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Fredholm determinant and the Sturm-Liouville problems in quantum mechanics
7 March 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 186 (1994) 51-58
Fredholm determinant and the Sturm-Liouville problems in quantum mechanics Mikio Nakahara Department of Physics, Faculty of Liberal Arts, Shizuoka University, Shizuoka 422, Japan, Department of Mathematics and Physics, Kinld University, Higashi-Osaka 577, Japan l Received 3 November 1993; revised manuscript received 6 January 1994; accepted for publication 13 January 1994 Communicated by A.P. Fordy
Abstract
The Fredholm determinant of a nonrelativistic Hamiltonian defined on a compact one-dimensional space is evaluated exactly. The Schr6dinger equation is rewritten as a first-order differential equation, which is integrated formally. Then a 2 x 2 eigenvalue equation is proved to be proportional to the Fredholm determinant. Our method turns out to be a powerful tool to solve eigenvalue problems in quantum mechanics. Finally several applications of our method are given.
G i v e n a H a m i l t o n i a n operator H , the Fredholm determinant Det ( H - z) is defined by D e t ( H - z) = 1-I(En - z ) ,
(1)
n
where the E~ are the spectrum o f H and z is a complex number. The Fredholm determinant is extensively used to obtain the effective field theory from a microscopic quantum field theory, see, e.g., Ref. [ 1 ]. This m e t h o d has been used in instanton calculus [2 ] and to evaluate the soliton creation energy in polyacetylene [3-5 ] and the vortex line profile in a type II superconductor [6]. Let us consider, e.g., the following action S [3],
S =f
f
(2)
where r is the imaginary time and H is a Hamiltonian o f either a Schr6dinger type or a Dirac type. The free energy F o f this system is obtained by a functional integral,
e -pF = f V ~ D ~ u e - s = D e t ( 0 , + H ) .
(3)
In the above equation and hereafter, Det and Tr denote a determinant and a trace, respectively, in a (infinitedimensional) functional space. We reserve det and tr for operations in a finite-dimensional vector space. In Ref. [ 1 ], a scattering theory analysis has been employed to evaluate a F r e d h o l m determinant and this is applied to polyactylene in Ref. [3]. In Refs. [ 4 - 6 ] , a systematic approximation scheme has been developed 1 Present address.
0375-9601/94/$07.00 (g) 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 ( 9 4 ) 0 0 0 4 4 - P
M. Nakahara / Physics Letters A 186 (1994) 51-58
52
by introducing the Schwinger's proper time and an expansion with respect to it (Minakshisundaram-Seeley expansion [7,8] ). In a recent paper, W a x m a n and Ivanova-Moser [9] conjectured, without a proof, a totally new method to evaluate a F r e d h o l m determinant o f a Dirac operator. In the present Letter, we study a F r e d h o l m determinant of a Schr6dinger H a m i l t o n i a n defined on a compact one-dimensional space and prove the conjecture in this case. We first rewrite the Schr6dinger equation in a form o f a 2 × 2 first order equation and state our conjecture on the equivalence o f the F r e d h o l m determinant with the formal solution of the first order equation. Then we give a sketch o f the p r o o f o f the conjecture. As a byproduct, we obtain a new method to solve boundary-value problems in quantum mechanics. Several examples that illustrate our method are given. Let us consider a H a m i l t o n i a n defined on a one-dimensional interval, d2
H -
(4)
dx 2 + U ( x ) .
Here U ( x ) is a background potential, x E [0, n] and we have put h 2 / 2 m = 1. To find the spectrum of the Hamiltonian, one has to solve the Schr6dinger equation with a relevant boundary condition. Suppose the system is a circle. Instead o f imposing an ordinary periodic boundary condition, we introduce a family of more general boundary conditions parametrised by or, ~ ' ( n ) = e i~n ~ ( 0 ) ,
(5)
~ ' ( ~ z ) = e i~n ~ ' ( 0 ) ,
where ~ ( x ) is the Schr6dinger wave function. The choice a = 0 ( m o d 2) corresponds to a periodic b o u n d a r y condition, that o f a = - 1 ( m o d 2) to an antiperiodic boundary condition. We denote the set o f the spectrum o f H with the boundary condition, Eq. (5), by S p e c ( H ; a ) . Let us rewrite the equation ( H - z ) q / ( x ) = 0 as a first order differential equation, (6)
OxtP(x) = L ( x , z ) ~ ( x ) ,
where L(x,z)
=
(0,)
U(x) - z 0
and
~(x) = (~,(x) ~, (x)) The solution ~U(x) depends linearly on the initial condition ~ ( 0 ) . Let us define a 2 x 2 matrix M ( x , z ) by ~(x)
(7)
= M(x,z)~(O).
By definition, M ( x , z ) satisfies the initial condition M ( 0 , z) = 1 and a differential equation
(8)
OxM(x,z) = L(x,z)M(x,z),
which is formally integrated to yield
M(x,z)
= Pexp
(/)~/X/ L ( t , z) dt
=
dxl
n=O 0
dx2 0
...
XnS1
dxn L ( x l , z ) L ( x 2 , z ) . . . L( xn , z) ,
0
(9)
M. Nakahara~PhysicsLettersA 186 (1994)51-58
53
where the path-ordering operator P is defined by
PA(x)B(y) = A(x)B(y), = B(y)A(x),
x > y, y > x.
The path-ordering is required since [L (x, z), L (y, z ) ] ~ 0 in general. Note that M ( x , z) is the fundamental solution of the Schr6dinger equation. Since the determinant of a fundamental solution is a Wronskian it is conserved; d e t M ( x , z) = 1 Vx E [0, 7t]. In fact, 0x l n d e t M = Ox tr l n M = t r L = 0, from which d e t M ( x , z) is found to be independent of x. Let us consider the boundary condition (5). In terms of M (x, z ) this is expressed as M ( 7 ~ , z ) ~ ( 0 ) = e iaTt ~ ( 0 ) .
(10)
The matrix M (n, z) is called the monodromy matrix and plays the central r61e in this Letter. For a nontrivial ~u(0) to exist, the parameter z must be so chosen that M ( n , z) has an eigenvalue e i~. That is, z E S p e c ( H ; a ) if and only if det[M(Tr, z) - e i ~ ] = 0.
(11)
As is seen from Eq. (9), d e t [ M ( n , z) - e i~ ] is (formally) a power series in z and has a zero of an integral order at E , E Spec(H; a). We conjecture that det[M(Tt, z) - e i~ ] c( H ( E n - z).
(12)
n
This is by no means a trivial statement; in the r.h.s, of Eq. (12) the degeneracy of the eigenvalue is properly counted but we know nothing about the power o f each factor E~ - z in the 1.h.s. The (ill-defined) proportionality constant in Eq. (12) may be eliminated by taking the ratio of two determinants, det[M(~t, z) - e i~ ] I'L (E. - z) = det[ (M0(tr, z) - e io" ] I'Ii(E°n-z) '
(13)
where M0 (x, z) is the fundamental solution for a reference Hamiltonian, Ho =
-Ox~,
(14)
and {E °} = Spec(H0; a). We now give the proof of the conjecture, which is based essentially on Floquet's theorem and the theory of Hill's equation, see, e.g., Ref. [10]. Given a Hamiltonian H = - 0 2 + U(x), we write the Fredholm determinant as Det(H-
z) = D e t [ ( m ; a [ - 02 + U ( x ) - z[n;a)],
(15)
where (x In;a) = ( 1/x/n) e i°~ e 2inx. Since Det ( H - z ) is a divergent infinite product, it is ill-defined and must be regnlarised. For this purpose, we introduce a reference Hamiltonian H0 = - 0 ] and regularise the determinant as
D e t ( H - z) R(z;~)=~-~
-
Det{[ (a + 2n)2- z]6mn + gm-n} ( gm-n Det{[(~+2n)2_z]6mn } =Det (a+2n)2_
+ ~mn) z
(16) ,
where g, is the Fourier transform of U (x), U (x) = )--~:o=_oog, e2inx and we assumed, without loss of generality, that go = 0. For a ---, ic~, we have R ( z ; i o o ) = 1. It is clear that the poles of R (z; a ) in the a-plane come from the zeros of Det (H0 - z). They are simple poles at ~ = - 2 n ± ,/2. For a given n, these two poles have the residue of the same magnitude and opposite signature,
54
M. Nakahara / Physics Letters A 186 (1994) 51-58
which we call K and - K , respectively. Moreover R ( z ; a ) is periodic in a with periodicity two and hence the residues + K are independent o f n. If we subtract all the poles from R (z; a ) , we have an entire function, E(a)
= R(z;a)
sin(nv/Z)
- nK
cos(ha)
(17)
- cos(nv~)"
The function E ( a ) is b o u n d e d in the infinite strip IRe a I < 1 and must be a constant E by the m a x i m u m principle. We take the limit a -~ ioo to find E = 1. Then we obtain K = c°s(nv/-Z) - c o s ( h a ) [ R ( z ; a ) - 1 ]. n sin (nv/-£)
(18)
N o w let us write relation (17) for a p a r a m e t e r fi and eliminate K using Eq. (18), 1 = R(z;fl)
-
cos(rive) -cos(ha) cos(nx/~)
[ R ( z ; a ) - 1].
(19)
- cos(nfi)
It is easy to see that // cos(v/-zx) ( 1 / v ~ ) sin(v/-zx) ) = ~ _v~sin(v/-£x ) cos(v/~X ) .
Mo(x,z)
(20)
F r o m this, we obtain det [M0 (n, z) - e ia~ ] = 2 e ip~ [ c o s ( v / z n ) - c o s ( f / n ) ] .
(21)
Then it follows that z E S p e c ( H ; fl) if and only if d e t [ M 0 ( n , fi) - e i a ~ ] R ( z ; f i ) (x [cos(x/'£n) - c o s ( f i n ) ] R ( z ; f i )
= 0.
(22)
Using the above condition in Eq. (19) we find that z C Spec(H; fi) if and only if z and fi are related by R(z;cz) =
cos(an) - cos(fig) cos(an)
- cos(v~n)
(23) "
Floquet's theorem states that cos (fin) = ½tr M (n, z ) and using this relation to eliminate fi in the above equation we finally arrive at R(z;~) = cos(an) - ½trM(n,z) c o s ( a n ) - cos(x/~n)
=
det[M(n,z) det[M0 (n, 7.) - -
e jan ] ei~n]
'
(24)
where we have used the identity d e t ( M - e i a n ) = e ia= [ 2 c o s ( a n ) - t r M ] . This concludes the p r o o f o f our conjecture. We summarise our result as the following theorem. T h e o r e m 1. Suppose H = - 0 2 + U ( x ) is a Schrrdinger operator defined on an interval [0, n ] with a boundary condition ~ ( n ) = e ian ~ ( 0 ) , g t ' ( n ) = e ia= e / ' ( 0 ) . They define the set ofeigenvalues {En} = S p e c ( H ; a ) . Then it follows that I-I(En - z ) d e t [ M ( n , z ) - e i~= ] I-[(E0n - z) - d e t [ M o ( n , z) - ei~=] '
(25)
where H0 = - 0 x2, {E0n} = Spec(H0, a ) and
M(x,z)
= Pexp
z-
U(x)
0
"
(26)
M. Nakahara/PhysicsLette~A 186(199~51-58
55
If we put a = 0 and a = 1 in the theorem above, we obtain the following corollary.
Corollary 1. For a periodic case (a = 0) and an antiperiodic case (~ = 1) the ratios of the Fredholm determinants are given by R ( z ; 0 ) = det[M(rt, z ) - l ] det[Mo(lt, z) - 1]
(27)
an d
R ( z ; l ) = det[M(Tt, z) + 1] det[M0(Tt, z) + 1]"
(28)
So far we have studied systems with boundary conditions (5). We may impose other interesting boundary conditions as well. Let us impose the Dirichlet boundary condition, ~ , ( 0 ) = ¥ ( r t ) = 0.
(29)
If we note that ~ (Tt) = M (7t, z ) ~ (0) we find this condition is consistent if and only if M (Tt, z)12 --- 0. The following theorem can be proved in a similar way to theorem 1.
Theorem 2. Suppose H is a SchrSdinger Hamiltonian defined on the interval [0, n]. If we impose a hardwell boundary condition ~u(0) = ~(rt) = 0 we have the spectrum {En} = Spec(H;Dirichlet). The Fredholm determinant in this case is R (z; Dirichlet) =
D e t ( H - z) M(n,z)12 Det(H0 - z) Mo (rt, z)12"
(30)
We have presented a new approach to Fredholm determinants. Once the matrix M (x, z) and hence 1-in ( E n - z) is obtained it is an easy matter to find zeros of the Fredholm determinant to obtain Spec(H; a). It is a highly nontrivial task to evaluate the monodromy matrix in general due to the path-ordering operation in Eq. (9). However there is a class of easily solvable potentials, namely, piecewise constant potentials for which the Pordering becomes a simple juxtaposition of finite number of matrices. We look at several simple examples of this type. In traditional quantum mechanics these problems are solved by painstaking matching method. The reader will find how powerful our method is. In the following our system will be defined on an arbitrary interval [a, b], for which the previous formulae remain essentially unchanged. (I) Constant potential. For a constant potential U (x) = U0, we easily solve Eq. (8) to find c o s h ( V/-ff0 - z x )
M(x,z) =
V/-~o - z sinh(v/-U0 - z x )
(1/~oz)sinh(x/Uo - zx)~ c o s h ( ~ o ~- z x ) J
(31)
for U0> z and
cos(vff-Uox) M(x,z) =
_V/-~_Uosin(v/-~_Uox)
(I/v/-~ - Uo)sin(v/~
-
cos(v/2--~x)
UoX)
'~
J
(32)
for z > U0. If we impose a periodic boundary condition on [ 0 , L ] , we have, for z > U0, d e t [ M ( L , z) - 1] = 211 - c o s ( V / ' z - UoL)] = 0, from which we obtain a well-known result
z = En =
+ Uo.
(33)
M. Nakahara/ PhysicsLettersA 186 (1994)51-58
56
If a hard-well condition is imposed at x = 0 and L we have
Mlz(L,z) -
1
s i n ( v / ' z - UoL) = 0.
(34)
This is satisfied if and only if sin (~/z - U0 L) = 0 namely, z = En -
+ U0.
(35)
(II) Square-well potential. Let an interval [0, L] be divided into n smaller intervals of lengths Axl, Ax2 . . . . . Ax, on each of which the potential takes a constant value U1, U2,..., U,. Then the path-ordering is easily carried out to yield
M(L, z) = Mn (Ax,, z ) . . . M2 (Ax2, z)Ml (Axl, z),
(36)
where ( cosh(v/-ffi - zAxi) ( l / v / - ~ i - z) sinh(x/rU/i - z A x i ) ) Mi(Axi, z) = \ V/-~-7._-zsinh(v/-U~/-zAx, ) cosh(v/~/-zAxi ) ,
(37)
for Ui > z and c o s ( v / ~ _ Ui Axi)
Mi(L~Xi, Z) =
(1/V/-~-L--Ui)sin(v/-~-UiAxi))
_V[-~_ U, s i n ( x / z _ UiAx,)
cos(v/~-
V~kXi)
,
(38)
f o r z > U,. Let us consider a square-weU potential defined on [ - L , L]
U(x) = - v 0 , = 0,
Ixl < a, a ~< Ixl ~< Z.
(39)
Here we will work out the negative eigenvalues only. The positive ones are obtained in a similar manner. The m o n o d r o m y matrix is given by
M(2L, z) = M3(L - a, z)M2(2a, z)Ml(L - a,z),
(40)
with f
cosh [v/L-z(L - a) ]
M l , 3 ( L - a , z ) = ~v/-L--~sinh[v/-L--~(L_ a) ]
(1/VC-z) s i n h [ v / Z z ( L - a ) ] ) cosh[ff=z(L-a)]
and
Mz(2a, z) =
cos(2v/-z + Voa) ( l / v / - z + Vo) sin(2x/-z + V o a ) ) -V/-z + Vosin(2v/-z + Voa) c o s ( 2 v / z + V0a)
If w e i m p o s e a periodic b o u n d a r y c o n d i t i o n tr M (2L, z ) = 2, we have
t a n ( a x / ~ + Vo) = _ ~ _ z
+z V0 t a n h [ x / - ~ ( L - a ) ]
(41)
M. Nakahara ~PhysicsLettersA 186 (1994) 51-58
57
and ¢
t a n ( a v / z + Vo) = , ] -
V
z tanh[v/-~(L - a)]. z+V0
(42)
If, instead, a hard-well condition is imposed, we have M12 (2L, z) = 0, which yields the eigenvalue equations t a n ( a v / ' z + V0) = - ~
z +z V0 tanh[vt-L--~(L _ a ) ]
(43)
and !
tan(av/-z + V0) = ~ /
z+v0Z coth[v/'L--z(L- a ) ] .
(44)
(III) Periodic molecules. Let us study a more interesting example. We apply our method to a periodic molecule to obtain an exact electronic spectrum. Both cyclic molecules and linear molecules are considered and a comparison between our result and the electronic spectrum of benzene is made. In the previous section, we studied the spectrum of a particle with a square-well potential. Let us write the matrix M (2L, z) o f Eq. (40) as Mi (2L, z) =
C
"
(45)
Now consider a periodic array of n such square-wells with the length 2nL. We denote the monodromy matrix for this case as
M(2nL, z ) = M , ( 2 L , z ) n = ( An B n ) Cn Dn "
(46)
Then we can prove the following theorem by induction.
Theorem 3. Suppose H is a Schr6dinger operator with a periodic array of n square-well potentials. Then the matrix M(2nL, z) is given by M(2nL, z) =
(
Tn(A, (B/n)T" (A) ) (C/n)T'(A) Tn(A) '
(47)
where Tn (A) is the Chebycheff polynomial o f order n and prime denotes the derivative with respect to A. The details o f the proof will be found in Ref. [ 13 ]. Now that we have obtained M (2 n L, z ), it is an easy matter to obtain the eigenvalues associated with the potential with a given boundary condition. We first look at a periodic boundary condition. We have z E Spec(H; 0) if and only if
Tn(A(z)) = 1.
(48)
Although Tn (x) does not have a simple factorization, it is known that Tn (x) - 1 does admit a factorization. As an example, let us look at benzene, for which we take n = 6. Since T6(A) - 1 = 2[(A + 1)(2A + 1) 2 x (2A - 1 )~ (A - 1 )], the spectra are determined by the conditions
A ( z ) = 1, A(z) = --~t
A (z ) = ½
(doubly degenerate),
(doubly degenerate),
A(z) = - 1 .
(49)
For benzene, it is known that the inter-CH distance is 2L = 1.3965 A. The other parameters, the potential depth V0 and the potential width 2a are determined by adjusting our lowest two solutions with those observed, namely,
58
M. Nakahara / Physics Letters A 186 (1994) 51-58
E1 = - 1 2 . 2 5 eV and E2 = - 9 . 2 4 eV. By numerical calculation we find the pair a = 0.29 A a n d ~ = 22.7 eV gives the desired spectra E1 and E2. F r o m these parameters we find E 3 = - 0 . 4 5 eV. It is very difficult to obtain E3 experimentally. A reasonable guess obtained from ETS (electron transmission spectroscopy) is approximately 1.12 eV, see Ref. [12]. Although our result is not quite close to this value, it is still better than the prediction o f the Hiickel theory with a = - 6 . 2 3 eV and fl = - 3 . 0 1 eV, which yields E3 = - 3 . 2 2 eV. This shows that the overlap o f the atomic orbitals between the next-nearest neighbours may not be negligible for this level. Similar analysis can be made for other b o u n d a r y conditions. I f we impose a hard-well b o u n d a r y condition, the spectrum is determined by BT'(A(z))
= 0.
(50)
The 1.h.s. is known to admit a simple factorization in A (z). In summary, we have developed a totally new method to evaluate a Fredholm determinant D e t ( H - z) for a Schr6dinger H a m i l t o n i a n H defined on a compact one-dimensional space. By introducing a 2 × 2 matrix M (x, z) we gave a formal expression o f the F r e d h o l m determinant in terms o f a two-dimensional determinant. F o r a certain class o f potentials we gave an explicit form o f this determinant and obtained the exact energy spectrum o f a first quantised H a m i l t o n i a n H without invoking to a tedious matching method. After giving simple applications, we used our analysis to obtain the energy spectra of periodic molecules such as benzene. The detailed analysis of the present work with some more examples will be published separately in Ref. [ 13 ]. We expect our method is quite powerful in analysing condensed matter systems with piecewise constant background potentials, such as s u p e r c o n d u c t o r - n o r m a l metal sandwiches, q u a n t u m dots and superlattices. Works on these systems are in progress. After completing this work, the author has received a preprint from W a x m a n [ 14], in which a similar conjecture for a ( 1 + 1 )-dimensional Dirac operator is proved using the Green's function. This work is supported in part by the G r a n t in Aid for Scientific Research. The author would like to thank Takashi Aoki for patiently explaining him Hill's equations and Jun-ichi Aihara and Shin Takagi for enlightening discussions. He also thanks D a v i d W a x m a n for useful comments.
References [1 ] [2 ] [3] [4] [5] [6] [7] [8] [9] [10] [ 11] [12]
B. Sakita, Quantum theory of many-variable systems and fields (World Scientific, Singapore, 1985 ). S. Coleman, in: The whys of subnuclear physics (Plenum, New York, 1979). M. Nakahara, D. Waxman and G. Williams, J. Phys. A 23 (1990) 5017. M. Nakahara and G. Williams, Prog. Theor. Phys. 86 (1991) 315. M. Nakahara, D. Waxman and G. Williams, J. Phys. Condens. Matter 3 (1991) 6743. M. Nakahara, D. Waxman and G. Williams, Prog. Theor. Phys. 88 (1992) 129. S. Minakshisundaram, J. Indian Math. Soc. 41 (1941) 6. R.T. Seeley, Proc. Symp. Pure Appl. Math. AMS10 (1967) 288. D. Waxman and K.D. Ivanova-Moser, Ann. Phys. (NY)226 (1993)271. W. Magnus and S. Winkler, Hill's equation (Dover, New York, 1979). F.L. Pilar, Elementary quantum chemistry, 2rid Ed. (McGraw-Hill, New York, 1990). P.D. Burrow, J.A. Michejda and K.D. Jordan, J. Chem. Phys. 86 (1987) 9; L. Sanche and G.J. Schulz, J. Chem. Phys. 58 (1973) 479. [13 ] M. Nakahara, in preparation. [14] D. Waxman, to be published in Ann. Phys. (NY).
ELSEVIER
Physics Letters A 186 (1994) 51-58
Fredholm determinant and the Sturm-Liouville problems in quantum mechanics Mikio Nakahara Department of Physics, Faculty of Liberal Arts, Shizuoka University, Shizuoka 422, Japan, Department of Mathematics and Physics, Kinld University, Higashi-Osaka 577, Japan l Received 3 November 1993; revised manuscript received 6 January 1994; accepted for publication 13 January 1994 Communicated by A.P. Fordy
Abstract
The Fredholm determinant of a nonrelativistic Hamiltonian defined on a compact one-dimensional space is evaluated exactly. The Schr6dinger equation is rewritten as a first-order differential equation, which is integrated formally. Then a 2 x 2 eigenvalue equation is proved to be proportional to the Fredholm determinant. Our method turns out to be a powerful tool to solve eigenvalue problems in quantum mechanics. Finally several applications of our method are given.
G i v e n a H a m i l t o n i a n operator H , the Fredholm determinant Det ( H - z) is defined by D e t ( H - z) = 1-I(En - z ) ,
(1)
n
where the E~ are the spectrum o f H and z is a complex number. The Fredholm determinant is extensively used to obtain the effective field theory from a microscopic quantum field theory, see, e.g., Ref. [ 1 ]. This m e t h o d has been used in instanton calculus [2 ] and to evaluate the soliton creation energy in polyacetylene [3-5 ] and the vortex line profile in a type II superconductor [6]. Let us consider, e.g., the following action S [3],
S =f
f
(2)
where r is the imaginary time and H is a Hamiltonian o f either a Schr6dinger type or a Dirac type. The free energy F o f this system is obtained by a functional integral,
e -pF = f V ~ D ~ u e - s = D e t ( 0 , + H ) .
(3)
In the above equation and hereafter, Det and Tr denote a determinant and a trace, respectively, in a (infinitedimensional) functional space. We reserve det and tr for operations in a finite-dimensional vector space. In Ref. [ 1 ], a scattering theory analysis has been employed to evaluate a F r e d h o l m determinant and this is applied to polyactylene in Ref. [3]. In Refs. [ 4 - 6 ] , a systematic approximation scheme has been developed 1 Present address.
0375-9601/94/$07.00 (g) 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 ( 9 4 ) 0 0 0 4 4 - P
M. Nakahara / Physics Letters A 186 (1994) 51-58
52
by introducing the Schwinger's proper time and an expansion with respect to it (Minakshisundaram-Seeley expansion [7,8] ). In a recent paper, W a x m a n and Ivanova-Moser [9] conjectured, without a proof, a totally new method to evaluate a F r e d h o l m determinant o f a Dirac operator. In the present Letter, we study a F r e d h o l m determinant of a Schr6dinger H a m i l t o n i a n defined on a compact one-dimensional space and prove the conjecture in this case. We first rewrite the Schr6dinger equation in a form o f a 2 × 2 first order equation and state our conjecture on the equivalence o f the F r e d h o l m determinant with the formal solution of the first order equation. Then we give a sketch o f the p r o o f o f the conjecture. As a byproduct, we obtain a new method to solve boundary-value problems in quantum mechanics. Several examples that illustrate our method are given. Let us consider a H a m i l t o n i a n defined on a one-dimensional interval, d2
H -
(4)
dx 2 + U ( x ) .
Here U ( x ) is a background potential, x E [0, n] and we have put h 2 / 2 m = 1. To find the spectrum of the Hamiltonian, one has to solve the Schr6dinger equation with a relevant boundary condition. Suppose the system is a circle. Instead o f imposing an ordinary periodic boundary condition, we introduce a family of more general boundary conditions parametrised by or, ~ ' ( n ) = e i~n ~ ( 0 ) ,
(5)
~ ' ( ~ z ) = e i~n ~ ' ( 0 ) ,
where ~ ( x ) is the Schr6dinger wave function. The choice a = 0 ( m o d 2) corresponds to a periodic b o u n d a r y condition, that o f a = - 1 ( m o d 2) to an antiperiodic boundary condition. We denote the set o f the spectrum o f H with the boundary condition, Eq. (5), by S p e c ( H ; a ) . Let us rewrite the equation ( H - z ) q / ( x ) = 0 as a first order differential equation, (6)
OxtP(x) = L ( x , z ) ~ ( x ) ,
where L(x,z)
=
(0,)
U(x) - z 0
and
~(x) = (~,(x) ~, (x)) The solution ~U(x) depends linearly on the initial condition ~ ( 0 ) . Let us define a 2 x 2 matrix M ( x , z ) by ~(x)
(7)
= M(x,z)~(O).
By definition, M ( x , z ) satisfies the initial condition M ( 0 , z) = 1 and a differential equation
(8)
OxM(x,z) = L(x,z)M(x,z),
which is formally integrated to yield
M(x,z)
= Pexp
(/)~/X/ L ( t , z) dt
=
dxl
n=O 0
dx2 0
...
XnS1
dxn L ( x l , z ) L ( x 2 , z ) . . . L( xn , z) ,
0
(9)
M. Nakahara~PhysicsLettersA 186 (1994)51-58
53
where the path-ordering operator P is defined by
PA(x)B(y) = A(x)B(y), = B(y)A(x),
x > y, y > x.
The path-ordering is required since [L (x, z), L (y, z ) ] ~ 0 in general. Note that M ( x , z) is the fundamental solution of the Schr6dinger equation. Since the determinant of a fundamental solution is a Wronskian it is conserved; d e t M ( x , z) = 1 Vx E [0, 7t]. In fact, 0x l n d e t M = Ox tr l n M = t r L = 0, from which d e t M ( x , z) is found to be independent of x. Let us consider the boundary condition (5). In terms of M (x, z ) this is expressed as M ( 7 ~ , z ) ~ ( 0 ) = e iaTt ~ ( 0 ) .
(10)
The matrix M (n, z) is called the monodromy matrix and plays the central r61e in this Letter. For a nontrivial ~u(0) to exist, the parameter z must be so chosen that M ( n , z) has an eigenvalue e i~. That is, z E S p e c ( H ; a ) if and only if det[M(Tr, z) - e i ~ ] = 0.
(11)
As is seen from Eq. (9), d e t [ M ( n , z) - e i~ ] is (formally) a power series in z and has a zero of an integral order at E , E Spec(H; a). We conjecture that det[M(Tt, z) - e i~ ] c( H ( E n - z).
(12)
n
This is by no means a trivial statement; in the r.h.s, of Eq. (12) the degeneracy of the eigenvalue is properly counted but we know nothing about the power o f each factor E~ - z in the 1.h.s. The (ill-defined) proportionality constant in Eq. (12) may be eliminated by taking the ratio of two determinants, det[M(~t, z) - e i~ ] I'L (E. - z) = det[ (M0(tr, z) - e io" ] I'Ii(E°n-z) '
(13)
where M0 (x, z) is the fundamental solution for a reference Hamiltonian, Ho =
-Ox~,
(14)
and {E °} = Spec(H0; a). We now give the proof of the conjecture, which is based essentially on Floquet's theorem and the theory of Hill's equation, see, e.g., Ref. [10]. Given a Hamiltonian H = - 0 2 + U(x), we write the Fredholm determinant as Det(H-
z) = D e t [ ( m ; a [ - 02 + U ( x ) - z[n;a)],
(15)
where (x In;a) = ( 1/x/n) e i°~ e 2inx. Since Det ( H - z ) is a divergent infinite product, it is ill-defined and must be regnlarised. For this purpose, we introduce a reference Hamiltonian H0 = - 0 ] and regularise the determinant as
D e t ( H - z) R(z;~)=~-~
-
Det{[ (a + 2n)2- z]6mn + gm-n} ( gm-n Det{[(~+2n)2_z]6mn } =Det (a+2n)2_
+ ~mn) z
(16) ,
where g, is the Fourier transform of U (x), U (x) = )--~:o=_oog, e2inx and we assumed, without loss of generality, that go = 0. For a ---, ic~, we have R ( z ; i o o ) = 1. It is clear that the poles of R (z; a ) in the a-plane come from the zeros of Det (H0 - z). They are simple poles at ~ = - 2 n ± ,/2. For a given n, these two poles have the residue of the same magnitude and opposite signature,
54
M. Nakahara / Physics Letters A 186 (1994) 51-58
which we call K and - K , respectively. Moreover R ( z ; a ) is periodic in a with periodicity two and hence the residues + K are independent o f n. If we subtract all the poles from R (z; a ) , we have an entire function, E(a)
= R(z;a)
sin(nv/Z)
- nK
cos(ha)
(17)
- cos(nv~)"
The function E ( a ) is b o u n d e d in the infinite strip IRe a I < 1 and must be a constant E by the m a x i m u m principle. We take the limit a -~ ioo to find E = 1. Then we obtain K = c°s(nv/-Z) - c o s ( h a ) [ R ( z ; a ) - 1 ]. n sin (nv/-£)
(18)
N o w let us write relation (17) for a p a r a m e t e r fi and eliminate K using Eq. (18), 1 = R(z;fl)
-
cos(rive) -cos(ha) cos(nx/~)
[ R ( z ; a ) - 1].
(19)
- cos(nfi)
It is easy to see that // cos(v/-zx) ( 1 / v ~ ) sin(v/-zx) ) = ~ _v~sin(v/-£x ) cos(v/~X ) .
Mo(x,z)
(20)
F r o m this, we obtain det [M0 (n, z) - e ia~ ] = 2 e ip~ [ c o s ( v / z n ) - c o s ( f / n ) ] .
(21)
Then it follows that z E S p e c ( H ; fl) if and only if d e t [ M 0 ( n , fi) - e i a ~ ] R ( z ; f i ) (x [cos(x/'£n) - c o s ( f i n ) ] R ( z ; f i )
= 0.
(22)
Using the above condition in Eq. (19) we find that z C Spec(H; fi) if and only if z and fi are related by R(z;cz) =
cos(an) - cos(fig) cos(an)
- cos(v~n)
(23) "
Floquet's theorem states that cos (fin) = ½tr M (n, z ) and using this relation to eliminate fi in the above equation we finally arrive at R(z;~) = cos(an) - ½trM(n,z) c o s ( a n ) - cos(x/~n)
=
det[M(n,z) det[M0 (n, 7.) - -
e jan ] ei~n]
'
(24)
where we have used the identity d e t ( M - e i a n ) = e ia= [ 2 c o s ( a n ) - t r M ] . This concludes the p r o o f o f our conjecture. We summarise our result as the following theorem. T h e o r e m 1. Suppose H = - 0 2 + U ( x ) is a Schrrdinger operator defined on an interval [0, n ] with a boundary condition ~ ( n ) = e ian ~ ( 0 ) , g t ' ( n ) = e ia= e / ' ( 0 ) . They define the set ofeigenvalues {En} = S p e c ( H ; a ) . Then it follows that I-I(En - z ) d e t [ M ( n , z ) - e i~= ] I-[(E0n - z) - d e t [ M o ( n , z) - ei~=] '
(25)
where H0 = - 0 x2, {E0n} = Spec(H0, a ) and
M(x,z)
= Pexp
z-
U(x)
0
"
(26)
M. Nakahara/PhysicsLette~A 186(199~51-58
55
If we put a = 0 and a = 1 in the theorem above, we obtain the following corollary.
Corollary 1. For a periodic case (a = 0) and an antiperiodic case (~ = 1) the ratios of the Fredholm determinants are given by R ( z ; 0 ) = det[M(rt, z ) - l ] det[Mo(lt, z) - 1]
(27)
an d
R ( z ; l ) = det[M(Tt, z) + 1] det[M0(Tt, z) + 1]"
(28)
So far we have studied systems with boundary conditions (5). We may impose other interesting boundary conditions as well. Let us impose the Dirichlet boundary condition, ~ , ( 0 ) = ¥ ( r t ) = 0.
(29)
If we note that ~ (Tt) = M (7t, z ) ~ (0) we find this condition is consistent if and only if M (Tt, z)12 --- 0. The following theorem can be proved in a similar way to theorem 1.
Theorem 2. Suppose H is a SchrSdinger Hamiltonian defined on the interval [0, n]. If we impose a hardwell boundary condition ~u(0) = ~(rt) = 0 we have the spectrum {En} = Spec(H;Dirichlet). The Fredholm determinant in this case is R (z; Dirichlet) =
D e t ( H - z) M(n,z)12 Det(H0 - z) Mo (rt, z)12"
(30)
We have presented a new approach to Fredholm determinants. Once the matrix M (x, z) and hence 1-in ( E n - z) is obtained it is an easy matter to find zeros of the Fredholm determinant to obtain Spec(H; a). It is a highly nontrivial task to evaluate the monodromy matrix in general due to the path-ordering operation in Eq. (9). However there is a class of easily solvable potentials, namely, piecewise constant potentials for which the Pordering becomes a simple juxtaposition of finite number of matrices. We look at several simple examples of this type. In traditional quantum mechanics these problems are solved by painstaking matching method. The reader will find how powerful our method is. In the following our system will be defined on an arbitrary interval [a, b], for which the previous formulae remain essentially unchanged. (I) Constant potential. For a constant potential U (x) = U0, we easily solve Eq. (8) to find c o s h ( V/-ff0 - z x )
M(x,z) =
V/-~o - z sinh(v/-U0 - z x )
(1/~oz)sinh(x/Uo - zx)~ c o s h ( ~ o ~- z x ) J
(31)
for U0> z and
cos(vff-Uox) M(x,z) =
_V/-~_Uosin(v/-~_Uox)
(I/v/-~ - Uo)sin(v/~
-
cos(v/2--~x)
UoX)
'~
J
(32)
for z > U0. If we impose a periodic boundary condition on [ 0 , L ] , we have, for z > U0, d e t [ M ( L , z) - 1] = 211 - c o s ( V / ' z - UoL)] = 0, from which we obtain a well-known result
z = En =
+ Uo.
(33)
M. Nakahara/ PhysicsLettersA 186 (1994)51-58
56
If a hard-well condition is imposed at x = 0 and L we have
Mlz(L,z) -
1
s i n ( v / ' z - UoL) = 0.
(34)
This is satisfied if and only if sin (~/z - U0 L) = 0 namely, z = En -
+ U0.
(35)
(II) Square-well potential. Let an interval [0, L] be divided into n smaller intervals of lengths Axl, Ax2 . . . . . Ax, on each of which the potential takes a constant value U1, U2,..., U,. Then the path-ordering is easily carried out to yield
M(L, z) = Mn (Ax,, z ) . . . M2 (Ax2, z)Ml (Axl, z),
(36)
where ( cosh(v/-ffi - zAxi) ( l / v / - ~ i - z) sinh(x/rU/i - z A x i ) ) Mi(Axi, z) = \ V/-~-7._-zsinh(v/-U~/-zAx, ) cosh(v/~/-zAxi ) ,
(37)
for Ui > z and c o s ( v / ~ _ Ui Axi)
Mi(L~Xi, Z) =
(1/V/-~-L--Ui)sin(v/-~-UiAxi))
_V[-~_ U, s i n ( x / z _ UiAx,)
cos(v/~-
V~kXi)
,
(38)
f o r z > U,. Let us consider a square-weU potential defined on [ - L , L]
U(x) = - v 0 , = 0,
Ixl < a, a ~< Ixl ~< Z.
(39)
Here we will work out the negative eigenvalues only. The positive ones are obtained in a similar manner. The m o n o d r o m y matrix is given by
M(2L, z) = M3(L - a, z)M2(2a, z)Ml(L - a,z),
(40)
with f
cosh [v/L-z(L - a) ]
M l , 3 ( L - a , z ) = ~v/-L--~sinh[v/-L--~(L_ a) ]
(1/VC-z) s i n h [ v / Z z ( L - a ) ] ) cosh[ff=z(L-a)]
and
Mz(2a, z) =
cos(2v/-z + Voa) ( l / v / - z + Vo) sin(2x/-z + V o a ) ) -V/-z + Vosin(2v/-z + Voa) c o s ( 2 v / z + V0a)
If w e i m p o s e a periodic b o u n d a r y c o n d i t i o n tr M (2L, z ) = 2, we have
t a n ( a x / ~ + Vo) = _ ~ _ z
+z V0 t a n h [ x / - ~ ( L - a ) ]
(41)
M. Nakahara ~PhysicsLettersA 186 (1994) 51-58
57
and ¢
t a n ( a v / z + Vo) = , ] -
V
z tanh[v/-~(L - a)]. z+V0
(42)
If, instead, a hard-well condition is imposed, we have M12 (2L, z) = 0, which yields the eigenvalue equations t a n ( a v / ' z + V0) = - ~
z +z V0 tanh[vt-L--~(L _ a ) ]
(43)
and !
tan(av/-z + V0) = ~ /
z+v0Z coth[v/'L--z(L- a ) ] .
(44)
(III) Periodic molecules. Let us study a more interesting example. We apply our method to a periodic molecule to obtain an exact electronic spectrum. Both cyclic molecules and linear molecules are considered and a comparison between our result and the electronic spectrum of benzene is made. In the previous section, we studied the spectrum of a particle with a square-well potential. Let us write the matrix M (2L, z) o f Eq. (40) as Mi (2L, z) =
C
"
(45)
Now consider a periodic array of n such square-wells with the length 2nL. We denote the monodromy matrix for this case as
M(2nL, z ) = M , ( 2 L , z ) n = ( An B n ) Cn Dn "
(46)
Then we can prove the following theorem by induction.
Theorem 3. Suppose H is a Schr6dinger operator with a periodic array of n square-well potentials. Then the matrix M(2nL, z) is given by M(2nL, z) =
(
Tn(A, (B/n)T" (A) ) (C/n)T'(A) Tn(A) '
(47)
where Tn (A) is the Chebycheff polynomial o f order n and prime denotes the derivative with respect to A. The details o f the proof will be found in Ref. [ 13 ]. Now that we have obtained M (2 n L, z ), it is an easy matter to obtain the eigenvalues associated with the potential with a given boundary condition. We first look at a periodic boundary condition. We have z E Spec(H; 0) if and only if
Tn(A(z)) = 1.
(48)
Although Tn (x) does not have a simple factorization, it is known that Tn (x) - 1 does admit a factorization. As an example, let us look at benzene, for which we take n = 6. Since T6(A) - 1 = 2[(A + 1)(2A + 1) 2 x (2A - 1 )~ (A - 1 )], the spectra are determined by the conditions
A ( z ) = 1, A(z) = --~t
A (z ) = ½
(doubly degenerate),
(doubly degenerate),
A(z) = - 1 .
(49)
For benzene, it is known that the inter-CH distance is 2L = 1.3965 A. The other parameters, the potential depth V0 and the potential width 2a are determined by adjusting our lowest two solutions with those observed, namely,
58
M. Nakahara / Physics Letters A 186 (1994) 51-58
E1 = - 1 2 . 2 5 eV and E2 = - 9 . 2 4 eV. By numerical calculation we find the pair a = 0.29 A a n d ~ = 22.7 eV gives the desired spectra E1 and E2. F r o m these parameters we find E 3 = - 0 . 4 5 eV. It is very difficult to obtain E3 experimentally. A reasonable guess obtained from ETS (electron transmission spectroscopy) is approximately 1.12 eV, see Ref. [12]. Although our result is not quite close to this value, it is still better than the prediction o f the Hiickel theory with a = - 6 . 2 3 eV and fl = - 3 . 0 1 eV, which yields E3 = - 3 . 2 2 eV. This shows that the overlap o f the atomic orbitals between the next-nearest neighbours may not be negligible for this level. Similar analysis can be made for other b o u n d a r y conditions. I f we impose a hard-well b o u n d a r y condition, the spectrum is determined by BT'(A(z))
= 0.
(50)
The 1.h.s. is known to admit a simple factorization in A (z). In summary, we have developed a totally new method to evaluate a Fredholm determinant D e t ( H - z) for a Schr6dinger H a m i l t o n i a n H defined on a compact one-dimensional space. By introducing a 2 × 2 matrix M (x, z) we gave a formal expression o f the F r e d h o l m determinant in terms o f a two-dimensional determinant. F o r a certain class o f potentials we gave an explicit form o f this determinant and obtained the exact energy spectrum o f a first quantised H a m i l t o n i a n H without invoking to a tedious matching method. After giving simple applications, we used our analysis to obtain the energy spectra of periodic molecules such as benzene. The detailed analysis of the present work with some more examples will be published separately in Ref. [ 13 ]. We expect our method is quite powerful in analysing condensed matter systems with piecewise constant background potentials, such as s u p e r c o n d u c t o r - n o r m a l metal sandwiches, q u a n t u m dots and superlattices. Works on these systems are in progress. After completing this work, the author has received a preprint from W a x m a n [ 14], in which a similar conjecture for a ( 1 + 1 )-dimensional Dirac operator is proved using the Green's function. This work is supported in part by the G r a n t in Aid for Scientific Research. The author would like to thank Takashi Aoki for patiently explaining him Hill's equations and Jun-ichi Aihara and Shin Takagi for enlightening discussions. He also thanks D a v i d W a x m a n for useful comments.
References [1 ] [2 ] [3] [4] [5] [6] [7] [8] [9] [10] [ 11] [12]
B. Sakita, Quantum theory of many-variable systems and fields (World Scientific, Singapore, 1985 ). S. Coleman, in: The whys of subnuclear physics (Plenum, New York, 1979). M. Nakahara, D. Waxman and G. Williams, J. Phys. A 23 (1990) 5017. M. Nakahara and G. Williams, Prog. Theor. Phys. 86 (1991) 315. M. Nakahara, D. Waxman and G. Williams, J. Phys. Condens. Matter 3 (1991) 6743. M. Nakahara, D. Waxman and G. Williams, Prog. Theor. Phys. 88 (1992) 129. S. Minakshisundaram, J. Indian Math. Soc. 41 (1941) 6. R.T. Seeley, Proc. Symp. Pure Appl. Math. AMS10 (1967) 288. D. Waxman and K.D. Ivanova-Moser, Ann. Phys. (NY)226 (1993)271. W. Magnus and S. Winkler, Hill's equation (Dover, New York, 1979). F.L. Pilar, Elementary quantum chemistry, 2rid Ed. (McGraw-Hill, New York, 1990). P.D. Burrow, J.A. Michejda and K.D. Jordan, J. Chem. Phys. 86 (1987) 9; L. Sanche and G.J. Schulz, J. Chem. Phys. 58 (1973) 479. [13 ] M. Nakahara, in preparation. [14] D. Waxman, to be published in Ann. Phys. (NY).