On the one-dimensional harmonic oscillator with a singular perturbation Текст научной статьи по специальности «Математика»

MSC 47B25, 47E05

DOI: 10.14529/mmp 160106

ON THE ONE-DIMENSIONAL HARMONIC OSCILLATOR WITH A SINGULAR PERTURBATION

V.A. Strauss, Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolívar, Caracas, Venezuela, str@usb.ve,

M.A. Winklmeier, Departamento de Matemáticas, Universidad de Los Andes, Bogotá, Colombia, mwinklme@uniandes.edu.co

In this paper we investigate the one-dimensional harmonic oscillator with a left-right boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point. The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.

Keywords: harmonic oscillator; singular perturbation; selfadjoint extensions; negative eigenvalues.

Introduction

In this paper, we use the following notations. Set R+ = (0, to) and R_ := (—to, 0) and for functions f : R ^ C define their restrictions f± := f |r±. The standard inner products on L2(R) and on Cn are both denoted by (•, •). There will never be danger of confusion. Given a sesquilinear form [ •, • ] on Cn, we call a subs pace L neutral if [u, v] = 0 for every u,v E L. A subspace L is called maximal neutral if it is neutral and not properly contained in any other neutral subspace (see [8] for details).

The one-dimensional harmonic oscillator is given by the formal differential expression

Af «== ( — ^ + 20 f

The aim of this paper is to investigate several possible realisations of A as a symmetric linear operator and determine all possible selfadjoint extensions.

By the Liouville - Green asymptotic formula [5, Theorem 2.2.1], it is known that there are solutions y± of Ay = Xy with the following asymptotic behaviour for Itl ^ to:

t .-

y±(t) ~ (p~2X)i eXK t2 — 2X + i( P—J*)' ds) «

a

for |a| large enough. This shows immediately that A is in the limit point case both at +to and —to.

In order to assign an operator to the differential expression A, we need to specify a domain of admissible functions. The minimal operator associated with A is

Aminf = Af, V(Amin) = (2)

Since A is in the limit point case both at and —ro, the operator Amin is essentially selfadjoint (see, e.g. [12, 7.1.3]). Its closure is the so-called maximal operator associated to A

Af = Af, D(A) = {f : R ^ C : f,f' abs. cont., f, Af E L2(R)}. (3)

Note that

Amin C Amin = A = A*

A

eigenvalues:

a(A) = ap(A) = {n + 1 : n E No}. (4)

The corresponding eigenfunctions are

^n(t) = e-t /2cnHn{t) where Hn is the nth Hermite polynomial of order n,

n

ynjt2 d „-t2

Hn(t) = —lTe* ^e

and the normalization factor cn := (n12nn!)-1 is chosen such that = Snm.

Remark 1. A straightforward calculation shows that if u is a solution of (A+A)u = 0, then (t — djt) u is a solution of (A+A — n)u = 0 and (t + dt) u is a solution of (A + A + n)u = 0.

A

Mt) = n-1 e-t2/2, Mt) = cn( t — ^ e-t2/2, n > 1. (5)

Note that (d; + t)n ^o = 0, in agreement with the fact that A has no negative eigenvalues.

From the recursion formula (5) it is clear that is an even function if n is even, and

n

In Section 1 we consider the restriction of the harmonic oscillator to the open half lines R±:

Aminf(t) := Af (t), D(Amin) := C^(R±), (6)

f, f' abs. cont.

!f,f' abs. cont., | f : R± ^ C : Af |R± E L2(R±)f

ATXf(t) := Af(t), V(ATX) :={ f : R± ^ C : Afl E . (7)

With them we define

Ao = Amin © Amin. (8)

Clearly, V(A*0) = D(Amax) © D(Am&x).

In Sections 2 and 3 we will study several restrictions of the operator A by imposing conditions on functions in its domain at t = 0. We define the closed symmetric operators

Bf := Af, D(B):= {f E D(A) : f(0) = 0}, (9)

Cf := Af, D(C) := {f E D(A) : f(0) = f(0) = 0}. (10)

So we have the following chain of operators

A0 с A0 = C EB с A = A* с B * с C* = A

With exception of the first one, all inclusions are one-dimensional. We will classify all selfadjoint extensions of B (Section 2) and C (Section 3) in terms of conditions on the behaviour at 0 of the functions in the corresponding domains. Slightly abusing language, we will call these conditions boundary conditions at 0. We will not use the von Neumann extension theory for symmetric operators, but will identify selfadjoint extensions with maximal neutral subspaces of C2, C^d C4, respectively, equipped with an inner product induced by the condition (Af, g) = (f, Ag) for f, g in appropriate spaces.

B

by one real parameter. Every selfadjoint extension is of the form

V(Be) = {f E V(B*) : V2 cos(9) f (0) = sin(9) [f(+0) - f'(-0)] } (11)

for 9 E [0, n). Any function of D(Be) is continuous at 0 (Lemma 4), but its derivative has a

0

between the constant of proportionality and the particular selfadjoint extension. The operators Be can also be interpreted as the classical harmonic oscillator with a ^-interaction 0

B f = (-^ + 1t2+rf)f

with c = There is a constant y > 0 such that if c > —7, then Be has only positive eigenvalues, if c = — 7, then 0 is an eigenvalue of B^, and if c < % then Be has exactly one negative eigenvalue. This eigenvalue decreases monotonically to as c tends to —<x>, or equivalently, 9 tends to n.

Finally, in Section 4 we give an interpretation of the operators Be and another selfadjoint extension denoted by CK as operators with a and ^'-interaction at 0 in a Hilbert space H- D L2(R). Some bibliographical notes are given in Section 5.

1. The Harmonic Oscillator on the Half Line

First we restrict the harmonic oscillator to the half lines R±. The corresponding minimal operators are given in (6). These operators are in the limit point case at and in the limit circle case at 0, hence they are not essentially selfadjoint. Their adjoint operators are the ones in (7). Note that for f E D(Amax) the one-sided limits f(+0) := lim f(t) and f'(+0) := lim f'(t) exist. Similarly, for f E D(Amax) the onesided limits f (—0) := lim f (t) and f'(—0) := lim f'(t) exist.

All selfadjoint extensions of Amin are given as restrictions of Amax by appropriate 0

is needed.

Recall that the defect index of a closed densely defined linear operator T with respect to z E p(T) is given by

n(T, z) := dim (ker(T* — z)).

It is well known that the defect indices are constant in the complement of the numerical range

W(T) := {(Tx,x) : x E D(T), ||x|| = 1},

see, e.g., [9, Ch. V, Theorem 3.2]. It is easy to see that W(T) C R for a symmetric operator T

them by

n+(T) = dim (ker(T* — z+)), n-(T) = dim (ker(T* — z-)),

for any z± E C with Im(z±) E R±. By the von Neumann theory, a symmetric operator has selfadjoint extensions if and only if its defect indices are equal (see for instance [14, Ch. 8.2]).

Lemma 1. The defect indices of Amin are n+(Amin) = n-(Amin) = 1 and

W(Amin) C R+. (12)

Proof. We show the lemma only for Af1™. For all f E D(Amin), integration by parts yields

(A^fJ) = -f'(x)f(x) 0 +J \f'\2 + x2\f|2 dx = J

0 0

\f \2 + x2\f\2 dx > 0

which shows (12). Hence the defect index of A™n is constant in C \ R+.

It can be easily verified that two pairs of independent solutions of (A + 1 )f = 0 are

Mt) = e2i2, 4+(t) = e2^ / e-s ds,

Mt) = e2

1 i2

1 i2 \ „-s2

U2

$-(t) = e24 e-s ds = M—)-

(13)

(14)

Observe that (p+ + = \fn Clearly |R+ E L2(R+^^ut 0+|r+ E L2(R+). Therefore ker(Amax + l) = span{(+|R+ ^d n+(Amin) = n^A1^) = 1.

Analogous calculations show that (Aminf,f) > 0 for all f E D(Amin), (1|R_ E L2(R-), (- |r_ E l2(R-^us ker(Amax + 2) = span{(-|R_ ^d n+(Amin) = n-(Amin) = 1.

□

The following result on selfadjoint extensions of Amin follows easily from the general theory of Sturm - Liouville operators. For the convenience of the reader, we present it here with a proof in order to illustrate the method of indefinite inner product spaces for the description of selfadjoint extensions.

oo

t

Lemma 2. All selfadjoint extensions of Afin are one-dimensional; they are restrictions of Afax of the form

V(A±e) = {f e D(Afax) : cos(9)f±(+0) = sin(9)f±(+0)} (15)

with 9 e [0, n).

Proof. We show the claim only for A™n since the corresponding assertions for A™n follow analogously.

By Lemma 1, the defect index of A™ is equal to 1 on C \ R+. Two fonctions f,g e D(Amax) belong to a particular selfadjoint extension of A™n if and only if (Amaxf, g) — (Amaxf, g) = 0

0 = (Afaxf, g) — (Afaxf, g) = f (+0)g' (+0) — f' (+0)g(+0). (16)

On C2 let us define the Hermitian inner product

X\

x2,

My;)] = ((0 o) CO - (t)>--У)-

Then f, g belong to a particular selfadjoint extension of A™" if and only if (f (+0), f'(+0))4 and (g(+0),g'(+0))4 belong to the same maximal neutral subspace of (C2, [•, •]). Clearly e+ = (1, i)4 is a positive and e- = (—1, i)4 is a negative vector and ||e+|| = \\e-1|. Hence all maximal neutral subspaces are given by

Le = span ^ ( i(11 +ee_2iö^ J = ^spa^ I ^) ) ¡> ) , 9 E [0,^).

Therefore all selfadjoint extensions of A™ are given by

ЧЛ+,0) = {f e D(Amax) : (/,(+)) e

= {f eD(Amax) : f(+O)(l+e-2i0) = -if (+0)(1 - e_2i*)} with 9 e [0,^). The last description yields (15).

□

Remark 2. Let f e L2(R+) and g e L2(R_) such that g(x) = f (—x) for x e R_. From formula (15) it is clear that f e D(A+,в) for some 9 e (0,^) if and only if g e D(A_,ж_д) and f e D(A+, 0) if and onl у if g e D(A_ ,0).

Recall that {фп : n e N}, the set of eigenfunctions of the harmonic oscillator on R (see (5)), is an orthonormal basis of L2(R). Denote

фп, + = фпк+, фп,_ = фп\ж- , n e No. (17)

Clearly both {ф2п ,± : n e N0^d {ф2п+1, ± : n e N0} form a complete orthogonal systems on R±. With these observations we can calculate the spectrum of the operators Л±>в for 9 = 0 and 9 = f. '

Corollary 1. Let Л±в be as in (15).

1. a(A±>0) = op(A±y0) = {2n + 2 : n E No} and the corresponding eigenfunctions are ^2n+1,±, n E No.

2. &(A±2) = ap(A±,n) = {2n + 1 : n E N0} and the corresponding eigenfunctions are

n E No.

Proof. We will prove the claim only for A+,0. All other statements are proved analogously. Since all are odd functions, it follows that ^2n+i(0) = 0 and therefore their

restrictions ^2n+1,+ belong to D(A+>0). Moreover, A+>0^2n+1,+ = (2n + |)^2n+1y+, hence {2n + 2 : n E N0}c op(A±y0). Now the claim follows from the completeness of the system {^2n+i,+} in ¿2(R+)-

□

From the asymptotic expansion (1) of solutions of the equation Ay = Xy it is clear that for every X E R there is exactly one solution y^ which is square integrable on R+. It belongs to the domain of exactly one selfadjoint extension A+0, namely the one with 9 E [0,^) such that cos9yA(0) = sin9y^(0). The expansion (1) also shows that all eigenvalues are simple.

Since W (Ar) ^ (0, to), the non-positive spectrum a(Ag) fi (—to, 0] of any selfadjoint extension Ad consists of at most one eigenvalue of multiplicity at most 1, see [14, Ch. 8.4, Cor. 2]. The lemma below deals with these eigenvalues, but as a first step we need to define some functions.

Let u > 0. On [0, to) consider the Cauchy problem

(—¿d* + 2*2) u(t,u) = —u2u(t,u), (18)

u(0,u) = 1, u't(0,u) = 0. (19)

Let us write u(t,u) as power series

u(t,u) = ^2 an(u)tn. (20)

n

n=0

Replacing (20) in (18) we obtain a2n+1(u) = 0 and a0(u) = 1, a2(u) = u2

a2n+2(u) = (2n + 2)1(2n + 1) (2u2a2n(u) + a2n-2(u)), n > 1 ^

Let us show that the series converges for all t > 0. To this end we will show that for every u > 0 there is a constant q(u) such that

a2n(u) - qU, n E N. (22)

Assume that this inequality is true for numbers n — 1 and n. Then, due to (21), we have

1 (2u2 q(u) + q(u) \

a2n+2 - (2n + 2)(2n + 1)V n! + (n — 1)iy q(u) ( u2 + n

(n + 1)^2n + 1 2(2n + 1) J '

Since ^n+i + 2(2ra+i) ^ 4 n ^ ro, there is a natural number n0(w) such that 2n+i + 2(2;n+1) < 1 for every n > n0(w). Thus, (22) holds if we take

q(w) = max{1, 2! • 02 (w),..., no! • a2n0(w)}.

Due to (22) the series (20) converges and u(t,w) < q(w)et2 for w > 0 t > 0.

Note that all a2n(w) are positive increasing functions of w, so for every t > 0 u(t,w) is an increasing function with respect to w and for every w > 0 u(t, w) is a positive increasing function with respect to t, so u(t,w) E L2(R+). For the special case w = 0 we obtain

u(t, 0) = 1 + £ ^ 4k(4k • (23)

t4n

1 nk=i 4k(4k — 1)

Due to the inequalities

111™

-77 < ^-77, n G N,

322n(2n - 1) 4n(4n - 1) 22 2n(2n - 1)

we have

3

Now let us define (compare with (13))

2 2

cosh (у ) < u(t, 0) < cosh (y) , t> 0. (24)

v(t,u) := u(t,u) J (u(s,w)) 2 ds (25)

t

and

G(w) := v(0,w) = J (u(s,w)) 2 ds. (26)

o

Lemma 3. Let A±0 be as in (15) and let aA = arctan(G(0)).

1. A+, 0 has the eigenv alue 0 if and only if 9 = n — aA. It has a negative eigenvalue if and only if 9 E (n — aA, n). Moreover, let \j be eigenvalues of A+,0j for j = 1, 2. If X2 < A1 < 0 then n — aA < 91 < 92 < n.

2. A-, 0 has the eigenv alue 0 if and, only if 9 = aA. It has a negative eigenvalue if and only if 9 E (0,aA). Moreover; let Aj be eigenvalues of A-,0j for j = 1, 2. If A2 < A1 < 0, then aA > 91 > 92 > 0.

Proof. We proof only the item 1. The claims in the item 2 follow from the item 1 and u(t, w) t w

that

X

r PX

v(t,w) <u(t,w) (u(t,w)) (u(s,w)) ds = (u(s,w)) ds

t

00

< J (u(s, 0)) ds < J (cosh(s2/3)) ds G L2(R+).

tt

t

It is easy to check that v(-,u) satisfies (18) and

vt(t,u) =--;-r+ u' (t,u) (u(s,u)) 2 ds ,

t u(t, u)

t

therefore, by (19), vt(0,u) = —1. It follows from (26) that

vt(0,u) = — Gg (27)

which is equivalent to the boundary condition (15) with 9 such that tan 9 = —G(u), that is, 9 = — arctan(G(u)) E (n/2, n). Observe that G(u) is decreasing and continuous in u and lim G(u) = 0. Hence 9 is increasing in u and tends to n as u ^ to. For the special

case u = 0 we obtain 9 = n/2 — aA where aA = arctan(G(0)).

□

2. One-Dimensional Restriction of the Harmonic Oscillator and Classification of All Selfadjoint Extensions

In this section we consider the harmonic oscillator B on the real line defined by (9). The operator B is closed and symmetric, but not selfadjoint. Let us recall also the definition

A0

Remark 3. The domain of A0 can be viewed as D(A0) = D(Am'n) © D(Amin), hence it is easy to see that its adjoint is given by

!

(t) = i Amax/+(t)' t>

0JW Mmax/_(t), t< 0,

D(A**) = {/ E L2(R) : /± E D(Amax)} = D(Amax) ©©(A1^).

It should be noted that A0 and the selfadjoint extensions B^d CK of B and C which will be calculated below are not differential operators on L2(R) in the classical

0

Lemma 4. We have the chain of extensions A0 c B c B* c A0 and B* / (t) = A*0/ (t),

D(B*) = {/ : R ^ C : /± E D(Amax), /(—0) = /(+0)}.

Proof. Note that A0 c ^^nce B* c A0^et / E D(B^d g E V(A*0)^^en / and /' are continuous in 0 /(0) = 0 and g± E D(Amax). Hence integration by parts yields

0 x 0 X

(B/,g) — (/,A0g) = — i /''g dt — i /''g dt + i /g'' dt + i /g'' dt

= /' (0K g(+0)—g(—0» •

Therefore g E D(B*) if and only if g is continuous in 0 and in this case B*g = A0g.

□

B*

0

B

Proposition 1. The defect indices of B are n+ (B) = n-(B) = 1. Hence all selfadjoint B B*

BA inclusion W(B) c W(A) c [0, to), so the defect index of B is constant in C \ [0, to) and it suffices to show that dim (ker(B* + 1)) = 1.

The proof of Lemma 1 shows that every L2-solution of (A + 2)/ = 0 must be of the form

0) + a+0+X(0,+() (28)

with a± E C and as in (13) and (14). Here X(-X,0) and X(0,+X) are the indicator functions of the sets (—to, 0) and (0, +to) respectively. Clearly every function (28) belongs to D(A*) and /(—0) = ^Vn and /(+0) = 0+Vn For / E D(B*) we must have that

/(—0) = /(+0)- Therefore ker B* + 2) = span{0-X(-x,0) + 0+x(0,+x)}- Thus n(B, — 2) = 1

B

all selfadjoint restrictions of B*. Let /,g E V(B*). Performing integration by parts we find {/, B*g) — (B*/, g) = [/'(—0) — /'(+0)]g(0) — [g'(—0) — g'(+0)]/(0).

Set

/0 —i r

G = I i 0 0

V—i 0 0

/, g B

(/(0), /'(—0), /'(+0))t and (g(0), g'(—0), g'(+0))t belong to a maximal neutral subspace of (C3, [■, ■]) where

<G Q • ©)

G I x2 I ' I У2 И = 1 [Х1(.Уз - У 2) + Vl{x2 - Хз)]

The eigenvalues of G are 0 and ±л/2 with eigenspaces

ker G = span {(0, 1, 1)*} , L+ := ker(G - л/2) = span {(iv^, -1, 1)*} ,

L- := ker(G + >/2) = span {(-iv^, -1, 1)*} .

Note that L± are maximal positive and maximal negative subspaces of C3 respectively. Hence any maximal neutral subspace of C3 has dimension 2. They are of the form ker G ф {v + Kv : v E ker(G — л/2)} where K is an isometry from L+ to L-. Clearly all such

Вестник ЮУрГУ. Серия «Математическое моделирование g]^

и программирование» (Вестник ЮУрГУ ММП). 2016. Т. 9, № 1. С. 73-91

isometries are of the form v+ M e 210v- where v± G L±. In summary, all maximal neutral subspaces are

L0 = span < 1

This can be rewritten as

L0 = span ,

-1 I + e

'>/2sin(6)N

- cos(6) cos(6)

210

—iV^ -1 1

6 G [0, n).

span

'v/2cos(6) sin(6) - sin(6)

It follows that f G D(B0) if and only if f G V(B*) and

v^cos 6f (0) + sin 6 [f' (-0) - f'(+0)] = 0.

(29)

□

Remark 4.

1. Observe that functions / in V(B)* which satisfy {(/ (0),f (—0),f (+0)} E kerG belong to D(B).

B

(a) 9 = In this case the boundary condition (29) is simplified to

f (—0) — f (+0) = 0. That is, / and /' are continuous and we obtain the classical harmonic oscillator:

B

n/2

A.

(b) 6 = 0. In this case the boundary condition (29) is simplified to f (0) = 0.

(c) 6 G (0,n) \ {n/2}. The boundary condition (29) can be written as

f(0)

tan(6)

f (+0) - f (-0^ .

Hence any function in D(B0) is continuous but its derivative has a jump in t = 0 which is proportional to the value of f in 0. Two different selfadjoint B

3. For every n G No, the function ^2ra+i from (5) is an eigenfunction of B0 with eigenvalue 2n+3/2. So the odd eigenvalues of the harmonic oscillator are not affected

0

An interpretation of these operators as a differential operator with a ^-interaction at

0

Let À G C. By the asymptotic expansion (1) the equation Ay = Xy has square integrable solutions y± on R± which are unique up to a constant factor. Let us define

y(x) =

{

y+(x), x > 0, y+(—x), x < 0.

(30)

Then clearly y E D(B*) and it is, up to a constant factor, the unique solution of (B*— A)y = 0. Moreover, y E D(Bo) where 0 is the unique number in [0,n) such that -\/2cos 0 f (0) = sin0 [f (+0) - f (-0)].

This shows that, as in the case of selfadjoint extensions of A0, every A E R appears as

B

Moreover, any given Bo can have at most one negative eigenvalue.

As in Lemma 3 we can identify all 0 such that Bo has a negative eigenvalue.

Lemma 5. Let Bo be as in Proposition 1 and let aB = arctan(GO-) with G as in (26). Then Bo has the eigenvalue 0 if and only if 0 = n — aB. It has a negative eigenvalue if and only if 0 E (n — aB, n). The corresponding eigenvalue A = —u2 can be found via the equality tan(0) = — Moreover, let Aj be eigenvalues of B0j forj = 1, 2. If A2 <A\ < 0, then n — aB < 0\ < 02 < n.

Proof. Let A = —u2 < 0 and y be as in (30) with y+ = v(-,u) (see (25)). Then y is an eigenfunction of Bo with eigenvalue A if and only if tan(0) = — pp Hence negative

eigenvalues occur if and only if 0 E (n — arctan(p)), n). Sinee G is decreasing in u with lim^^, G(u) = 0, also the last claim follows.

□

For A E — (2N — 2) we can calculate the corresponding eigenfunctions by a recursion formula.

Lemma 6. Let n E N and 4>± be as in (13) and (14). Set 4>±,n(t) := (dt +1)n $±(t) for t E R± and

U+,n(t), t> 0, Un(t) := <

(4>-,n(t), t< 0.

Then u2n E V(B*). That is, u2n defines a selfadjoint extension Bo of B and it is an eigenfunction of Bo with eigenvalu,e —2n —

Proof. Clearly un E D(Amax)- ft is easy to check that 0+(t) = 0_(—t) for t > 0. Moreover, a straightforward calculation shows

(a + n + 0 ф± n = 0.

Hence ф+;n(t) = (—1)гаф_,n(-1) for t > ^^o u2n is continuous in 0, and therefore it belongs to V(B*). If in addition we had U2n(—0) = u'2n(+0), the n u2n E V(A),&a.à —2n — 2 would be an eigenvalue of A, in contradiction to (4).

□

3. Two-Dimensional Restriction of the Harmonic Oscillator and Classification of Its Selfadjoint Extensions

Let us further restrict the harmonic oscillator on the real line. We consider the following restriction С of the selfadjoint operator A:

Cf := Af, D(C) := {f E D(A) : f(0) = f'(0) = 0}.

The operator C is closed and symmetric, but not selfadjoint. It is easy to see that

C * = A0.

The operator C is closely related to the harmonic oscillator on the half lines R± because

C = (C*)* = (A*0)* = A0 = Amin © A™n.

(31)

Analogously to Lemma 2 and Proposition 1 we now classify all selfadjoint extensions of C. Observe that the selfadjoint extensions of C are exactly those of A0.

Recall that U(2) is the set of all unitary 2 x 2 matrices.

Proposition 2. The defect indices n+(C) and n-(C) of C are such that n+(C) = n-(C) = 2. Hence all selfadjoint extensions of C are two-dimensional restrictions of C*.

U(2) C

for every K = (kim)fm=1 G U(2), the operator

Ck f = Af,

D(Ck )

k ) =

f G D(C*)

0 = (1 - kn)f (—0) + i(1 + kn)f (—0) 1 + ifcl2f (+0) — ki2f' (+0),

0 = —k2if (—0)+ik2if (—0)

+ i(1 + k22 )f (+0) + (1 — k22)f (+0)J

(32)

is a selfadjoint extension of C. There are no other selfadjoint extensions and CK = C^ if and only if K = K.

For a parametrization of the selfadjoint extensions with four real parameters, see the corollary after the proof of this proposition.

Proof. From Lemma 2 we know that n+(Amin) = n-(A™) = 1, so

dim (ker(A^ax — i)) = dim (ker(A^ax + i)) = 1.

Hence there are functions = 0 such that ker(Amax — i) = span{^±}. From Remark 3 we have n E ker(A0 — i) if and only if E ker(Amax — i). Therefore (compare with (28))

ker(A* — i) = span{X(-^ ,o)^-, X(o,

and n+(A0) = 2. Analogously n-(A0) = 2 can be shown.

Ao

A *o

ff,A*g) — (A*0/,g) = / (+0)g' (+0) — f (+0)g(+0) — / (—0)g' (—0) + f (—0)g(—0)

for all /, g E D(A0).

/, g A o

(/(—0), /'(—0), /(+0), /'(+0))4 and (g(—0), g'(—0), g(+0), g'(+0))4 belong to a maximal

neutral subspace of (C4, [-, -]) where

X1 Xf Хз X4

У1 У2 Уз У4

:=

(G -i

i G G G

G G G i

\G G -i G/

G G X1

Xf Хз

X4

У1 У2 Уз У4

)

i(xiy2 - Х2У1 - ХзУ4 + Х4Уз)•

Every maximal neutral subspace has dimension 2. Let

1 /

vi = ^(l, i, G, G)4,

1 / 44

V2 = ^2(G, G, i, l)4,

w1 = -^(1, -i, 0, oy, w2 = -U 0, -i, iy. v 2 V 2

Then L+ = span{v1; v2} is a maximal positive and L_ = span{w1, w2} is a maximal negative subspace of (C4, [•, •]) and all maximal neutral subspaces are of the form

Lk = {v + Kv : v G L+} = {w + K*w : w G L_}[±1 = {w - K*w : w G L_}x

where K is a unitary operator from L+ to L_ and [±] denotes the orthogonal complement with respect to the inner product [ •, • ]. With respect to the basis vectors v1,v2,w1,w2, K can be written as quadratic matrix

K

Í kii ki2\ \kfl kff J

(33)

with kjk G C (for the form of these numbers see the corollary after this proof). With respect to the standard unit vectors ei, e2, e3, e4 in C^, the space LK can be written as

LK = span

Í 1 + kii \ i(l - kii) -ikfi kfi

( ki2 \

-ikl2 i(l - kff) V 1 + kf2 )

span

V

( 1 - kll \ -i(l + kll) -ikl2 -kl2

( -k_2l \ -ik2¿ -i(l +kf2 ) V 1 - kf2 )

±

/

(34)

where K = (kjj)2j=1 as ^n From (34) it follows that every selfadjoint extension of C is given by (32).

□

It is well-known that U(2) is parameterized % four real parameters G R:

Every K G U(2) is given by

K=e

for fixed ф, a, ßl,ß2.

ф( ei ei

eißl sin a eiß2 cos a

e iß2 cos a

—e ißl sin a

sa in a

(35)

Therefore the boundary conditions in (32) can be rewritten as follows:

Corollary 2. Let K G U(2) be as ¿n (35). Then f G D(Ck) if and only if f G D(C*) and f satisfies the boundary conditions

0 = (1 - sin a) f (-0) + i(1 +ei<V^ sin a) f (-0)

+ iei(V№ cos af (+0) - ei(V№ cos a f (+0),

0 = -e^2 cos a f (-0) + ie^2 cos a f (-0)

+i(1 - e^e-№ sin a) f (+0) + (1 + e^e-№ sin a) f (+0).

(36)

K

3.1. Classical Harmonic Oscillator

Let K = ^ 0"^ • For instance, we can choose a = [1 = [2 = 0 fi = n. Then the boundary conditions (32) reduce to

f (-0) = f (+0) and f (-0) = f (+0).

Hence CK = A is the classical harmonic oscillator.

3.2. Boundary Conditions such that CK = Be

Let [1 = [2 = 0 a G (0,n), fi = a + n/2. Then

. a ( sin a cos a \ K = ieia .

cos a - sin a

(37)

and the boundary conditions (36) become

0 = (1 — ieia sin a) f (—0) + i (l + ieia sin a) f'(—0) — eia cos af (+0) + if'(+0)) ,

0 = —ieia cos af (—0) — if'(—0))

+ i (1 — ieia sin a) f (+0) + (l + ieia sin a) f'(+0),

which, for a = n/2 is true if and only if

f(—0) = f(+0)=: f(0) and f'(—0) — f'(+0) = 2tanaf(0). (38)

Choose 0 E [0,n) such that cot 0 = — v^tana. Then CK = Bo with K as above. For a = n/2, the conditions (37) are equivalent to f (+0) = f (—0) = 0.

3.3. Boundary Conditions with Continuous Derivative

Let a G (0,n) and let [1 = [2 = 0 fi = n/2 - a. Note that e^ = ie-ia and

K = ie-iaisin a cos a\ ycos a — sin ay

(39)

Then equations (36) become

0 = e-ia cos a f (-0) + i(1 + ie-ia sin a) f (-0) - e-ia cos af (+0) + if (+0))

0 = -e-ia cos a (if (-0) + f (-0))

+ ie-ia cos a f (+0) + (1 +ie-ia sin a)f (+0). If a = n/2, then (39) is equivalent to

f (-0) = f (+0)=: f (0) and f (+0) - f (-0) = -2 tan a f'(0). (40)

If a = n/2, then (39) is equivalent to f'(-0) = f'(+0) = 0.

4. Interpretation of Some Extensions Via 8- and 8'-Type Interactions

Observe that the operator A from (3) is closed. Hence the set H+ := D(A) becomes a Hilbert space with the norm

\\f 11+ := \\f \\a := (IfII2 + A||2)2 , f e H+. Let H0 := L2 (R). In addition to the usual norm on H, define

\- :=sup{|(f,g>| : g e H+, \\g\\ + < 1}, f e H0,

and we define H_ to be the closure of H0 with respect to the norm || ■ ||_. Then (H-, || - ^_)

is a Hilbert space and it can be viewed as the dual space of H+. Observe that we have the continuous inclusions

H+ c Ho C H_.

On says that H0 is rigged by H+ and H_, see, for instance, [2], Chapter 14.

If T : H+ ^ Ho is a bounded linear operator, then define its adjoint operator T* : H0 ^ H_ as the unique bounded linear operator that satisfies

(Tf,g) = (f, T *g), f E H+, g E Ho,

where (■, ■) denotes the inner product on H0. Let us define the functions

\v(t, 0), if t> 0 ^ \v(t, 0), if t> 0,

Wl(t) = < \ W2(t) = < ' '

1W \v(—t, 0), if t< 0, \—v(—t, 0), if t< 0,

with v as in (25). Clearly w1,w2 G H0 C H—. Observe that

Wi(+0) = wi(—0) = W2(+0) = W2( 0) = v(0, 0) = G(0) (41)

and wi(+0) = — wi(—0) = w2(+0) = w2(—0) = v'(0, 0) = —1. (42)

Lemma 7. The linear Junctionals

5 : H+ ^ C, 5f = f (0), S' : H+ ^ C, 5'f = f'(0)

are bounded and

5f = 1 {Af,wi), 5'f =2G(0y(Af,w2), f G H+.

Proof. Note that for any f G H+ = D(A) and j = 1, 2, using integration by parts twice, we have

+ <X

(Af,wj) = j(Af)(t) • wj(t)dt

—x

0 +x

= J (—f''(t)+ t2f (t))wj(t)dt + J (—f''(t) + tf (t))wj(t)dt

— x 0

0 +x

= J f (t)(—wj(t)+ t2w3(t))dt + ^ f (t)(—wj(t)+ t2w3(t))dt

— x 0

+ f' (0) {wj (+0) — wj (—0)} + f (0) {wj (—0) — wj (+0)} = f (0) {wj (+0) — wj (—0)} + f (0) {wj (—0) — wj (+0)},

55

\5f \ = 12{Af,wi)\ < 2\\Af || ||wi|| < 2 f || + |wi^ and analogously \5'f | < ^ f ll + l|wi||.

□

Recall that in our case, H+ C D(A0) C H0 C H—. By definition of H+, the operator

1: H+ ^ H0, Af = Af

is bounded. Let us calculate how A* acts on elements g G D(A0). As in the proof of Lemma 7, integration by parts gives for f G H+

(Af, g) = {g(+0) — g(—0)} f' (0) + {g' (—0) — g' (+0)} f (0) + (f, A*g)

= {g(+0) — g(—0)} ^G^y (Af,w2) + {g' (—0) — g' (—0)} 2 (Afw) + (f,A0g)

= {g(+0) — g(—0)} ^(f,A*w2) + {g'(—0) — g'(+0)} 1 (f,A*wi) + (f,A*g).

математическое моделирование

So by Lemma 7, we obtain

j* g(+0) — g(—0) j* + g (—0) — g (+0) - + .*

A g =-2g(0)-A w2 +-2-A wi + Aog,

or, if we identify H— and (H+)',

A*g = {g(+0) — g(—0)}5' — {g(+0) — g(—0)}5 + A0g g (h+)'.

Hence A0 can be seen as a perturbation of A*-.

A*g = A*g — {g(+0) — g(—0)}5' + {g(+0) — g(—0)}5 G (H+)' (43)

for g G D(A*). Recall that the operators Be from Section 2 and CK from Section 3 satisfy B C Be C A** and C C CK C A0- So we obtain the following:

• Any function g G D(CK) with K as in Subsection 3.1 satisfies g(—0) = g(+0) and g (—0) = g (+0)

Ck g = A*g = A0g.

• Any funct ion g G D(CK) wit h K as in Subsection 3.2 and a = n/2 satisfi es g(—0) = g(+0) and g'(—0) — g'(+0) = 2 tan a g(0), hence

CKg = A0g = A*g — 2 tan a g(0)5.

If we take 6 such that cot 6 = — v^2tan a, we obtain

Beg = Ckg = v^cot 6 g(0)5 + A*g.

Note that Be has exactly one negative eigenvalue if 6 G (n/2 + aA, n) and this eigenvalue decreases monotonically to —to as 6 n, that is v/2cot(6) —>• — oo.

• Any fonction g G D(CK^th K as in Subsection 3.3 satisfies g'(—0) = g'(+0) and g(+0) — g(—0) = —2 tan a g'(0). Hence, for a = n/2

CKg = A0g = A*g + 2 tan a g'(0)5'.

5. Closing Remarks

The 3-dimensional point potential for the Schrodinger operator was considered by Zeldovich [16] and Berezin and Faddeev [3]. The free Schrodinger operator with 10

by Kurasov in [10]. Both use von Neumann's extension theory to obtain selfadjoint extensions of a given differential operator on R \ {0} and interpret their results in terms 5 5 0

5

by Gadella, Glasser and Nieto in [7] and Viana-Gomes and Peres in [13]. In both works the eigenfuntions are calculated in terms of confluent hypergeometric functions. Moreover, it is shown that the eigenvalues with odd eigenfunctions are not changed, whereas the

eigenvalues with even eigenfunctions increase (for c > 0) от decrease (for c < 0) when compared with the eigenvalues of the harmonic oscillator without singular perturbation.

Acknowledgment. We are grateful to the "Fondo de Investigaciones de la Facultad de Ciencias de la Universidad de los Andes, Convocatoria 2014-1 para la Financiación de proyectos de Investigación Categoría: Profesores De Planta ", project "Operadores lineales en espacios con producto interno indefinido", for its financial support.

References / Литература

1. Albeverio S., Gesztesy F., H0egh-Krohn R., Holden H. Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Providence, RI, 2005.

2. Berezansky Y.M., Sheftel Z.G., Us G.F. Functional Analysis. Vol. II, Volume 86 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1996.

3. Berezin F.A., Faddeev L.D. A Remark on Schrödinger's Equation with a Singular Potential. Soviet Mathematics. Doklady, 1961, no. 2, pp. 372-375.

4. Chernoff P.R., Hughes R.J. A New Class of Point Interactions in One Dimension. Journal of Functional Analysis, 1993, vol. Ill, no. 1, pp. 97-117. DOI: 10.1006/jfan. 1993.1006

5. Eastham M.S.P. The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem. Oxford, Clarendon Press, 1989.

6. Gadella M., Glasser M.L., Nieto L.M. The Infinite Square Well with a Singular Perturbation. International Journal of Theoretical Physics, 2011, vol. 50, no. 7, pp. 2191-2200. DOI: 10.1007/sl0773-011-0690-5

7. Gadella M., Glasser M.L., Nieto L.M. One Dimensional Models with a Singular Potential of the Type —aó(x) + ßö'(x). International Journal of Theoretical Physics, 2011, vol. 50, no. 7, pp. 2144-2152.

8. Gohberg I., Lancaster P., Rodman L. Indefinite Linear Algebra and Applications. Birkhäuser Verlag, Basel, 2005.

9. Kato T. Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. New York, Springer-Verlag New York, 1966.

10. Kurasov P. Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients. Journal of Mathematical Analysis and Applications, 1996, vol. 201, no. 1, pp. 297-323. DOI: 10.1006/jmaa.l996.0256

11. Seba P. The Generalized Point Interaction in One Dimension. Czechoslovak Journal of Physics B, 1986, vol. 36, no. 6, pp. 667-673. DOI: 10.1007/BF01597402

12. Triebel H. Higher Analysis. Hochschulbücher für Mathematik. [University Books for Mathematics]. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.

13. Viana-Gomes J., Peres N.M.R. Solution of the Quantum Harmonie Oscillator Plus a Delta-Function Potential at the Origin: the Oddness of its Even-Parity Solutions. European Journal of Physics, 2011, vol. 32, no. 5, pp. 1377-1384. DOI: 10.1088/0143-0807/32/5/025

14. Weidmann J. Linear Operators in Hilbert Spaces, Volume 68 of Graduate Texts in Mathematics. New York, Berlin, Springer-Verlag, 1980.

15. Weidmann J. Spectral Theory of Ordinary Differential Operators, Volume 1258 of Lecture Notes in Mathematics. Berlin, Springer-Verlag, 1987.

16. Zeldovic Ya.B. Scattering by a Singular Potential in Perturbation Theory and in the Momentum Representation. Soviet Physics. Journal of Experimental and Theoretical Physics, 1960, no. 11, pp. 594-597.

Received June 17, 2015

УДК 517.921.25+517.984.5 Б01: 10.14529/ттр16010б

ОДНОМЕРНЫЙ ГАРМОНИЧЕСКИЙ ОСЦИЛЯТОР С СИНГУЛЯРНЫМ ВОЗМУЩЕНИЕМ

В. А. Штраус, М.А. В и и к л ъш а й ер

В настоящей работе исследуется одномерный возмущенный гармонический осциллятор с лево-правосторонними граничными условиями в нуле. На рассматриваемый объект можно смотреть как на классический самосопряженный гармонический осциллятор с сингулярным возмущением, сосредоточенным в одной точке. Указанное возмущение порождается дельта-функцией Дирака и/или ее производной. Описываются все самосопряженные реализации этой схемы в терминах указанных граничных условий. Показывается, что при некоторых ограничениях на возмущение (или, что эквивалентно, на граничные условия) у соответствующего дифференциального оператора может появиться ровно одно неположительное собственное значение, и приводится аналитическое выражение для соответствующей собственной функции. Указанное собственное значение пробегает всю неотрицательную полуось когда коэффициент возмущения пробегает установленный промежуток. Для некоторых случаев приводится непосредственная зависимость между подходящими граничными условиями, соответствующим неотрицательным собственным значением и его собственной функцией.

Ключевые слова: гармонический осциллятор; сингулярные возмущения; самосопряженные расширения; отрицательные собственные значения.

Владимир Абрамович Штраус, доктор физико-математических наук, профессор, департамент чистой и прикладной математики, университет Симон Боливар, Каракас, Венесуэла, str@usb.ve.

Моника Винкльмайер, доктор естественных наук, доцент, департамент математики, университет Лос Андес, Богота, Колумбия, mwinklme@uniandes.edu.со.

Поступила в редакцию 17 июня 2015 г.