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A characterization of the Hamiltonian
VO1. 53 (2004)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
A CHARACTERIZATION OF THE HAMILTONIAN FRANCISZEK HUGON SZAFRANIEC Instytut Matematyki, Uniwersytet Jagiellofiski ul. Reymonta 4, PL-30059 Krak6w, Poland ([email protected]) (Received September 29, 2003 - Revised January 19, 2004)
The classical Hamiltonian Z ( _ a2 + x 2) of the very classical quantum harmonic oscil2 dx2 lator, which is regarded as a germ of the most of what comes about in quantum mechanics, can be sublimed to an abstract operator in a separable Hilbert space. Having this done one may ask for a condition which would allow it to be identified among operators of a suitable class. This class is that corresponding to three diagonal matrices and the property which makes the action successful is a kind of diagonal invariance (up to change of basis) within the class in question. Keywords and phrases: unbounded operator, self-adjoint operator, Jacobi (matrix) operator, abstract Hamiltonian, Hermite polynomials, Charlier polynomials.
One of the intriguing features of the creation operator of the quantum harmonic oscillator is that its translational invariance distinguishes it from all the weighted shifts; this was discovered in [4] as a result of detailed analysis of what has been done in [6]. The proof in [4] makes use of geometrical aspects of weighted shifts. Afterwards, it turned out that the Charlier polynomials fit well into the circumstances making the argument more natural [7] and shedding new light on the creation operator itself (cf. [8]). In this paper we show that alike description is possible for the Hamiltonian as well. A bit of introduction oo 0 be a sequence of Hermite functions and let {c,(a)}~ = 0 be that of Let {h n}n= Charlier sequences (the definitions as well as further details are in the main body of the paper). Set oo
=
c
(n)hg,
n=0,1 .....
(1)
k=0
oo and {Sn(a)}~ Then both {h n}n=0 _-o are orthonormal bases in £a(I~). The research resulting in this paper was supported at its final stage by the KBN grant #2P03A 037 24. [3931
394
F.H. SZAFRANmC Comparing (3) and (17) we get Ua* ( ~~( ~ t d~ 2
: x2):)Ua
= ~ ( ~
..2~1 _ V~ax + a
(2)
on lin{sn~a); n = 0, 1 . . . . }, where Ua stands for the unitary operator of £2(I~) sending
s~a) into hn for every n. This sets up the following observation: the classical Hamiltonian 12 r~.- a 2dx 2 + x 2) is unitarily equivalent to itself modulo a Jacobi matrix operator ~/2-ax + a and this holds for any a > 0. This striking finding is the essence of our approach and it suggests the characterization of the Hamiltonian we are going to put into effect in this paper. More precisely, the equality (2) becomes a condition for an operator from the class of Jacobi matrices put into the greyish box to be identified as the classical Hamiltonian operator; notice for this it is enough (2) to hold for at least two a's (including 0). Abstract Hamiltonian: the definition
The Hamiltonian of the quantum harmonic oscillator is an operator !2 t~_ ddx2 2 .~_x 2) acting in £2(~) with domain equal to the linear span lin{hn; n = 0, 1 . . . . }, say; hn is the n-th Hermite function
hn = 2-n/E(n!)-l/27~-l/4e-x2/E Hn with Hn, the n-Hermite polynomial, defined as nn(x)
dn
=(-1)
n e x 2 ~--~e - x 2.
It is an essentially self-adjoint operator with discrete spectrum, that is I t ' - - d2 "-~ x 2 ) h n = ( n + 2', dx 2
1)hn,
n = 0, 1, . . . .
(3)
oo in it. Suppose we are given a separable Hilbert space 7-/ and a basis e =df {e n}~=0 Then we say that an operator H is a Hamiltonian with respect to e (or rather an abstract Hamiltouian with respect to this basis) if it is closabte,
Hen = (n + ~)en,
n = 0, 1 . . . . .
and lin{e~; n = 0, 1 . . . . } is its core 1. Such an H is apparently essentially seff-adjoint (that is, its closure /-/ is self-adjoint). l D contained in the domain D(A) of a closable operator A is its core if (AI29) = A with the dash -standing for the closure.
395
A CHARACTERIZATION OF THE HAMILTONIAN
Charlier
polynomials
and the discrete
Itamfltonian
The Charlier polynomials 2 {c~a)}~=0, a > 0, are determined by oo
e-az(1 q- Z) x
n
~ c(a)(x) ~-f-. =Z__, n n[ n=0
They are orthogonal with respect to a nonnegative integer supported measure according to
oo e_a ax E c- - m( a ) (v -~ cz - -' n~ ( 7 ( a ) ( x )X - !
--
Smnann!,
m, n = O, 1 . . . . .
x=0
Define the Charlier functions (or, rather, the Charlier sequences) c~a), n = O, 1 . . . . in a discrete variable x as z ~ a X c n(a )(x) d_.f ( _ l ) n a _ 7 (n n )_~c~a)(x)e_~aT(x! ) - - ½,
for x _> 0.
The sequence "f'~(a)/°° t~'n Jn=0 satisfies
c~a)(x) = c~xa)(n),
n, x = 0, 1 . . . .
(4)
and hence (cf. [2]) is a basis in e 2. Because
- n C ( a ) ( x ) : aC(na)(x q- 1) - (x + a)C(na)(x)
-]- x C ( a ) ( x
-
1),
the sequence {c(a)}~= o satisfies the second-order difference equation 3
-nc(a)(x) = v/a(x + 1)c(a)(x + 1) - (x + a)c~a)(x) +
~/-'~£(a)(x
--
1),
n, x = O, 1 . . . . .
(5)
Setting for the domain
D(H(a)) ~- - lln{{c~ . (a) }~=o, oo . n = O, 1 . . . . } and for the operator 4
(H(a) f ) ( x ) df --~/a(x + 1 ) f ( x + 1) + (x + a + ½ ) f ( x ) -- V'-d-xf(x - 1), f ~ D(H(~)),
n, x = O, 1 . . . . .
2See either [1] or [3]; their definitions differ by a factor. 3If ~ c g2 we write ~(n) for its n-th coordinate. 4Notice that f ( - 1 ) at the end of the right-hand side of the formula (6) is irrelevant.
(6)
396
F.H. SZAFRANIEC
we get, by (5), according to the previous definition, a Hamiltonian H (a) with respect to the basis {cn(a)}~=o which is nothing but a discrete Hamiltonian acting in ~2 a s a second-order difference operator. Because e2-convergence yields coordinatewise one we get that f is in D(H~ a)) if and only if the sequence + 1 ) f ( x + 1) + (x + a + 1 ) f ( x ) - ~/-d-~f(x - 1)
x ~ -,¢/~
is in e 2. Consequently, the canonical zero-one basis {c~°~}~=0, say, of in D(H(a)). Thus we have
H~a)e~°) = H(°)e(n°) - ~ - ~
~2
is contained
.. (o) _(o) m (o) + l)en+ 1 + ae, - ~/anen_ 1,
(7)
where H ~°) is a Hamiltonian with respect to t,,~0~/o~ t~n m=0" (7) says roughly that the two Hamiltonians (apparently with respect to different bases) differ by a Jacobi matrix. This motif is the starting point for our programme. Hamiltonian
as a Jacobi
matrix
Come back to the Hilbert space 7-/ and set O0
e~na) d~ ~-,c~a~(n)ek;
n =O, 1 . . . . .
(8)
k=0
Then, according to Proposition 2 of [7], e (a) de to(a)~ l-~n J n ~ 0 is another basis in ~ and the reciprocity V TM r(a) (lr'~a(a). L . a ~ n v~J"k , k=O
en
n
f~ v , 1~, . . . ,
holds. Let H be the Hamiltonian with respect to e. For a > 0 we have due to (4) and (6) N
N
H y ~ c~a)(n)ek = ~ k=0
c~'O(n)(k + 1)ek = Y~Lo(H(alc~a))(n)ek
k=0
N
= ~--~(_~/c~
-{- 1)C(ka)( n "1-
1)+ (n + a
+ 1)c~a)(n)-
q"-a-ffc~a)(n -
1))ek
k=0 N
= - v / a - ~ + 1) ~
c(a)(n + 1)ek + (n + a + 1) )--~k=0Nc~a)(n)ek
k=0 N
- vran Z k=O
c~a)(n - 1 ) e k .
(9)
397
A CHARACTERIZATION OF THE HAMILTONIAN
Because every ingredient in the very last expression above converges in £2 as N ~ c~ and H is closable we get that en(a) ~ 79(//) and .~ (a) 1 / / e (a) = --x/a(n d- l)en+ 1 q- (n + ~ q- a)e (a) - ~-d-ffe,_ (a)1,
n = 0, 1, . . . .
(10)
n = 0, 1 , . . .
(11)
This means that the operator l~-(a) - ae(na) q- C~h-ffe~(a_)1, H ( a ) : e(~a) ~ / / e ( ~ a) + v / a ( n + "J%+1
is a Hamiltonian with respect to the basis {e(a)}~=0. So having a formula (a) _ ae(a) + ~ / - ~ e n(a) _l, + ~. l)en+l
H(a)e~a) = //e(na) + ~
n = 0, 1. . . . .
(12)
we do also its dual 5, which resembles (7), H(a)en = H e n - ~/a(n W 1)en+l q- aen - ~fa-ffen-1,
n = O, 1 . . . .
Anyway, a glimpse of (1) leads to the rough statement: after passing from one basis to the other, the old Hamiltonian becomes a Jacobi matrix in which the new one is included as a natural ingredient (or the other way); this is the message we want to bring forth here. CONCLUSION. If H is a Hamiltonian with respect to e then so is the operator in (11) with respect to e (a) f o r every a > O. Moreover, (X)
~-~(H(a)c(ka))(n)ek= [-le(na) -- ~/a(n -]- .l)en+ . (a) 1 Jr ae(na) -- ~d'ffe(na)_l k=0 OO
n=0,1
....
y ~ ,'" ( l-l(a)o(a)~[~ "~(a)= H e n q- v / a ( n q- 1)en+l - aen q- ~¢rd-nen_1. ~k ,v.J~k k=0
A look at the third and the fifth member of the chain of equalities (9) gives us the first of the above, the other goes the same way. This is what we are going exploit later on as well. REMARK. Notice that 7 9 ( / - / ( a ) ) = 7 9 ( / - / ) regardless of a > 0.
Diagonal invariance and the eharaeterization Suppose we are given ~ ~ r(a)~¢~ 1° two bases e = {e n}n=0 and f(a) = tJn J~=0, 2 ° two closable operators H and G (") such that e is a core for H and so is e (a) for G ("), and such that e (a) C D ( H ) and e, f(a) C D(G(a)). 5The reader is asked to forgive us each time we apply the selfduality relation (4) without mentioning it; here is this case.
398
F. I~. SZAFRANmC
Suppose, moreover,
G(a)en = Hen - v/a(n + 1)en+l + aen - ~fa-nen-1,
n = 0, 1 . . . . .
(13)
Under these circumstances the first part of Conclusion gains its converse which rounds up the characterization in question. THEOREM 1. Assume what is in the preamble to this section. If H acts as a diagonal matrix with respect to e, that is Hen = )~nen, and for some a > 0 so does G (a) with respect to f(a), that is G(a) f(a) : t~nJntt rr(a), where {~.n}n= 0o0 and to {/z,}n=0 are arbitrary sequences of complex numbers, then Xn = lZn = n + 1,
provided )~o = #o = ~1 and f ( a ) = e(a). Consequently, H is a Hamiltonian with respect to e (cf. (8)) and G(a) = H (a) (that is, it is a Hamiltonian with respect to f(a) = e(a)). fn (a) = e (a), n ---- 0 , 1 . . . .
Proof: Write fn(a) = Y-~k=O~k,nek to in G(a) fr~a ) = lZnfn(a), engage (13), play with closability like in (10) and finally use the fact that e is orthonormal and complete so as to get ()~k "~ a -- [Zn)~k,n -- ~lra-n~k_l, n -- v / - ~ - ~
- 1)~k,n+l :
0,
k, n = 0, 1 . . . . .
(14)
Starting to compare (14) with (5) for n = 0, in which case (14) is just ) ~ k - a~k,O Jr ~/-d~k,1 = 0 and ~k,O = c~a)(o), we go on to deduce that ~,n=l~n=n"[
-1 ,
~k,n=c~a)(n),
k,n =0,1 .....
Because ~k,n = (f~("), ek), the second condition in the above identifies f ( " ) as e ("). [] Interaction between the discrete Hamiltonian and an abstract one Suppose F is an operator in £2 such that hn{cn " (a) }n=0' ~ . n = 0, 1 , . . . } is its core and H is an operator in 7-[ such that lin{en; n = 0, 1 . . . . } is its core for some basis {e n}n=0" Suppose moreover that Wn t . , ( a )Jn=0 ~ C ~)(H). Consider the following conditions:
~(Fc~a))(n) ek =
lte(na) -v/-d-~
. (a) _t.ae(a) ~. a/r'~(a) + Il)en+ln --'V .... n--l'
n = 0, 1 .....
(15)
n = 0, 1 . . . . .
(16)
k=0
to
--
-
-~(Fc(~a)(n)e---(ka) t t n + ~/a(n + 1)en+l -- aen + v f ~ e n - 1 ,
k=0
These formulae are generic, one can get recovered from any o f them the Hamiltonians as in Conclusion. THEOREM 2. Suppose H acts as a diagonal matrix with respect to {en}n=O and F acts as a diagonal matrix with respect to t,.(a)~ t~n in=0" I f for some a > 0 they
A CHARACTERIZATION OF THE HAMILTONIAN
399
oo and satisfy either (15) or (16) then H is a Hamiltonian with respect to {e n}rt_-o so is F, is a Hamiltonian with respect to ~p(a)too /_/(a)) provided tt'n Jn=0 (that is, F ;LO=/zO=
1 ~.
Proof: Let {Xn]~-0 and {/zn}~-o be the diagonal entries for H and F, respectively. Suppose (15) holds. Then o~
oo
E.kc~a'(n)ek=E(Fc~a')(n)ek k=0
k=0
+ l)en+l . \ (a) + ae(a) _ ~e(na)_l
= He( a~ _ ~ oo
= --~ (Zrtc(ka)(n)- x/a(n + 1)c~a)(n)+ ac~a)(n)- ~/"~c~a) (n ) )en_a . k=0
Equating the coefficients at ei we get
tzic~a)(n) = ~.nc~a)(n) - ~/a(n + 1)c~a)(n) + ac~a)(n) - ~'a-ffc~a)(n), and comparing this with (5) (recall X0 = / z 0 = ½) we come to the conclusion.
[]
Theorem 2 is a sample of what is possible to carry out from the duality relations (15) or (16). Others can be guessed after looking at [8] where a number of combinations for the creation operator is considered.
Back to the configuration space In the case of 7-/=/~2(I~) and en = hn, where hn is the n-th Hermite function, the basis t~(~)~oo t°rt Jn~..~.0 drafted after (8) is of the form (1), that is oo S~ a) =df E c
~a)(n)hk,
n = 0 , 1. . . . .
k=0
Remind that the three-term recurrence relation for the Hermite functions is
xhn = ~/l(n +
1)hn+l +
V~nhn-1.
Taking into account this and the following two relations, cf. [5] 6, n ~ - ~ C ( ~ X ( X ) = ~ ¢ / a c ( a ) ( x ) -- ~4/"xc(a)(x -- 1),
= V dc a)(x) --
c(a~ df 0,
XgT-4--fc a (x + 1)
6There is a difference (by the factor ( - 1 ) n) in defining Charlier functions there and here.
400
F.H. SZAFRANIEC
w e get t h e t h r e e - t e r m r e c u r r e n c e r e l a t i o n for s~a)'s,
XS(na) = - - / l ( n
. (a) 1 -I- ~/~as(na) __ ~/~ ~nSn-l" (a) + tl)Sn+
Then, by (12), ( ! c - d2 + x 2) - ~/2-dx + a)S~na) = (n + 1-~s(a) 2', dx 2 2j n ,
n = O, 1 . . . . .
(17)
and =
1"~ ^(a)
_as(na)~'~-ffs(a)__X,
n = O, 1 . . . . .
(18)
Consequently, the operator
!c-d~ +x 2)- ~x 2 ~ dx 2
+a
is a H a m i l t o n i a n w i t h r e s p e c t to t°n t~(a)~oo J n = 0 a n d its p a r t
-~f-~x
+ a
is a Jacobi matrix operator with respect to the same basis. REFERENCES [1] T. S. Chihara: An Introduction to Orthogonal Polynomials, Gordon and Breach, New York 1978. [2] G. K. Eagleson: A duality relation for discrete orthogonal systems, Studia Sc. Math. Hung. 3 (1968), 127. [3] R. Koekoek and R. F. Swarttouw: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Report of the Department of Technical Mathematics and Informatics no. 98-17, 1998. [4] J. Stochel and F. H. Szafraniec: A peculiarity of the creation operator, Glasgow Math. J. 44 (2001), 137. [5] F. H. Szafraniec: Yet another face Of the creation operator, Operator Th. Adv. Appl. 80 (1995), 266. [6] F. H. Szafraniec: Subnormality in the quantum harmonic oscillator, Commun. Math. Phys. 210 (2000), 323. [7] E H. Szafraniec: Charlier polynomials and translational invariance in the quantum harmonic oscillator, Math. Nachtr. 241 (2002), 166. [8] E H. Szafraniec: Duality in the quantum harmonic oscillator, J. Phys. A: Math. Gen. 34 10487 (2001).
REPORTS ON MATHEMATICAL PHYSICS
No. 3
A CHARACTERIZATION OF THE HAMILTONIAN FRANCISZEK HUGON SZAFRANIEC Instytut Matematyki, Uniwersytet Jagiellofiski ul. Reymonta 4, PL-30059 Krak6w, Poland ([email protected]) (Received September 29, 2003 - Revised January 19, 2004)
The classical Hamiltonian Z ( _ a2 + x 2) of the very classical quantum harmonic oscil2 dx2 lator, which is regarded as a germ of the most of what comes about in quantum mechanics, can be sublimed to an abstract operator in a separable Hilbert space. Having this done one may ask for a condition which would allow it to be identified among operators of a suitable class. This class is that corresponding to three diagonal matrices and the property which makes the action successful is a kind of diagonal invariance (up to change of basis) within the class in question. Keywords and phrases: unbounded operator, self-adjoint operator, Jacobi (matrix) operator, abstract Hamiltonian, Hermite polynomials, Charlier polynomials.
One of the intriguing features of the creation operator of the quantum harmonic oscillator is that its translational invariance distinguishes it from all the weighted shifts; this was discovered in [4] as a result of detailed analysis of what has been done in [6]. The proof in [4] makes use of geometrical aspects of weighted shifts. Afterwards, it turned out that the Charlier polynomials fit well into the circumstances making the argument more natural [7] and shedding new light on the creation operator itself (cf. [8]). In this paper we show that alike description is possible for the Hamiltonian as well. A bit of introduction oo 0 be a sequence of Hermite functions and let {c,(a)}~ = 0 be that of Let {h n}n= Charlier sequences (the definitions as well as further details are in the main body of the paper). Set oo
=
c
(n)hg,
n=0,1 .....
(1)
k=0
oo and {Sn(a)}~ Then both {h n}n=0 _-o are orthonormal bases in £a(I~). The research resulting in this paper was supported at its final stage by the KBN grant #2P03A 037 24. [3931
394
F.H. SZAFRANmC Comparing (3) and (17) we get Ua* ( ~~( ~ t d~ 2
: x2):)Ua
= ~ ( ~
..2~1 _ V~ax + a
(2)
on lin{sn~a); n = 0, 1 . . . . }, where Ua stands for the unitary operator of £2(I~) sending
s~a) into hn for every n. This sets up the following observation: the classical Hamiltonian 12 r~.- a 2dx 2 + x 2) is unitarily equivalent to itself modulo a Jacobi matrix operator ~/2-ax + a and this holds for any a > 0. This striking finding is the essence of our approach and it suggests the characterization of the Hamiltonian we are going to put into effect in this paper. More precisely, the equality (2) becomes a condition for an operator from the class of Jacobi matrices put into the greyish box to be identified as the classical Hamiltonian operator; notice for this it is enough (2) to hold for at least two a's (including 0). Abstract Hamiltonian: the definition
The Hamiltonian of the quantum harmonic oscillator is an operator !2 t~_ ddx2 2 .~_x 2) acting in £2(~) with domain equal to the linear span lin{hn; n = 0, 1 . . . . }, say; hn is the n-th Hermite function
hn = 2-n/E(n!)-l/27~-l/4e-x2/E Hn with Hn, the n-Hermite polynomial, defined as nn(x)
dn
=(-1)
n e x 2 ~--~e - x 2.
It is an essentially self-adjoint operator with discrete spectrum, that is I t ' - - d2 "-~ x 2 ) h n = ( n + 2', dx 2
1)hn,
n = 0, 1, . . . .
(3)
oo in it. Suppose we are given a separable Hilbert space 7-/ and a basis e =df {e n}~=0 Then we say that an operator H is a Hamiltonian with respect to e (or rather an abstract Hamiltouian with respect to this basis) if it is closabte,
Hen = (n + ~)en,
n = 0, 1 . . . . .
and lin{e~; n = 0, 1 . . . . } is its core 1. Such an H is apparently essentially seff-adjoint (that is, its closure /-/ is self-adjoint). l D contained in the domain D(A) of a closable operator A is its core if (AI29) = A with the dash -standing for the closure.
395
A CHARACTERIZATION OF THE HAMILTONIAN
Charlier
polynomials
and the discrete
Itamfltonian
The Charlier polynomials 2 {c~a)}~=0, a > 0, are determined by oo
e-az(1 q- Z) x
n
~ c(a)(x) ~-f-. =Z__, n n[ n=0
They are orthogonal with respect to a nonnegative integer supported measure according to
oo e_a ax E c- - m( a ) (v -~ cz - -' n~ ( 7 ( a ) ( x )X - !
--
Smnann!,
m, n = O, 1 . . . . .
x=0
Define the Charlier functions (or, rather, the Charlier sequences) c~a), n = O, 1 . . . . in a discrete variable x as z ~ a X c n(a )(x) d_.f ( _ l ) n a _ 7 (n n )_~c~a)(x)e_~aT(x! ) - - ½,
for x _> 0.
The sequence "f'~(a)/°° t~'n Jn=0 satisfies
c~a)(x) = c~xa)(n),
n, x = 0, 1 . . . .
(4)
and hence (cf. [2]) is a basis in e 2. Because
- n C ( a ) ( x ) : aC(na)(x q- 1) - (x + a)C(na)(x)
-]- x C ( a ) ( x
-
1),
the sequence {c(a)}~= o satisfies the second-order difference equation 3
-nc(a)(x) = v/a(x + 1)c(a)(x + 1) - (x + a)c~a)(x) +
~/-'~£(a)(x
--
1),
n, x = O, 1 . . . . .
(5)
Setting for the domain
D(H(a)) ~- - lln{{c~ . (a) }~=o, oo . n = O, 1 . . . . } and for the operator 4
(H(a) f ) ( x ) df --~/a(x + 1 ) f ( x + 1) + (x + a + ½ ) f ( x ) -- V'-d-xf(x - 1), f ~ D(H(~)),
n, x = O, 1 . . . . .
2See either [1] or [3]; their definitions differ by a factor. 3If ~ c g2 we write ~(n) for its n-th coordinate. 4Notice that f ( - 1 ) at the end of the right-hand side of the formula (6) is irrelevant.
(6)
396
F.H. SZAFRANIEC
we get, by (5), according to the previous definition, a Hamiltonian H (a) with respect to the basis {cn(a)}~=o which is nothing but a discrete Hamiltonian acting in ~2 a s a second-order difference operator. Because e2-convergence yields coordinatewise one we get that f is in D(H~ a)) if and only if the sequence + 1 ) f ( x + 1) + (x + a + 1 ) f ( x ) - ~/-d-~f(x - 1)
x ~ -,¢/~
is in e 2. Consequently, the canonical zero-one basis {c~°~}~=0, say, of in D(H(a)). Thus we have
H~a)e~°) = H(°)e(n°) - ~ - ~
~2
is contained
.. (o) _(o) m (o) + l)en+ 1 + ae, - ~/anen_ 1,
(7)
where H ~°) is a Hamiltonian with respect to t,,~0~/o~ t~n m=0" (7) says roughly that the two Hamiltonians (apparently with respect to different bases) differ by a Jacobi matrix. This motif is the starting point for our programme. Hamiltonian
as a Jacobi
matrix
Come back to the Hilbert space 7-/ and set O0
e~na) d~ ~-,c~a~(n)ek;
n =O, 1 . . . . .
(8)
k=0
Then, according to Proposition 2 of [7], e (a) de to(a)~ l-~n J n ~ 0 is another basis in ~ and the reciprocity V TM r(a) (lr'~a(a). L . a ~ n v~J"k , k=O
en
n
f~ v , 1~, . . . ,
holds. Let H be the Hamiltonian with respect to e. For a > 0 we have due to (4) and (6) N
N
H y ~ c~a)(n)ek = ~ k=0
c~'O(n)(k + 1)ek = Y~Lo(H(alc~a))(n)ek
k=0
N
= ~--~(_~/c~
-{- 1)C(ka)( n "1-
1)+ (n + a
+ 1)c~a)(n)-
q"-a-ffc~a)(n -
1))ek
k=0 N
= - v / a - ~ + 1) ~
c(a)(n + 1)ek + (n + a + 1) )--~k=0Nc~a)(n)ek
k=0 N
- vran Z k=O
c~a)(n - 1 ) e k .
(9)
397
A CHARACTERIZATION OF THE HAMILTONIAN
Because every ingredient in the very last expression above converges in £2 as N ~ c~ and H is closable we get that en(a) ~ 79(//) and .~ (a) 1 / / e (a) = --x/a(n d- l)en+ 1 q- (n + ~ q- a)e (a) - ~-d-ffe,_ (a)1,
n = 0, 1, . . . .
(10)
n = 0, 1 , . . .
(11)
This means that the operator l~-(a) - ae(na) q- C~h-ffe~(a_)1, H ( a ) : e(~a) ~ / / e ( ~ a) + v / a ( n + "J%+1
is a Hamiltonian with respect to the basis {e(a)}~=0. So having a formula (a) _ ae(a) + ~ / - ~ e n(a) _l, + ~. l)en+l
H(a)e~a) = //e(na) + ~
n = 0, 1. . . . .
(12)
we do also its dual 5, which resembles (7), H(a)en = H e n - ~/a(n W 1)en+l q- aen - ~fa-ffen-1,
n = O, 1 . . . .
Anyway, a glimpse of (1) leads to the rough statement: after passing from one basis to the other, the old Hamiltonian becomes a Jacobi matrix in which the new one is included as a natural ingredient (or the other way); this is the message we want to bring forth here. CONCLUSION. If H is a Hamiltonian with respect to e then so is the operator in (11) with respect to e (a) f o r every a > O. Moreover, (X)
~-~(H(a)c(ka))(n)ek= [-le(na) -- ~/a(n -]- .l)en+ . (a) 1 Jr ae(na) -- ~d'ffe(na)_l k=0 OO
n=0,1
....
y ~ ,'" ( l-l(a)o(a)~[~ "~(a)= H e n q- v / a ( n q- 1)en+l - aen q- ~¢rd-nen_1. ~k ,v.J~k k=0
A look at the third and the fifth member of the chain of equalities (9) gives us the first of the above, the other goes the same way. This is what we are going exploit later on as well. REMARK. Notice that 7 9 ( / - / ( a ) ) = 7 9 ( / - / ) regardless of a > 0.
Diagonal invariance and the eharaeterization Suppose we are given ~ ~ r(a)~¢~ 1° two bases e = {e n}n=0 and f(a) = tJn J~=0, 2 ° two closable operators H and G (") such that e is a core for H and so is e (a) for G ("), and such that e (a) C D ( H ) and e, f(a) C D(G(a)). 5The reader is asked to forgive us each time we apply the selfduality relation (4) without mentioning it; here is this case.
398
F. I~. SZAFRANmC
Suppose, moreover,
G(a)en = Hen - v/a(n + 1)en+l + aen - ~fa-nen-1,
n = 0, 1 . . . . .
(13)
Under these circumstances the first part of Conclusion gains its converse which rounds up the characterization in question. THEOREM 1. Assume what is in the preamble to this section. If H acts as a diagonal matrix with respect to e, that is Hen = )~nen, and for some a > 0 so does G (a) with respect to f(a), that is G(a) f(a) : t~nJntt rr(a), where {~.n}n= 0o0 and to {/z,}n=0 are arbitrary sequences of complex numbers, then Xn = lZn = n + 1,
provided )~o = #o = ~1 and f ( a ) = e(a). Consequently, H is a Hamiltonian with respect to e (cf. (8)) and G(a) = H (a) (that is, it is a Hamiltonian with respect to f(a) = e(a)). fn (a) = e (a), n ---- 0 , 1 . . . .
Proof: Write fn(a) = Y-~k=O~k,nek to in G(a) fr~a ) = lZnfn(a), engage (13), play with closability like in (10) and finally use the fact that e is orthonormal and complete so as to get ()~k "~ a -- [Zn)~k,n -- ~lra-n~k_l, n -- v / - ~ - ~
- 1)~k,n+l :
0,
k, n = 0, 1 . . . . .
(14)
Starting to compare (14) with (5) for n = 0, in which case (14) is just ) ~ k - a~k,O Jr ~/-d~k,1 = 0 and ~k,O = c~a)(o), we go on to deduce that ~,n=l~n=n"[
-1 ,
~k,n=c~a)(n),
k,n =0,1 .....
Because ~k,n = (f~("), ek), the second condition in the above identifies f ( " ) as e ("). [] Interaction between the discrete Hamiltonian and an abstract one Suppose F is an operator in £2 such that hn{cn " (a) }n=0' ~ . n = 0, 1 , . . . } is its core and H is an operator in 7-[ such that lin{en; n = 0, 1 . . . . } is its core for some basis {e n}n=0" Suppose moreover that Wn t . , ( a )Jn=0 ~ C ~)(H). Consider the following conditions:
~(Fc~a))(n) ek =
lte(na) -v/-d-~
. (a) _t.ae(a) ~. a/r'~(a) + Il)en+ln --'V .... n--l'
n = 0, 1 .....
(15)
n = 0, 1 . . . . .
(16)
k=0
to
--
-
-~(Fc(~a)(n)e---(ka) t t n + ~/a(n + 1)en+l -- aen + v f ~ e n - 1 ,
k=0
These formulae are generic, one can get recovered from any o f them the Hamiltonians as in Conclusion. THEOREM 2. Suppose H acts as a diagonal matrix with respect to {en}n=O and F acts as a diagonal matrix with respect to t,.(a)~ t~n in=0" I f for some a > 0 they
A CHARACTERIZATION OF THE HAMILTONIAN
399
oo and satisfy either (15) or (16) then H is a Hamiltonian with respect to {e n}rt_-o so is F, is a Hamiltonian with respect to ~p(a)too /_/(a)) provided tt'n Jn=0 (that is, F ;LO=/zO=
1 ~.
Proof: Let {Xn]~-0 and {/zn}~-o be the diagonal entries for H and F, respectively. Suppose (15) holds. Then o~
oo
E.kc~a'(n)ek=E(Fc~a')(n)ek k=0
k=0
+ l)en+l . \ (a) + ae(a) _ ~e(na)_l
= He( a~ _ ~ oo
= --~ (Zrtc(ka)(n)- x/a(n + 1)c~a)(n)+ ac~a)(n)- ~/"~c~a) (n ) )en_a . k=0
Equating the coefficients at ei we get
tzic~a)(n) = ~.nc~a)(n) - ~/a(n + 1)c~a)(n) + ac~a)(n) - ~'a-ffc~a)(n), and comparing this with (5) (recall X0 = / z 0 = ½) we come to the conclusion.
[]
Theorem 2 is a sample of what is possible to carry out from the duality relations (15) or (16). Others can be guessed after looking at [8] where a number of combinations for the creation operator is considered.
Back to the configuration space In the case of 7-/=/~2(I~) and en = hn, where hn is the n-th Hermite function, the basis t~(~)~oo t°rt Jn~..~.0 drafted after (8) is of the form (1), that is oo S~ a) =df E c
~a)(n)hk,
n = 0 , 1. . . . .
k=0
Remind that the three-term recurrence relation for the Hermite functions is
xhn = ~/l(n +
1)hn+l +
V~nhn-1.
Taking into account this and the following two relations, cf. [5] 6, n ~ - ~ C ( ~ X ( X ) = ~ ¢ / a c ( a ) ( x ) -- ~4/"xc(a)(x -- 1),
= V dc a)(x) --
c(a~ df 0,
XgT-4--fc a (x + 1)
6There is a difference (by the factor ( - 1 ) n) in defining Charlier functions there and here.
400
F.H. SZAFRANIEC
w e get t h e t h r e e - t e r m r e c u r r e n c e r e l a t i o n for s~a)'s,
XS(na) = - - / l ( n
. (a) 1 -I- ~/~as(na) __ ~/~ ~nSn-l" (a) + tl)Sn+
Then, by (12), ( ! c - d2 + x 2) - ~/2-dx + a)S~na) = (n + 1-~s(a) 2', dx 2 2j n ,
n = O, 1 . . . . .
(17)
and =
1"~ ^(a)
_as(na)~'~-ffs(a)__X,
n = O, 1 . . . . .
(18)
Consequently, the operator
!c-d~ +x 2)- ~x 2 ~ dx 2
+a
is a H a m i l t o n i a n w i t h r e s p e c t to t°n t~(a)~oo J n = 0 a n d its p a r t
-~f-~x
+ a
is a Jacobi matrix operator with respect to the same basis. REFERENCES [1] T. S. Chihara: An Introduction to Orthogonal Polynomials, Gordon and Breach, New York 1978. [2] G. K. Eagleson: A duality relation for discrete orthogonal systems, Studia Sc. Math. Hung. 3 (1968), 127. [3] R. Koekoek and R. F. Swarttouw: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Report of the Department of Technical Mathematics and Informatics no. 98-17, 1998. [4] J. Stochel and F. H. Szafraniec: A peculiarity of the creation operator, Glasgow Math. J. 44 (2001), 137. [5] F. H. Szafraniec: Yet another face Of the creation operator, Operator Th. Adv. Appl. 80 (1995), 266. [6] F. H. Szafraniec: Subnormality in the quantum harmonic oscillator, Commun. Math. Phys. 210 (2000), 323. [7] E H. Szafraniec: Charlier polynomials and translational invariance in the quantum harmonic oscillator, Math. Nachtr. 241 (2002), 166. [8] E H. Szafraniec: Duality in the quantum harmonic oscillator, J. Phys. A: Math. Gen. 34 10487 (2001).