method, but only with a considerable amount of computation. 577 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
578
EDWARD L. GREEN
When the perturbation is nonnegative and T 0 is semibounded, methods have been developed for obtaining lower bounds for the bottom of the spectrum w7]13x. Along with the Rayleigh-Ritz upper bounds, a bracketing of these eigenvalues may be obtained. Some methods only apply to the case where the perturbation is of the form Ý i,N js1, i/ j Ž1r< x i y x j <. w12, 13x. Aronszajn’s method w9, 10x and the method of truncation which is a modification of Aronszajn’s method w7, 8x require that the spectrum of some intermediate operator be obtained which may be difficult. Although the bracketing method developed in this paper only applies to bounded perturbations, nonnegativity is not required and the numerical scheme is very simple. The results here also give insight into situations where the first few eigenvalues may be only slightly perturbed and yet the operator norm of V is large. When 5 V 5 o p - `, the spectrum of the operator T 0 q V can be expected to be shifted from that of T 0 by no more than 5 V 5 o p . A lemma by Skriganov w14x which proves that in certain situations the size of the perturbation is much less than 5 V 5 o p may be improved to yield even tighter bounds on the perturbation.
II. MAIN RESULTS On a Hilbert space H let T be a self-adjoint operator of the form T s T 0 q V, where T 0 and V are also self-adjoint and 5 V 5 o p - `. Let Ej 4 and Ej0 4 respectively denote the eigenvalues of T and T 0 . If P 0 denotes any spectral projection of T 0 , we define V s P 0 VP 0 q Ž I y P 0 . VP 0 q P 0 V Ž I y P 0 . q Ž I y P 0 . V Ž I y P 0 . ' V11 q V21 q V12 q V22 ,
Ž 1.
respectively. THEOREM 1. With the preceding notation suppose for j s J that P 0 VP 0 s 0 Ž i.e., V11 s 0., where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' Ž E J0 y r, E J0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p F a 5 V 5 o p where 0 - a F 1. Then if there exists a positi¨ e number q such that Ž2 q q .5 V 5 o p F r, then < E J y E J0 < F
a2 4
ž
Ž q q 2 . ln 1 q
ž
2 q
/
q
2 q
y 1 5V 5 op .
/
Proof. Using the methods of regular perturbation theory w1, 2, 15x, make a one-parameter family of self-adjoint operators by T Ž t . s T 0 q tV,
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BOUNDS ON PERTURBATIONS
0 F t F 1. Let Ej Ž t .4 denote the eigenvalues of T Ž t ., with normalized eigenfunctions w j Ž t .. The functions Ej Ž t . are continuous piecewise differentiable functions and dEj dt
s Ž w j Ž t . , Vw j Ž t . . .
The Hilbert space may be decomposed using P 0 into H1 s P 0 H and H2 s Ž I y P 0 . H. Thus
w J Ž t . s P 0w J Ž t . q Ž I y P 0 . w J Ž t . ' F 1 Ž t . q F 2 Ž t . ,
Ž 2.
T 0 s P 0 T 0 P 0 q Ž I y P 0 . T 0 Ž I y P 0 . ' T1 q T2 .
Ž 3.
and
Also note that V s V12 q V21 q V22 , since P 0 VP 0 s 0. Applying P 0 on the left on both sides of the eigenvalue equation ŽT 0 q tV . f J Ž t . s E J Ž t . f J Ž t . and using Ž1. ] Ž3., one obtains T1F 1 Ž t . q tV12 F 2 Ž t . s E J Ž t . F 1 Ž t . . If one applies Ž I y P 0 . on the left on both sides of the eigenvalue equation, one obtains T2 F 2 Ž t . q tV21F 1 Ž t . q tV22 F 2 Ž t . s E J Ž t . F 2 Ž t . , and so tV21 F 1Ž t . s w E J Ž t . y T2 y tV22 xF 2 Ž t .. Solving for F 2 Ž t . gives F 2 Ž t . s yt T2 q tV22 y E J Ž t .
y1
V21F 1 Ž t . ' ytR Ž t . V21F 1 Ž t . , Ž 4 .
where RŽ t . is the resolvent of T2 q tV22 . Now T2 Ž H1 . s 0 and V22 Ž H1 . s 0, thus the spectrum of T2 has the property that s T 2 l Ž E J0 y r, E J0 q r . s B. Now turn on the perturbation V22 . Then the spectrum of T2 q tV22 is shifted less than t 5 V 5 o p . This implies that the spectrum of T2 q tV22 on H2 does not intersect the interval Ž E J0 y r q t 5 V 5 o p , E J0 q r y t 5 V 5 o p .. Since E J0 y t 5 V 5 o p F E J Ž t . F E J0 q t 5 V 5 o p this implies
Ž I y P 0 . RŽ t . Ž I y P 0 . F
1 r y 2 t 5V 5 op
-
1
Ž q q 2 y 2 t . 5V 5 op
. Ž 5.
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EDWARD L. GREEN
Now we have dEJ Ž t .
s Ž F 1 q F 2 , Ž V12 q V21 q V22 . Ž F 1 q F 2 . .
dt
s Ž F 1 , V12 F 2 . q Ž F 2 , V21F 1 . q Ž F 2 , V22 F 2 . s yt Ž F 1 , V12 R Ž t . V21F 1 . y t Ž R Ž t . V21F 1 , V21F 1 . q t 2 Ž V22 R Ž t . V21F 1 , R Ž t . V21F 1 . .
Ž 6.
But because of the self-adjointness, the first and second terms in the last U expression are identical Žnote V21 s V12 ., and thus dEJ Ž t . dt
s y2 t Ž F 1 , V12 R Ž t . V21F 1 . q t 2 Ž F 1 , V12 R Ž t . V22 R Ž t . V21F 1 . .
Ž 7. Using < E J y
E J0 <
< E J y E J0 < F
F 1
ž
H0
H01 < dEJ Ž t .rdt < 2t
a 2 5 V 5 2o p a 2 5 V 5 3o p q t2 2 Ž q q 2 y 2 t . 5V 5 op Ž q q 2 y 2 t . 5 V 5 2o p
s a 2 5V 5 op s
a2 4
ž
dt and 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p gives
1
H0
ž
2t q q 2 y 2t
Ž q q 2 . ln 1 q
ž
2 q
/
t2
q q
Ž q q 2 y 2t. 2 q
y 2 5V 5 op .
/
2
/
/
dt
dt
Ž 8.
Note. Skriganov assumes that r ) 4 5 V 5 o p with a s 1 and proves that < E J y E J0 < F 4 5 V 5 2o prr. If q s 2 and a s 1, we may compare with Skriganov’s result, which is that the perturbation is bounded by 4 5 V 5 2o prr 5 V 5 o p . With the improvement one gets approximately 0.44315 5 V 5 o p which is a better result. If q ) 1 then f Ž q . s Ž1r4.Ž q q 2.lnŽ1 q 2rq . q 1r2 q y 1r2 is less than 1rq. Since lim q ª`Ž f Ž q . y 1rq . s 0, it may be more practical to use < E J y E J0 < F Ž a 2rq .5 V 5 o p for large q. We also have the following immediate result: THEOREM 2. Suppose for j s J that 5 P 0 VP 0 5 s e , where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' Ž E J0 y r, E J0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p where 0 - a F 1. Then if there exists a positi¨ e number q such that Ž2 q q .Ž5 V 5 o p q e . - r, then < E J y E J0 < F
a2 4
ž
Ž q q 2 . ln 1 q
ž
2 q
/
q
2 q
y2
/Ž
5 V 5 o p q e . q e . Ž 9.
581
BOUNDS ON PERTURBATIONS
Proof. Consider V˜ s V y P 0 VP 0 and note that 5 V˜ 5 o p F 5 V 5 o p q e . ˜ 0 s 0, V˜12 s V12 , V˜21 s V21 , and Thus Theorem 1 applies to V˜ since P 0 VP ˜ q is such that r ) Ž2 q q .5 V 5 o p . Thus < E˜J y E J0 < F F
a2 4
a2 4
ž ž
2
Ž q q 2 . ln 1 q
ž ž
q 2
Ž q q 2 . ln 1 q
q
/ /
q q
2 q 2 q
y 2 5 V˜ 5 o p y2
/ /Ž
5V 5 op q e . .
Since 5 P 0 VP 0 5 o p s e , then 5 E J y E˜J < F e and hence the result follows. THEOREM 3. Let T 0 be a self-adjoint operator which is bounded from below. For j G 1, let Ej 4 and Ej0 4 denote the eigen¨ alues of T 0 q V and T 0 , respecti¨ ely, in increasing order, counting multiplicities. Suppose for j F J that P 0 VP 0 s 0, where P 0 denotes the spectral projection of T 0 onto the inter¨ al I ' w E10 , Ej0 q r . for some r ) 0. Also suppose that s ŽT 0 . l I consists of a finite number of eigen¨ alues, and that 5 V12 5 o p s 5 V21 5 o p - a 5 V 5 o p where 0 - a F 1. If there exists a positi¨ e number q such that Ž2 q q .5 V 5 o p - r, then Ej0 y Ej F
a 2 5V 5 op qq1
.
Ž 10 .
Proof. Let T 0f j0 s Ej0f j0 and thus the eigenspace, H1 s P 0 H, associJ ated with I has eigenbasis f j0 4js1 . Since P 0 VP 0 s 0, then for all c g H1 , 0 0 Ž c , ŽT q V . c . s Ž c , T c .. Thus the eigenvalues of T1 s P 0 TP 0 are also J Ej0 4js1 . By the Rayleigh-Ritz technique, the eigenvalues Ej Ž t . of T 0 q tV satisfy Ej Ž t . F Ej0 , that is, the eigenvalues are not perturbed upward. Now follow the logic in Theorem 1 up to Ž5.. Since the spectrum of H2 does not intersect w E10 , Ej0 q r y t 5 V 5 o p ., then, in place of Ž5., we have
Ž I y P 0 . RŽ t . Ž I y P 0 . F
1 r y t 5V 5 op
s
1
Ž q q 2 y t . 5V 5 op
. Ž 11 .
Therefore Ej0 y Ej F
s
1
H0
ž
2t
a2 qq1
a 2 5 V 5 2o p a 2 5 V 5 3o p q t2 2 Ž q q 2 y t . 5V 5 op Ž q q 2 y t . 5 V 5 2o p 5V 5 op .
/
dt
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EDWARD L. GREEN
III. DISCUSSION AND APPLICATIONS A. Comparison with Temple’s Inequality and Generalized Temple’s Inequality. Suppose the spectrum of s ŽT 0 . intersected with interval I s Ž E10 y r, E10 q r . is only E10 4 and that r ) 5 V 5 o p . Then if f 10 is used as a test function and Ž f 10 , Vf 10 . s 0, Temple’s inequality indicates that E10 y E1 F
Ž f 10 , V 2f 10 . Ž q q 1. 5 V 5 o p
.
Ž 12 .
The value of q is determined by the condition Ž2 q q .5 V 5 o p - r Žthus Ž q q 1.5 V 5 o p is a lower bound for < E10 y m 2 < where m 2 - E2 .. Generally we may bound Ž f 10 , V 2f 10 . since for any normalized f ,
Ž f , V 2f . F 5 V 5 2o p .
Ž 13 .
When additionally Ž13. is used, then Theorem 3 and Temple’s inequality give the same result with a s 1. One of the advantages of these results over Temple’s inequality is that they give similar bounds when E10 is degenerate. The other advantage is that if P 0 VP 0 s 0 or is small on a larger interval about E10 , then tighter bounds are possible. Now let J ) 1. If Ž f J0 , Vf J0 . s 0, f J0 is used as a test function, the interval I s Ž E J0 y r, E J0 q r . contains only the eigenvalue E J0 , a s 1, and r - 5 V 5 o p , then the generalized Temple’s inequality w4, 5x gives a better bound than Theorem 1, indicating when Ž13. is used that < E J y E J0 < F 5 V 5 o prŽ q q 1., where Ž q q 2.5 V 5 o p - r. Again, one advantage of Theorem 1 is that it applies when the eigenvalue is degenerate. If P 0 VP 0 s 0 or is small on an interval about E J0 which includes other eigenvalues, or if a - 1, these results give improved bounds on the size of the perturbation. B. Perturbation of a One-Dimensional Laplacian Operator. Let T 0 s yd 2rdx 2 on the interval w0, 1x with Dirichlet boundary conditions. Let V s V Ž x . s 100 cos 100p x so that 5 V 5 o p s 100. In applying Temple’s inequality, Ž12. cannot be used since r - 5 V 5 o p . Instead we use f 10 s '2 sin p x as a trial function in E10 y E1 F Ž f , V 2f .r< E10 y m 2 <. An upper bound for < E10 y m 2 < is 3p 2 q 100. Thus the bound from Temple’s inequality is greater than 5000rŽ3p 2 q 100. f 38.58. Thus with f 10 as a trial function. Temple’s inequality does not give a small bound on the perturbation. Let P 0 be the projection of T 0 onto the eigenspace associated with the open interval Ž0, 2500p 2 .. The condition P 0 VP 0 s 0 is easily verified since the eigenfunctions are sine functions. Thus Theorem 3 applies and since 5 V12 5 o p s 5 V21 5 o p s 50 we may choose a s 1r2. The condition r ) Ž2 q q .5 V 5 o p will also be satisfied for eigenvalues p 2 . . . 2401p 2 for different values of q. For example, apply Theorem 3 to E10 s p 2 . Then q
583
BOUNDS ON PERTURBATIONS
may be chosen to be 244.64 and hence E10 y E1 F Ž1r4.Ž0.004071.Ž100. s 0.1018. Using numerical methods Žshooting method with Runge-Kutta., E1 f 9.82 and so the actual shift is E10 y E1 f 0.05. Table I compares the theoretical bound of the shift with the approximate value found numerically by the shooting method. C. In¨ erse Spectral Theory. The inverse spectral problem for operators of the form y
d2 dx 2
c Ž x . q V Ž x . c Ž x . s lc Ž x .
with self-adjoint boundary conditions at x s 0 and x s 1, consists of determining the potential V Ž x . from the spectrum w16]18x. In general, due to isospectral sets V Ž x . cannot be constructed unless additional information is given. If the potential is symmetric with Dirichlet boundary conditions a unique solution for V Ž x . exists w14x. A practical version of this problem is the reconstructing of V Ž x . from finite spectral data w19x. The implications of Example B to this problem is this: 5 V 5 o p cannot be determined with finite spectral data. ŽThis fact is probably known, but Example B makes it clear.. D. One-Dimensional Harmonic Oscillator w20x. Let T 0 s ydrdx 2 q x 2 on L2 Ž R ., whose eigenvalues are given by l0n s 2 n q 1, n s 0, 1, . . . , ` 2 and whose eigenfunctions are fnŽ x . s eyŽ1 r2. x HnŽ x . where HnŽ x . is a Hermite polynomial. Let P 0 be the projection onto the eigenspace associ4 of T 0 , i.e., the interval Žy`, 103.. ated with eigenvalues l00 , l10 , . . . , l50 0 TABLE I
n 1 2 3 4 5 10 20 30 40 49 50 51 60
En f 9.82 39.43 88.78 157.86 246.69 986.91 3947.79 8882.59 15791.32 23695.80 24624.01 25672.0 35530.6
Shifts of eigenvalues En0 s n2p 2 Theoretical bound 9.87 39.48 88.83 157.91 246.74 986.96 3947.84 8882.64 15791.37 23696.92 24674.01 25670.84 35530.58
0.1018 0.1019 0.1021 0.1024 0.1028 0.1060 0.1212 0.1593 0.2847 2.850 50.00 2.788 0.2399
Actual shift y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y0.05 y1.12 y50.00 1.16 0.02
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EDWARD L. GREEN
Let 2
VŽ x. s
½
e x P101 Ž x . , 0,
if < x < F 1 if < x < ) 1,
where P101Ž x . is a Legendre polynomial w21x. The condition P 0 VP 0 is easily verified since products of polynomials of degree less than or equal to fifty can be written as a linear combination of P0 Ž x ., P1Ž x ., . . . , P100 Ž x .4 a n d h e n c e t h e i n t e g r a l 1 1 Hy1 Hn Ž x . V Ž x . Hm Ž x . eyx dx s Hy1 Ý100 is0 a i Pi Ž x . P101 Ž x . dx s 0. 2
In this example 5 V 5 o p s e. For the first eigenvalue l00 s 1 we can let q s 35.52. Applying Theorem 3 with a s 1, gives l00 y l 0 F 0.0745. E. Separable Potentials in R n . Let V Ž x 1 , . . . , xn . s
n
Ý ai cos 100p x i is1
and T s yD on w0, 1 with Dirichlet boundary conditions. Let P 0 be the projection of T 0 on the interval Žy`, 2500p 2 q Ž n y 1.p 2 .. Then one easily verifies the condition P 0 VP 0 s 0. In this case En0 s Ýnis1Ž ji . 2p 2 , where ji is a positive integer. This eigenvalue is often degenerate. Consider the perturbation of one of these eigenvalues En0 , where En0 - 2500p 2 . If this eigenvalue is p-fold degenerate, the perturbation will possibly split it into p nondegenerate eigenvalues. Let En be one of these perturbed eigenvalues. Then with a s 1r2 we may use Theorem 3 with 5 V 5 o p s Ýnis1 < a i <. For example, consider the perturbation of the lowest eigenvalue with n s 3 and a1 s a2 s a3 s 10. Then 5 V 5 o p s 30 and q f 830 so that 3p 2 y E1 - 0.00903. 0
xn
F. Radially Symmetric Potential in R 3. Consider the Schrodinger equa¨ tion ŽyD y 1rr . c s Ec on R 3. One can separate variables w20x taking c Ž r, u , f . s uŽ r .Yl mŽ u , f . resulting in yŽ1rr 2 .Ž drdr .Ž r 2 Ž durdr .. y Ž1rr . u q Ž l Ž l q 1.rr 2 . u s Eu. The eigenvalues do not depend on l and are given by En0 s y1r4n2 for n s 1, 2, . . . , and the unnormalized eigenlq1 Ž y1 ' functions are given by cn0 s r l L2nql n 2 r . eyr r '2 . The degree of l 2 lq1 Ž y1 ' r L nq l n 2 r . is n y 1. Consider now the perturbation y '2 '2 r U e P2001 Ž r . , V Ž x . s ke 0,
½
if r - 1 if r G 1,
U Ž r . is the shifted Legendre polynomial w21x of degree 2001. Let where P2001 0 P be the projection onto the eigenspace associated with En0 for n s
BOUNDS ON PERTURBATIONS
585
1, 2, . . . , 1000. The condition P 0 VP 0 s 0 is easily verified since the product of two functions in this eigenspace times the factors of ey '2 r and r 2 would yield a polynomial with degree less than or equal to 2000. Note that 5 V 5 o p s k. For E10 take r s 0.24999975 and assume that k 0.24999975r4. Thus q s 0.24999975rk y 2 is greater than or equal to 2. Under these conditions with a s 1, we have E10
y E1 F
k2 0.24999975 y k
.
For example, if k s 0.01 then E10 y E1 F 0.000417. This perturbation function oscillates rapidly with 2001 zeros between r s 0 and r s 1. Thus this shows that small fluctuations in the potential may have little effect on the bottom of the spectrum.
IV. METHOD FOR BRACKETING EIGENVALUES Let T 0 be a self-adjoint operator bounded from below and let V be a bounded, self-adjoint perturbation of T 0 . Let the bottom of the spectrum of T 0 consist of eigenvalues E10 F E20 F ??? F E J0 with corresponding eigenfunctions f 10 , f 20 , . . . , f J0 . Let P 0 be the spectral projection of the separable Hilbert space H onto the eigenspace of T 0 associated with w E10 , E J0 x. Step One. Find the spectrum of T˜ s T 0 q P 0 VP 0 on P 0H . This is a finite dimensional matrix problem. Denote this spectrum by E˜1 F E˜2 F ??? F E˜J . By the Rayleigh-Ritz theory, it is known that the eigenvalues of Ei of T 0 q V satisfy the relations Ei F E˜i provided there are i eigenvalues of T 0 q V. Step Two. Shift the spectrum by replacing V with Vk s V y k in such a way that the spectrum of Tk0 s T 0 q P 0 Vk P 0 does not intersect the 0 . interval w E Jq 1 , ` . The value of k may also be chosen in order to reduce 5 V y k 5 o p. Step Three. Compute 5 Vk y P 0 Vk P 0 5 o p , 5Ž Vk y P 0 Vk P 0 .12 5 o p , and a . For eigenvalue E˜i , compute r and q. Theorem 3 may be used to obtain bounds on the perturbation of Tk s T 0 q Vk s Tk0 q Ž Vk y P 0 Vk P 0 ., since the perturbation term satisfies P 0 Ž Vk y P 0 Vk P 0 . P 0 s 0. Thus a bound to the quantity Eik y Ž Ei y k . may be attained. Since Eik s E˜i y k we have a bound on the quantity E˜i y Ei . In Step 3, one may use the inequality 5 Vk y P 0 Vk P 0 5 o p F 5 Vk 5 o p q 5 P 0 Vk P 0 5 o p . The value of 5 P 0 Vk P 0 5 o p can be approximated numerically.
586
EDWARD L. GREEN
EXAMPLES OF METHOD. Let T 0 s yd 2rdx 2 on wy1, 1x with Dirichlet boundary conditions and let V s x 2 . Let P 0 be the projection onto the eigenspace associated with the first four eigenvalues of T 0 which is associated with the interval wp 2r4, 4p 2 x. Using the inverse power method the value of E˜1 is approximately 2.5969325. Applying the method with k s 0.5 yields 5 Vk y P 0 Vk P 0 5 o p - 0.88. For E˜1 , r s 59.58809, and thus q s 65.7137. Using a s 1 yields E˜1 y E1 - 0.0132. By the shooting method the actual shift is approximately 10y5 . For the same T 0 let V s < x < and let P 0 project onto the eigenspace associated with wp 2r4, 9p 2 x. Use k s 0.5. Then 5 Vk y P 0 Vk P 0 5 o p - 0.886. Using a s 1 yields the following bracketing: 2.75602086 F E1 F 2.76268727.
ACKNOWLEDGMENT It is a pleasure to thank my thesis advisor Evans M. Harrell, II for suggestions and discussions.
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