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fCW YORK tMVBtSTY

AFC RC-TN-55-40

Division of Electromagnetic Researdi

RESEARCH REPORT No. CX-19

The Determination of the Scattering Potential

from the Spectral Measure Function

Part II: Point Eigenvalues and Proper Eigenfunctions

IRVIN KAY and HARRY E. MOSES

CONTRACT NO. A F - 1 9( 1 2 2 ) -463

JUNE 1955

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No, CX-19

THE DETERMINATION OF THE SCATTERING POTENTIAL

FROM THE SPECTRAL MEASURE FUNCTION

Part II: Point Eigenvalues and Proper Eigenfunctions

I. Kay and H. E. Moses

r-rk^-i^^ l/jr. ;e^^

H. E.lJW^

r. o?to7(H^), W^^(H^) - W^ 7

B. The eigenfunctions of the discrete spectrum j the operator

U >?(-Hq) 11

C. The completeness theorem for Hj definition of H in the

extended space 13

U. More detailed discussion of the Q-representation and Q-

extended operators l5

5Â» Theorems on the inverse problem. The equation for K, K 17

6. Conditions on the eigenfunctions IH ,A :E. ,a > 20

o o 1

References 22

- 1 -

1Â» Introduction

In Part I of this paper^ J we restricted ourselves to the consideration of

weight operators which led to Hamiltonians H whose spectra were identical to the

spectrum of a piven unperturbed Hamiltonian H , In this part of the paper we pro-

pose to show how one may choose weight operators which lead to Hamiltonians H

having^ spectra different from that of H . To make the discussion more concrete

we shall take the case where H has a purely continuous spectrum extending from

to 00 and where H has a spectrum which has a continuous part which coincides

with the spectrum of H , and, in addition, has negative point eigenvalues. How-

ever, the procedure which is discussed is capable of being generalized consider-

ably, at least in a formal faf^hion.

It is our objective to find operators U and U such that

(1.1) WU* - U

o

(1.2) U^ - U'-'-

where W is the weight operator. We wish H to be obtained from

(1.3) H - UH U - UH U"-^ ,

o

Clearly (1.3) cannot hold if H and H have a different spectrum, which is the

case being considered here. We shall have to define a vector space larger

than Hilbert space such that the spectrum of H in this larger space is the

same as that of H in this space.

It will be possible to carry out the extension of the Hilbert spiace to

the larger space in terms of the Q-representation which is the representation

in which K is triangular in the sense described in Part I, i.e.,

(l.U) < q|U|qÂ« > - 6(q-q') â™¦ 6 < q|K|q' > ,

where

- 2 -

(1.5) < q|Klq' > - for q' > q.

Like Part I this paper has two main divisions. In the first (Sections 2

and 3) we shall characterize the weight operator assuming the operator H and

its spectrum given. In the second division (Section 5) we shall show how H can

be obtained from the weight operator. We shall see that in addition to pres-

cribing the weight operator which gives the weight function of the eigenfunction

of the continuous spectinim and prescribing the boundary conditions which these

eigenf unctions must satisfy, we shall have to give the normalization constants

and eigenvalues of the proper eigenfunctions. Furthermore, we shall have to

specify a set of 'eigenfunctions' of H which span the vector space which must

be added to the Hilbert space to assure us that the spectrum of H coincides

with that of H.

2, The eigenfunctions of the unperturbed Hamiltonian and the extended vector

space

Since we shall work with a vector space larger than Hilbert space, it will

be useful to introduce a projection operator which shall be denoted by ^(H^)Â»

This operator is to have the property that

(2.1) ^i%)\9 >'{

if the state \ is in the Hilbert space, and that

(2.2) >((H^j)|9 > =

if 1^ > is in that part of the vector space which is orthogonal to the Hilbert

space â€¢

As ijQ Part I we shall denote the eigenstates of H^, A^ belonging to the

eigenvalues E, a, respectively, by |HQ,AQ}E,a >. Since we are assuming that

the spectrum of H is continuous and ranges from zero to +oo, we shall express

- 3 -

the fact that the |H ,A }E,a > for E > are a complete set in the Hilbert space

(2.3) jj |H^,A^;E,a > >^(E)dEda< H^,A^jE,a| - Wo^>

>7(E) being the usual Heaviside step function.

We shall extend the definition of the operator H so that it has eigenvalues

which correspond to the negative point eigenvalues E. of the operator H, Toward

this end we introduce 'eigenfunctions â€¢ IH ,A :E,a > which are defined for E <

in the vicinity Ae. of each eigenvalue E. of H. These eigenfunctions are to

span a vector space orthogonal to the Hilbert space. We shall characterize this

extended space by working in the Q-representation. Every operator A defined in

Hilbert space can be represented as an integral operator with the kernel < q|A|q' >.

If the operator A as defined in the extended space has the same form as an integral

operator in the Q-representation when operating on vectors in the extended space,

we shall call the operator A 'a Q-extended operator'. We shall discuss Q-extended

operators in more detail in Section U.

In particular we shall take H as a Q-extended operator. Hence for E < 0,

the eigenfunctions |H ,A }E,a > are defined by

(2.U) J < q|H^|q' > dq- < q'|H^,A^jE,a > - E < q|H^,A^jE,a >

%

for E in the vicinity Ae. of each point eigenvalue E. of H, If, for example,

H = - â€” X (where the superscript Q means that Hâ€ž is expressed in the Q-representa-

J from

- 1^ < q|H^,A^jE,a > - E < q|H^,A^}E,a > .

dq

Not every solution of (2.U) will be used to define an eigenfunction of H in

the extended space, for we shall require that < q|H^,AQ}E,a > be quadraticaUy

integrable functions of q in the vicinity of q^. If q^ â– -oo then a necessary

- u -

condition is

(2.5) llm < q|H ,A jE,a > - .

q->-oo

As we shall see in Section 6, this condition is a necessary condition for the

bound states of H to be quadratically integrable functions of q.

We shall designate the function < q|H ,A jE,a > which is a solution of

(2.U) subject to the above condition by the ket |H ,A jE,a >. The corresponding

bra < H ,A jE,a| will be used to denote the function < H ,A }E,ajq > where

(2.6) < H^,A^jE,a|q > - < q|H^,A^;E,a >*

where the asterisk denotes complex conjugate

â€ž< H ,A

B 0* o

Let us denote by â€ž< H ,A ;E,a|q > another set of functions which satisfy both

(?.l4) and q

(2.7) I I H^,A^;E,a|q> dq< q|H^,A^;E',aÂ« > . 6(E-E')6(a,a').

K

We also require that the eigenf unctions of H belonging to the positive

spectrum be orthogonal to the eigenf unctions of the extended space, i.e.,

(2.7a) >^(-E)>7(E') d4< q|H^,A^jESa' > - 0.

%

(2.7b) >](-Â£) >|(E') < H^,A^jE,a|q > dq < q|H^,A^;E' ,a' > - 0.

We can write

(2.8) < qlH |q' > < qMHo,A^iE,a >b Â« E < q|H^,A^;E,a >g ,

%

where < qlH ,A ;E,a >â€ž is defined by

^' o o' ' B

(2.9) < q|H^,A^,E,a >g - | H^,A^;E,alq >*,

The brag: H ,A ;lr:,a| is used to denote and the ket |H ,A jE,a >

is used to denote < q|H ,A ;E,a ^ . The subscript B is used to denote the fact

th^t^i; H ,A }E,a| is the 'bi-orthogonal ' bra to |H ,A }E,a >,

If I is the identity operator of the entire vector space, we write

(2.1D) Iq > dq < q| = I,

(2.10a) < q|q' > = 5(q-q') .

Then (2.7) can be written abstractly as

(2.11) ^ H^,A^}E,a|HQ,A^}ESa' > - 5(E-B' )5(a,a' ),

< H^,A^;E,a|H jA^jESa' >, " 5(E-E' )5(a,a') , for(E,E'->|(-E)dEda < H^,A^,E,a|

where the integration is carried out over the intervals Ae. . The extended

space is defined by

(2.13) ^(%) * ^(-Hq) ' I.

The operator ^(-H ) can be shown to be a projection operator which maps the

Hilbert space into zero and the space added to the Hilbert space to form the

full space into itself. A general vector | ^ > in the full space can be

written as

- 6 -

(2.1ii) -(j< q|H^,A^jE,a > y|(E)dEd* < H^,A^jE,a| J >

* I j < qlH^,A^jE,*>^(-E)dEd*| H^,A^,E,a| J>,

vhare the first intafr*! goÂ«* o^J* thB pcsltivv I Â«ti8 and the atcond intSj^al

!Â«â€¢â€¢ ovsr the IntevviAt Li't^

In (2. Ill) tlÂ» H^ reprÂ«Â«*iÂ«tliw o' I J > whiÂ«* iÂ» < H^,A^;E,Â»| } > 1* â€¢

quadratically integrable function of E, a for E > 0. The function g< H^,A^ ji;,a|(? >

is an integrable function of E, a in the intervals Ae^ for E < 0. The defini-

tion of < H ,A ;E,a| 5 > for E < outside the intervals Af^ is immaterial.

Boo "*

Let us now consider some properties of the projection operators ^(H^),

>{(-H). First of all we note

(2.15) >^(H^)|H^,A^;E,a> - >|(E) |H^,A^;E,a >

>|(-H^)|H^,AQ;E,a > - >|(-E) |H^,A^jE,a >.

The projection operator >^(-H ) maps the Hilbert space into zero and maps

that part of the vector space orthogonal to the Hilbert space into itself. By-

definition (2.12) and properties (2.7), (2.7a) and (2.7b)

(2.16) 'li-^^^)^(ii^) = 0, >^^(H^) - )^(H^), >^^(-H^) - ^(-H^)*

If T is any operator defined in the whole extended space then

T = T>((H^) + T >] (-H^),

where T >] (H ) is an operator which is zero when it operates on vectors orthogonal

to the Hilbert space and where T ^(-H^) is zero when it operates on the Hilbert

space. All the operators defined in Part I may be considered to be of the form

T'>7(H ) and hence most of the properties of the various operators discussed in

^ o

-7 -

Part I will have analogues to operators of this chao'acter appearing here. The

principal purpose of the present paper is to define operators over the full space,

particularly the operators U, U , and W, i.e., to define D >7(-H ), etc. It will

not be necessary, however, to extend L^(H ), M (H ), as will be clear from the

subsequent work.

The formal Hermitian adjoint of T '^(H^) is V|(H )T*. If T commutes with

H , it can be shown foraally that

(2.17) ^("o^^* â– '^*^("o^'

Hence if such an ocerator operates on Hilbort epace^ its adjoint will still operate

on the Hilbert space and maps vectors orthogonal to that space into zero.

If R is another operator trtiich commutes with H , one can show

(2.18) Rn(Ho)r>((H^) - RT>](H^).

Hence products of such operators R and T acting on Hilbert space can be obtained

by projecting on Hilbert space the product of the operators defined on the com-

plete space. We shall use the properties (2.17) and (2.18) often without re-

ferring to them explicitly,

3, The eigenfunctions of the perturbed Hamiltonian and the transformation operator

A. The eigenfunctions of the continuous spectrum; the operators

U>((H^), U^^(H^), W>|(H^) - W^

Let us consider the eigenfunctions |H,AjE,a > of H = H^ + eV. As in Part I

we introduce the operator U such that

(3.1) |H,A}E,a > - U|H^,A^;E,a >.

When E > the eigenfunctions |H ,A jE,a > are those which span the Hilbert space.

- 8 -

But >rtien E Â« E., where E. is one of the point eigenvalues of H, we have

(3.2) |H,AjE^,a > . UlH^,A^}E^,a >,

where |H ,A jE. ,a > is one of the vectors added to form the extended space. In

fact these elements were added in order that (3.1) should hold for all values of

E in the spectrum of H.

Let us consider the continuous spectmam of H, We shall obtain expressions

for U "^(H ) U >| (H ), S, etc., analogous to those obtained in Part I where the

continuous part of the spectrum which we are considering was the entire spectnun.

The various expressions that are given below are obtained in an almost identical

manner as those for the analogous operators of Part I. Therefore, instead of

carrying out the derivations in detail, we shall indicate only the more important

relationships â€¢

For the case of the continous spectrura, (3.1) may be wiritten

(3.2) |H,A}E,a > - ljV](E)|H^,A^iE,a >

- U r|(H^)|H^,AjE,a > (E > O).

In a manner similar to that used in Section 3 of Part I it can be shown that

U^(H ) satisfies the equation

^

(3.3) UV|(H^) -L>1(H^) W -L_ VU>](H^)6(E-H^)dE,

where L is an arbitrary operator which commutes with H , and where as in Part I,

r3.3a) 5(E-H^) - |H^,A^}E,a > da < HQ,A^;E,a|.

)

The integral in (3,3) is formally taken over the whole spectrum of the extended

operator H . The factor >j (H ) in the integral, hoirever, in effect cuts out

the negative part.

Now, as in Part I, there are two operators, U V|(H^), which are particularly

interesting and whose integral equations can be obtained by selecting L>](H^) pro-

- 9 -

perly. These operators are defined by the conditions

IH t .â€ž.

(3.U) lim e Â° e"^^ U_)| (H^) |gr > - V( (H^) !?( > ,

t-^>-oo

iH t

(3.Ua) lim e Â° e'^"* D^ v| (h^) |9( > - >|(H ) I?? > ,

t-i>*00

vrtiere >|(H ) |p > is an arbitrary state in Hilbert space. As in Part I, U/>7 (H )

satisfy

(3.5) U^n(Â«o^ " ^^"o^ * ^ J Y^(E-H^,)VU^>((H^)5(E-H^)dE.

The scattering operator S is defined by

iH t J TT.

(3.6) lim e Â° e ^"^ U_ V|(H^) |;2(> . S v^ (H^) |^ > ,

t^>+oo

where, as before, >|(H )\(jf > is an arbitrary state in Hilbert space. It can be

shown that

(3.7) S* >|(H^)-2nit J 5(E-H^)VU_>|(H^)6(E-H^)dE

and that its inverse is given by

(3.8) S~^ = ^(H^) + 2Td6 j 6(E-H^)VU^ >|(H^)6(E-H^)dE.

From (3.7) and (3.8) it is clear that S and S' can be written

(3.9) S = SV](H^),

(3.10) S-^ = S"^^(H^)

and therefore that S and S maps the space orthogonal to the Hilbert space into

zero.

The general operator U "^(H ) may be expressed in terms of U^ ^(Hq) as follows:

(3.11) Uv^(H^) - \^i%K\^V.^), .

where

- 10 -

(3.12) \^("o^ " L^l^^o^ * ^^ I 6(E.K^)VUV|(H^^)6(E-H^)dE.

From (3.12) it is clear that M >| (H ) commutes with H . It can be shown that

M 'h(H ) has an inverse in Hilbert space which connButes with H and which we

shall denote by M^-^>i (H^):

(3.33) M;^>((H^)M^r|(H^) - K^>|(H^)m;^Y|(H^) - >| (H^);

alternatively, noting that M commutes with H and using (2.13), we can write

(3.13a) M;-*- M_^n(H^) - M_^ M+"^^(Ho) " ^(H^).

It can also be shown that

Â«

(3.1ii) ^(HJU^U^-^(H^) . V|(H^),

which is a generalization of (U.21) of Part I. Equation (3.1U) is essentially

the normalization condition on the eigenfunctions

(3.lUa) lH,A}E,a>^ - U^ v|(h^) |H^,A^jE,a >

(see Part I), nsimely

(3.mb) V|(E)V|(E')^< H,A}E,a|H,AiE',a' >^ - Y^(E)5(E-E')6(a,a' ) ,

Frrm (3.n) and (3.13) the corresponding relation for U is

(3.15) W^>((H^)U*Un(H^) - V|(H^)

wheiÂ«

(3.16) W^v^(H^) - i^'^\(%)\^\(^o^ ' ^'^l'^^(\^ ' n^Ho^^c-

It is to be noted that W Vi (H ) is a positive definite operator which has an

inverse in Hilbert space. Equation (3.15) expresses the orthogonality rela-

tions between the eigenfvmctions of the continuous spectmm of H associated

- n -

with the vreight operator W 'â€¢^(H ).

Two expressions for S whose analogues appear in Part I are

(3.17) S - V|(H^) U* U_V((H^),

(3.18) S - M^n(Ho^^I''''l^"o^ ' "X"''^^"Â©^'

B. Ihe eigenfunctions of the discrete spectrum; the operator U>](-H )

Let us now consider the eigenfunctions of the discrete spectrum of H, namely

|H,A;E, ,a >, VJe should first note that the nature of the degeneracy operator A

and its eigenvalues may be completely different for the discrete spectinun of H

and for the continuous spectrum. In the case of the continuous spectrum the

operator A may be chosen so that its eigenvalues are not countable (i.e., A may

have a continuous spectnun). However, in the case of point eigenvalues E. of

H, the degeneracy is always countable, and in most applications is finite for

a given value of E. . Hence the operator A, when operating on that subspace of

Hilbert space spanned by the eigenstates of the discrete spectrum of H, will

have to be defined so as to have countable eigenvalues.

In order that the spectjrum of the extended operator H have the same spectrum

as H, it is necessary that the degeneracy operator A and its eigenvalues of addi-

tional eigenvectors |H ,A ;E,a > which are introduced for values of E in the

neighborhood ^jE. of each eigenvalue E. of H, have the same point spectrum as

the eigenvalues of A in the eigenstate |H,A}E. ,a >.

We may then write, since E. < 0,

|H,AjE^,a > - U|HQ,A^jE^,a >

(3.19) - U V|(-E)|HQ,A^jE^,a >

- Uv^(-H^)|H^,A^}E^,a>.

NE^ YOPK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

25 WÂ»verly Pbce, New York 3, N. Y.

- 12 -

For each value of a and E. the operator maps an 'eigenvector' of the extended

operator H into an eigenvector H with the same value of a as a degeneracy-

label.

Generally the eigenfunctions of H corresponding to the discrete spectrum

satisfy the following orthogonality relation:

(3.20) < H,AiE^,a|H,A}Ej,b > - 6(i,j) < alai^-'-(E^) |b >,

where 5(i,j) is the Kronecker 5. The matrix < a|co" (E.) |b > is a positive definite

Hermitian matrix in the space of eigenfunctions of the operator A corresponding to

a fixed eigenvalue E. of H. In the case where the eigenfunctions belonging to the

same eigenvalue E but having different degeneracy labels have been made orthogonal

to each other, < a|w~ (E, ) |b > woiald have the form

< a|(o^-'-(E^)|b > = C^g6(a,b) ,

The constant C. (which is always positive) is the normalization constant for the

ia

eigenfunction lH,A}E.,a >, that is,

< H,AiE^,a|H,A}E^,a > - C^^^ .

The operator < a|co' (E. )|b > has a positive-definite inverse which we denote by

< a|oo (E. ) |b > } the latter, by definition, satisfies the relation

(3.21)

J2 < a|o3^(E^)|b > < b|co^^(E^)|b >

b

-1

y < a|cA)^ (E^) |b > < b|w^(E^) |c > - 6(a,c).

There is one other orthogonality relation which we should note, namely the

relation which expresses the orthogonality between the eigenfunctions of the

discrete and continuous spectrum of H. It is

(3.22) >1(E)>|(-E^) < H,A;E,a|H,A;Ej_,b > - 0.

- 13 -

C. The completeness theorem for H; definition of H in the extended space

The principal difference between the results of Parts I and II arises from

the fact that in the present case the Hilbert space is spanned by the eigen-

f\mcticna belonging to the discrete as well as the continuous eigenfunctions of

H.

Let us consider any state ^(H ^JJJ > in Hilbert space. Further, let us take

for the eigenfunctions of the continuous spectrvun either the outgoing or incoming

eigenstates |H,A}E,a > - U v|(H ) |H^,ApjE,a >. The arbitrary state Vi(H ) \

can then be expanded in the following way:

Y}(H^)|9( > - J J |H,AjE,a >^>((E)dEda< H,A}E,a| r| (H^) |9( >

(3.23)

+ r E |H,AjE.,a > < a|a)^(E^) |b >< H,AiE^,b| V|(h^) jjl >.

T" a,b

The coefficients of |H,AjE,a > and |H,AjE.,a > in the expansion (3,23) follow

from the normalization conditions (3.20) and (3.lUb). Since ^(H )|P > is an

arbitrary state in Hilbert space, (3.23) is equivalent to

j |H,A}E,a > >|(E)dEda^< H,A}E,a| >|(Hq)

(3.2U)

*ril |H,A;E ,a>< a|co^(E^)|b >< H,AjE.,b|v^(H^) - V^(H^).

1 a,b

It is useful to define the positive definite matrtces < a|co,(E) |b > in the

previously described intervals Ae. so that for E - E. the operators < a|co^(E)(b >

are the matrices < a|a),(E.)|b > introduced earlier. On using (3.19), (3.lUa)

'a 1

and (2,3) we have

^.Y%KY%^ * EE UV((-Kâ€ž)6(Ei-H^)

i a,b

(3.25)

. I |H^,A^jE,a>< a|c.^(E)|b > dE < H^,A^}E,b | V|(-k^)U* r|(H^)- )^(H^),

where the integral is taken over each of the intervals Ae^^. Let us introduce

- lU -

the operator

(3.26) W^^(-H^) - E l^o'^o*^*^ ^ "= a|a3^(E)|b > dE < H^,A^jE,b | >^(-H^) .

a,b^

It is clear that W, ">^(-H ) is a positive definite operator in the space ortho-

gonal to the Hilbert space and that it commutes with H â€¢

On using (3Â»11) and (3.16), EqÂ« (3.2$) becomes

(3.27) UWU*V^(H^) - ^(H^),

where

(3.28) W - W^v^(H^) + ^ 5(E.-H^)W^V^(-H^).

Eq. (3.28) is the weight operator for the case iriiere point eigenvd. ues exist*

As can be seen, the weight operator is positive definite and has an inverse in

the vector space. Eq. (3.28) provides a decomposition of W into two parts: W ,

c

which characterizes the orthogonality relations of the continuous spectrom of H

(hence the subscript c for 'continuous')? and W., which characterizes the ortho-

gonality relations of the discrete spectrum (hence the subscript d for 'discrete').

We have not yet discussed how H is to be defined in the extended space. The

extension of H can be carried out by giving the vectors which span the extended

space. Since we want the spectrum of the extended operator H to have the sane

spectrum as the spectrum of the extended operator H and hence of the original

operator H, it is seen that the eigenstates of H which span the Hilbert space

must also span the extended space. It is clear that the spectrum of H is thus

preserved.

The effect of the extension of the definition of H is that equation (3.2?)

is replaced by

(3.29) UWU* - I,

- 15 -

i4Â« MorÂ» d>t>ilÂ«d discueaion of the Q-rÂ»pi-eaÂ«ntation and Q-extrtdsd operators

As in Part I wb shall introduca ths Q-rÂ«presentatlon and require that the

operators K and K be triangular in tenwj of this repiresentaticn, i.e.,

< q|K|q' > - 0, q' > q,

(U.l)

q(I^UÂ« > - 0, q' > q,

K and K are ^Â«Â«n by

U - I + eK

(U.2) .

U - WU - I + eK ,

Generally, the Q is defined only in the Hilbert space (i.e., only Q^(H ) is de-

fined). Its eigenstates |q > satisfy the completeness relation

(U.3) |q> dq< q|>((H^) - >](H^),

K

where q^ and q^ are the upper and lower limits of the eigenvalues of Q, To

extend the definition of Q into the whole vector space we write the complete-

ness relation for the eigenf unctions |q > as

(U.U) |q> dq < q| - I.

From (li.U) we have

(U.5) < qlq' > - 5(q-q') = , i.e., if |^ > is a state in

the extended space < qlA|p > can be written as (Using (U.U))

(a.6) < q|A|9f > . I < q|A|q' > dq' < q' |.

^%

As in ordinary Hilbert space theory, the operator A is defined if the kernel

< q|A|q' > is given.

Let us consider now an operator of the form A )](H ). This operator

operates in a non-trivial way only on the Hilbert space. It maps vectors

orthogonal to the Hilbert space into zero. The kernel < q|A>] (H )|q' > is,

of course, a known function of q and q'.

One might wish to extend the definition of the operator A>?(H ) to an

operator A which can be applied in a non-trivial way to the whole vector

space and \rfiich equals A >? (H ) when applied to the Hilbert space. One possible

way of defining the extended operator A is to define the kernel < q|A|q' > as a

AFC RC-TN-55-40

Division of Electromagnetic Researdi

RESEARCH REPORT No. CX-19

The Determination of the Scattering Potential

from the Spectral Measure Function

Part II: Point Eigenvalues and Proper Eigenfunctions

IRVIN KAY and HARRY E. MOSES

CONTRACT NO. A F - 1 9( 1 2 2 ) -463

JUNE 1955

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No, CX-19

THE DETERMINATION OF THE SCATTERING POTENTIAL

FROM THE SPECTRAL MEASURE FUNCTION

Part II: Point Eigenvalues and Proper Eigenfunctions

I. Kay and H. E. Moses

r-rk^-i^^ l/jr. ;e^^

H. E.lJW^

r. o?to7(H^), W^^(H^) - W^ 7

B. The eigenfunctions of the discrete spectrum j the operator

U >?(-Hq) 11

C. The completeness theorem for Hj definition of H in the

extended space 13

U. More detailed discussion of the Q-representation and Q-

extended operators l5

5Â» Theorems on the inverse problem. The equation for K, K 17

6. Conditions on the eigenfunctions IH ,A :E. ,a > 20

o o 1

References 22

- 1 -

1Â» Introduction

In Part I of this paper^ J we restricted ourselves to the consideration of

weight operators which led to Hamiltonians H whose spectra were identical to the

spectrum of a piven unperturbed Hamiltonian H , In this part of the paper we pro-

pose to show how one may choose weight operators which lead to Hamiltonians H

having^ spectra different from that of H . To make the discussion more concrete

we shall take the case where H has a purely continuous spectrum extending from

to 00 and where H has a spectrum which has a continuous part which coincides

with the spectrum of H , and, in addition, has negative point eigenvalues. How-

ever, the procedure which is discussed is capable of being generalized consider-

ably, at least in a formal faf^hion.

It is our objective to find operators U and U such that

(1.1) WU* - U

o

(1.2) U^ - U'-'-

where W is the weight operator. We wish H to be obtained from

(1.3) H - UH U - UH U"-^ ,

o

Clearly (1.3) cannot hold if H and H have a different spectrum, which is the

case being considered here. We shall have to define a vector space larger

than Hilbert space such that the spectrum of H in this larger space is the

same as that of H in this space.

It will be possible to carry out the extension of the Hilbert spiace to

the larger space in terms of the Q-representation which is the representation

in which K is triangular in the sense described in Part I, i.e.,

(l.U) < q|U|qÂ« > - 6(q-q') â™¦ 6 < q|K|q' > ,

where

- 2 -

(1.5) < q|Klq' > - for q' > q.

Like Part I this paper has two main divisions. In the first (Sections 2

and 3) we shall characterize the weight operator assuming the operator H and

its spectrum given. In the second division (Section 5) we shall show how H can

be obtained from the weight operator. We shall see that in addition to pres-

cribing the weight operator which gives the weight function of the eigenfunction

of the continuous spectinim and prescribing the boundary conditions which these

eigenf unctions must satisfy, we shall have to give the normalization constants

and eigenvalues of the proper eigenfunctions. Furthermore, we shall have to

specify a set of 'eigenfunctions' of H which span the vector space which must

be added to the Hilbert space to assure us that the spectrum of H coincides

with that of H.

2, The eigenfunctions of the unperturbed Hamiltonian and the extended vector

space

Since we shall work with a vector space larger than Hilbert space, it will

be useful to introduce a projection operator which shall be denoted by ^(H^)Â»

This operator is to have the property that

(2.1) ^i%)\9 >'{

if the state \ is in the Hilbert space, and that

(2.2) >((H^j)|9 > =

if 1^ > is in that part of the vector space which is orthogonal to the Hilbert

space â€¢

As ijQ Part I we shall denote the eigenstates of H^, A^ belonging to the

eigenvalues E, a, respectively, by |HQ,AQ}E,a >. Since we are assuming that

the spectrum of H is continuous and ranges from zero to +oo, we shall express

- 3 -

the fact that the |H ,A }E,a > for E > are a complete set in the Hilbert space

(2.3) jj |H^,A^;E,a > >^(E)dEda< H^,A^jE,a| - Wo^>

>7(E) being the usual Heaviside step function.

We shall extend the definition of the operator H so that it has eigenvalues

which correspond to the negative point eigenvalues E. of the operator H, Toward

this end we introduce 'eigenfunctions â€¢ IH ,A :E,a > which are defined for E <

in the vicinity Ae. of each eigenvalue E. of H. These eigenfunctions are to

span a vector space orthogonal to the Hilbert space. We shall characterize this

extended space by working in the Q-representation. Every operator A defined in

Hilbert space can be represented as an integral operator with the kernel < q|A|q' >.

If the operator A as defined in the extended space has the same form as an integral

operator in the Q-representation when operating on vectors in the extended space,

we shall call the operator A 'a Q-extended operator'. We shall discuss Q-extended

operators in more detail in Section U.

In particular we shall take H as a Q-extended operator. Hence for E < 0,

the eigenfunctions |H ,A }E,a > are defined by

(2.U) J < q|H^|q' > dq- < q'|H^,A^jE,a > - E < q|H^,A^jE,a >

%

for E in the vicinity Ae. of each point eigenvalue E. of H, If, for example,

H = - â€” X (where the superscript Q means that Hâ€ž is expressed in the Q-representa-

J from

- 1^ < q|H^,A^jE,a > - E < q|H^,A^}E,a > .

dq

Not every solution of (2.U) will be used to define an eigenfunction of H in

the extended space, for we shall require that < q|H^,AQ}E,a > be quadraticaUy

integrable functions of q in the vicinity of q^. If q^ â– -oo then a necessary

- u -

condition is

(2.5) llm < q|H ,A jE,a > - .

q->-oo

As we shall see in Section 6, this condition is a necessary condition for the

bound states of H to be quadratically integrable functions of q.

We shall designate the function < q|H ,A jE,a > which is a solution of

(2.U) subject to the above condition by the ket |H ,A jE,a >. The corresponding

bra < H ,A jE,a| will be used to denote the function < H ,A }E,ajq > where

(2.6) < H^,A^jE,a|q > - < q|H^,A^;E,a >*

where the asterisk denotes complex conjugate

â€ž< H ,A

B 0* o

Let us denote by â€ž< H ,A ;E,a|q > another set of functions which satisfy both

(?.l4) and q

(2.7) I I H^,A^;E,a|q> dq< q|H^,A^;E',aÂ« > . 6(E-E')6(a,a').

K

We also require that the eigenf unctions of H belonging to the positive

spectrum be orthogonal to the eigenf unctions of the extended space, i.e.,

(2.7a) >^(-E)>7(E') d4< q|H^,A^jESa' > - 0.

%

(2.7b) >](-Â£) >|(E') < H^,A^jE,a|q > dq < q|H^,A^;E' ,a' > - 0.

We can write

(2.8) < qlH |q' > < qMHo,A^iE,a >b Â« E < q|H^,A^;E,a >g ,

%

where < qlH ,A ;E,a >â€ž is defined by

^' o o' ' B

(2.9) < q|H^,A^,E,a >g - | H^,A^;E,alq >*,

The brag: H ,A ;lr:,a| is used to denote and the ket |H ,A jE,a >

is used to denote < q|H ,A ;E,a ^ . The subscript B is used to denote the fact

th^t^i; H ,A }E,a| is the 'bi-orthogonal ' bra to |H ,A }E,a >,

If I is the identity operator of the entire vector space, we write

(2.1D) Iq > dq < q| = I,

(2.10a) < q|q' > = 5(q-q') .

Then (2.7) can be written abstractly as

(2.11) ^ H^,A^}E,a|HQ,A^}ESa' > - 5(E-B' )5(a,a' ),

< H^,A^;E,a|H jA^jESa' >, " 5(E-E' )5(a,a') , for(E,E'->|(-E)dEda < H^,A^,E,a|

where the integration is carried out over the intervals Ae. . The extended

space is defined by

(2.13) ^(%) * ^(-Hq) ' I.

The operator ^(-H ) can be shown to be a projection operator which maps the

Hilbert space into zero and the space added to the Hilbert space to form the

full space into itself. A general vector | ^ > in the full space can be

written as

- 6 -

(2.1ii) -(j< q|H^,A^jE,a > y|(E)dEd* < H^,A^jE,a| J >

* I j < qlH^,A^jE,*>^(-E)dEd*| H^,A^,E,a| J>,

vhare the first intafr*! goÂ«* o^J* thB pcsltivv I Â«ti8 and the atcond intSj^al

!Â«â€¢â€¢ ovsr the IntevviAt Li't^

In (2. Ill) tlÂ» H^ reprÂ«Â«*iÂ«tliw o' I J > whiÂ«* iÂ» < H^,A^;E,Â»| } > 1* â€¢

quadratically integrable function of E, a for E > 0. The function g< H^,A^ ji;,a|(? >

is an integrable function of E, a in the intervals Ae^ for E < 0. The defini-

tion of < H ,A ;E,a| 5 > for E < outside the intervals Af^ is immaterial.

Boo "*

Let us now consider some properties of the projection operators ^(H^),

>{(-H). First of all we note

(2.15) >^(H^)|H^,A^;E,a> - >|(E) |H^,A^;E,a >

>|(-H^)|H^,AQ;E,a > - >|(-E) |H^,A^jE,a >.

The projection operator >^(-H ) maps the Hilbert space into zero and maps

that part of the vector space orthogonal to the Hilbert space into itself. By-

definition (2.12) and properties (2.7), (2.7a) and (2.7b)

(2.16) 'li-^^^)^(ii^) = 0, >^^(H^) - )^(H^), >^^(-H^) - ^(-H^)*

If T is any operator defined in the whole extended space then

T = T>((H^) + T >] (-H^),

where T >] (H ) is an operator which is zero when it operates on vectors orthogonal

to the Hilbert space and where T ^(-H^) is zero when it operates on the Hilbert

space. All the operators defined in Part I may be considered to be of the form

T'>7(H ) and hence most of the properties of the various operators discussed in

^ o

-7 -

Part I will have analogues to operators of this chao'acter appearing here. The

principal purpose of the present paper is to define operators over the full space,

particularly the operators U, U , and W, i.e., to define D >7(-H ), etc. It will

not be necessary, however, to extend L^(H ), M (H ), as will be clear from the

subsequent work.

The formal Hermitian adjoint of T '^(H^) is V|(H )T*. If T commutes with

H , it can be shown foraally that

(2.17) ^("o^^* â– '^*^("o^'

Hence if such an ocerator operates on Hilbort epace^ its adjoint will still operate

on the Hilbert space and maps vectors orthogonal to that space into zero.

If R is another operator trtiich commutes with H , one can show

(2.18) Rn(Ho)r>((H^) - RT>](H^).

Hence products of such operators R and T acting on Hilbert space can be obtained

by projecting on Hilbert space the product of the operators defined on the com-

plete space. We shall use the properties (2.17) and (2.18) often without re-

ferring to them explicitly,

3, The eigenfunctions of the perturbed Hamiltonian and the transformation operator

A. The eigenfunctions of the continuous spectrum; the operators

U>((H^), U^^(H^), W>|(H^) - W^

Let us consider the eigenfunctions |H,AjE,a > of H = H^ + eV. As in Part I

we introduce the operator U such that

(3.1) |H,A}E,a > - U|H^,A^;E,a >.

When E > the eigenfunctions |H ,A jE,a > are those which span the Hilbert space.

- 8 -

But >rtien E Â« E., where E. is one of the point eigenvalues of H, we have

(3.2) |H,AjE^,a > . UlH^,A^}E^,a >,

where |H ,A jE. ,a > is one of the vectors added to form the extended space. In

fact these elements were added in order that (3.1) should hold for all values of

E in the spectrum of H.

Let us consider the continuous spectmam of H, We shall obtain expressions

for U "^(H ) U >| (H ), S, etc., analogous to those obtained in Part I where the

continuous part of the spectrum which we are considering was the entire spectnun.

The various expressions that are given below are obtained in an almost identical

manner as those for the analogous operators of Part I. Therefore, instead of

carrying out the derivations in detail, we shall indicate only the more important

relationships â€¢

For the case of the continous spectrura, (3.1) may be wiritten

(3.2) |H,A}E,a > - ljV](E)|H^,A^iE,a >

- U r|(H^)|H^,AjE,a > (E > O).

In a manner similar to that used in Section 3 of Part I it can be shown that

U^(H ) satisfies the equation

^

(3.3) UV|(H^) -L>1(H^) W -L_ VU>](H^)6(E-H^)dE,

where L is an arbitrary operator which commutes with H , and where as in Part I,

r3.3a) 5(E-H^) - |H^,A^}E,a > da < HQ,A^;E,a|.

)

The integral in (3,3) is formally taken over the whole spectrum of the extended

operator H . The factor >j (H ) in the integral, hoirever, in effect cuts out

the negative part.

Now, as in Part I, there are two operators, U V|(H^), which are particularly

interesting and whose integral equations can be obtained by selecting L>](H^) pro-

- 9 -

perly. These operators are defined by the conditions

IH t .â€ž.

(3.U) lim e Â° e"^^ U_)| (H^) |gr > - V( (H^) !?( > ,

t-^>-oo

iH t

(3.Ua) lim e Â° e'^"* D^ v| (h^) |9( > - >|(H ) I?? > ,

t-i>*00

vrtiere >|(H ) |p > is an arbitrary state in Hilbert space. As in Part I, U/>7 (H )

satisfy

(3.5) U^n(Â«o^ " ^^"o^ * ^ J Y^(E-H^,)VU^>((H^)5(E-H^)dE.

The scattering operator S is defined by

iH t J TT.

(3.6) lim e Â° e ^"^ U_ V|(H^) |;2(> . S v^ (H^) |^ > ,

t^>+oo

where, as before, >|(H )\(jf > is an arbitrary state in Hilbert space. It can be

shown that

(3.7) S* >|(H^)-2nit J 5(E-H^)VU_>|(H^)6(E-H^)dE

and that its inverse is given by

(3.8) S~^ = ^(H^) + 2Td6 j 6(E-H^)VU^ >|(H^)6(E-H^)dE.

From (3.7) and (3.8) it is clear that S and S' can be written

(3.9) S = SV](H^),

(3.10) S-^ = S"^^(H^)

and therefore that S and S maps the space orthogonal to the Hilbert space into

zero.

The general operator U "^(H ) may be expressed in terms of U^ ^(Hq) as follows:

(3.11) Uv^(H^) - \^i%K\^V.^), .

where

- 10 -

(3.12) \^("o^ " L^l^^o^ * ^^ I 6(E.K^)VUV|(H^^)6(E-H^)dE.

From (3.12) it is clear that M >| (H ) commutes with H . It can be shown that

M 'h(H ) has an inverse in Hilbert space which connButes with H and which we

shall denote by M^-^>i (H^):

(3.33) M;^>((H^)M^r|(H^) - K^>|(H^)m;^Y|(H^) - >| (H^);

alternatively, noting that M commutes with H and using (2.13), we can write

(3.13a) M;-*- M_^n(H^) - M_^ M+"^^(Ho) " ^(H^).

It can also be shown that

Â«

(3.1ii) ^(HJU^U^-^(H^) . V|(H^),

which is a generalization of (U.21) of Part I. Equation (3.1U) is essentially

the normalization condition on the eigenfunctions

(3.lUa) lH,A}E,a>^ - U^ v|(h^) |H^,A^jE,a >

(see Part I), nsimely

(3.mb) V|(E)V|(E')^< H,A}E,a|H,AiE',a' >^ - Y^(E)5(E-E')6(a,a' ) ,

Frrm (3.n) and (3.13) the corresponding relation for U is

(3.15) W^>((H^)U*Un(H^) - V|(H^)

wheiÂ«

(3.16) W^v^(H^) - i^'^\(%)\^\(^o^ ' ^'^l'^^(\^ ' n^Ho^^c-

It is to be noted that W Vi (H ) is a positive definite operator which has an

inverse in Hilbert space. Equation (3.15) expresses the orthogonality rela-

tions between the eigenfvmctions of the continuous spectmm of H associated

- n -

with the vreight operator W 'â€¢^(H ).

Two expressions for S whose analogues appear in Part I are

(3.17) S - V|(H^) U* U_V((H^),

(3.18) S - M^n(Ho^^I''''l^"o^ ' "X"''^^"Â©^'

B. Ihe eigenfunctions of the discrete spectrum; the operator U>](-H )

Let us now consider the eigenfunctions of the discrete spectrum of H, namely

|H,A;E, ,a >, VJe should first note that the nature of the degeneracy operator A

and its eigenvalues may be completely different for the discrete spectinun of H

and for the continuous spectrum. In the case of the continuous spectrum the

operator A may be chosen so that its eigenvalues are not countable (i.e., A may

have a continuous spectnun). However, in the case of point eigenvalues E. of

H, the degeneracy is always countable, and in most applications is finite for

a given value of E. . Hence the operator A, when operating on that subspace of

Hilbert space spanned by the eigenstates of the discrete spectrum of H, will

have to be defined so as to have countable eigenvalues.

In order that the spectjrum of the extended operator H have the same spectrum

as H, it is necessary that the degeneracy operator A and its eigenvalues of addi-

tional eigenvectors |H ,A ;E,a > which are introduced for values of E in the

neighborhood ^jE. of each eigenvalue E. of H, have the same point spectrum as

the eigenvalues of A in the eigenstate |H,A}E. ,a >.

We may then write, since E. < 0,

|H,AjE^,a > - U|HQ,A^jE^,a >

(3.19) - U V|(-E)|HQ,A^jE^,a >

- Uv^(-H^)|H^,A^}E^,a>.

NE^ YOPK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

25 WÂ»verly Pbce, New York 3, N. Y.

- 12 -

For each value of a and E. the operator maps an 'eigenvector' of the extended

operator H into an eigenvector H with the same value of a as a degeneracy-

label.

Generally the eigenfunctions of H corresponding to the discrete spectrum

satisfy the following orthogonality relation:

(3.20) < H,AiE^,a|H,A}Ej,b > - 6(i,j) < alai^-'-(E^) |b >,

where 5(i,j) is the Kronecker 5. The matrix < a|co" (E.) |b > is a positive definite

Hermitian matrix in the space of eigenfunctions of the operator A corresponding to

a fixed eigenvalue E. of H. In the case where the eigenfunctions belonging to the

same eigenvalue E but having different degeneracy labels have been made orthogonal

to each other, < a|w~ (E, ) |b > woiald have the form

< a|(o^-'-(E^)|b > = C^g6(a,b) ,

The constant C. (which is always positive) is the normalization constant for the

ia

eigenfunction lH,A}E.,a >, that is,

< H,AiE^,a|H,A}E^,a > - C^^^ .

The operator < a|co' (E. )|b > has a positive-definite inverse which we denote by

< a|oo (E. ) |b > } the latter, by definition, satisfies the relation

(3.21)

J2 < a|o3^(E^)|b > < b|co^^(E^)|b >

b

-1

y < a|cA)^ (E^) |b > < b|w^(E^) |c > - 6(a,c).

There is one other orthogonality relation which we should note, namely the

relation which expresses the orthogonality between the eigenfunctions of the

discrete and continuous spectrum of H. It is

(3.22) >1(E)>|(-E^) < H,A;E,a|H,A;Ej_,b > - 0.

- 13 -

C. The completeness theorem for H; definition of H in the extended space

The principal difference between the results of Parts I and II arises from

the fact that in the present case the Hilbert space is spanned by the eigen-

f\mcticna belonging to the discrete as well as the continuous eigenfunctions of

H.

Let us consider any state ^(H ^JJJ > in Hilbert space. Further, let us take

for the eigenfunctions of the continuous spectrvun either the outgoing or incoming

eigenstates |H,A}E,a > - U v|(H ) |H^,ApjE,a >. The arbitrary state Vi(H ) \

can then be expanded in the following way:

Y}(H^)|9( > - J J |H,AjE,a >^>((E)dEda< H,A}E,a| r| (H^) |9( >

(3.23)

+ r E |H,AjE.,a > < a|a)^(E^) |b >< H,AiE^,b| V|(h^) jjl >.

T" a,b

The coefficients of |H,AjE,a > and |H,AjE.,a > in the expansion (3,23) follow

from the normalization conditions (3.20) and (3.lUb). Since ^(H )|P > is an

arbitrary state in Hilbert space, (3.23) is equivalent to

j |H,A}E,a > >|(E)dEda^< H,A}E,a| >|(Hq)

(3.2U)

*ril |H,A;E ,a>< a|co^(E^)|b >< H,AjE.,b|v^(H^) - V^(H^).

1 a,b

It is useful to define the positive definite matrtces < a|co,(E) |b > in the

previously described intervals Ae. so that for E - E. the operators < a|co^(E)(b >

are the matrices < a|a),(E.)|b > introduced earlier. On using (3.19), (3.lUa)

'a 1

and (2,3) we have

^.Y%KY%^ * EE UV((-Kâ€ž)6(Ei-H^)

i a,b

(3.25)

. I |H^,A^jE,a>< a|c.^(E)|b > dE < H^,A^}E,b | V|(-k^)U* r|(H^)- )^(H^),

where the integral is taken over each of the intervals Ae^^. Let us introduce

- lU -

the operator

(3.26) W^^(-H^) - E l^o'^o*^*^ ^ "= a|a3^(E)|b > dE < H^,A^jE,b | >^(-H^) .

a,b^

It is clear that W, ">^(-H ) is a positive definite operator in the space ortho-

gonal to the Hilbert space and that it commutes with H â€¢

On using (3Â»11) and (3.16), EqÂ« (3.2$) becomes

(3.27) UWU*V^(H^) - ^(H^),

where

(3.28) W - W^v^(H^) + ^ 5(E.-H^)W^V^(-H^).

Eq. (3.28) is the weight operator for the case iriiere point eigenvd. ues exist*

As can be seen, the weight operator is positive definite and has an inverse in

the vector space. Eq. (3.28) provides a decomposition of W into two parts: W ,

c

which characterizes the orthogonality relations of the continuous spectrom of H

(hence the subscript c for 'continuous')? and W., which characterizes the ortho-

gonality relations of the discrete spectrum (hence the subscript d for 'discrete').

We have not yet discussed how H is to be defined in the extended space. The

extension of H can be carried out by giving the vectors which span the extended

space. Since we want the spectrum of the extended operator H to have the sane

spectrum as the spectrum of the extended operator H and hence of the original

operator H, it is seen that the eigenstates of H which span the Hilbert space

must also span the extended space. It is clear that the spectrum of H is thus

preserved.

The effect of the extension of the definition of H is that equation (3.2?)

is replaced by

(3.29) UWU* - I,

- 15 -

i4Â« MorÂ» d>t>ilÂ«d discueaion of the Q-rÂ»pi-eaÂ«ntation and Q-extrtdsd operators

As in Part I wb shall introduca ths Q-rÂ«presentatlon and require that the

operators K and K be triangular in tenwj of this repiresentaticn, i.e.,

< q|K|q' > - 0, q' > q,

(U.l)

q(I^UÂ« > - 0, q' > q,

K and K are ^Â«Â«n by

U - I + eK

(U.2) .

U - WU - I + eK ,

Generally, the Q is defined only in the Hilbert space (i.e., only Q^(H ) is de-

fined). Its eigenstates |q > satisfy the completeness relation

(U.3) |q> dq< q|>((H^) - >](H^),

K

where q^ and q^ are the upper and lower limits of the eigenvalues of Q, To

extend the definition of Q into the whole vector space we write the complete-

ness relation for the eigenf unctions |q > as

(U.U) |q> dq < q| - I.

From (li.U) we have

(U.5) < qlq' > - 5(q-q') = , i.e., if |^ > is a state in

the extended space < qlA|p > can be written as (Using (U.U))

(a.6) < q|A|9f > . I < q|A|q' > dq' < q' |.

^%

As in ordinary Hilbert space theory, the operator A is defined if the kernel

< q|A|q' > is given.

Let us consider now an operator of the form A )](H ). This operator

operates in a non-trivial way only on the Hilbert space. It maps vectors

orthogonal to the Hilbert space into zero. The kernel < q|A>] (H )|q' > is,

of course, a known function of q and q'.

One might wish to extend the definition of the operator A>?(H ) to an

operator A which can be applied in a non-trivial way to the whole vector

space and \rfiich equals A >? (H ) when applied to the Hilbert space. One possible

way of defining the extended operator A is to define the kernel < q|A|q' > as a

1 2

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