von Neumann-Wigner theorem:
level’s repulsion and degenerate
eigenvalues.
Yu.N.Demkov and P.Kurasov
Abstract. Spectral properties of Schrödinger operators with point
interactions are investigated. Attention is focused on the interplay
between the level’s repulsion (von Neumann-Wigner theorem) and
the symmetry of the points configuration. Explicit solvability of
the problem allows to observe level’s repulsion for two centers. For
larger number of centers the families of point interactions leading
to the highest possible degeneracy is investigated.
1. Introduction
The method of operator extensions in mathematics and its special case called in physics the method of zero range potentials (ZRP),
is rapidly developing in the last decades due to its universality, applicability to many physical problems leading to essential simplifications (usually algebraization). See [5] where physical applications are
considered and [2] where the same model is studied from the mathematical standpoint. Relative to other approximative methods this one
contains from the very beginning continuous spectrum and preserves
such properties of the exact problems as unitarity (in contrast to the
Born approximation). In many problems appearing in physics short
range objects are separated by large distances (galaxies, stars, planets,
atoms and molecules, nuclei and elementary particles). Apart from
very exotic cases only solid state doesn’t satisfy this property, but even
there we can consider excitations as quasiparticles again returning to
the same description.
In zero range potential models we have a unique possibility to solve
quantum problems exactly. The method is a particular case of the
nonlocal projection operators method when the projection functions
1
2
YU.N.DEMKOV AND P.KURASOV
are δ−functions and this allows one to preserve the locality of the
operator.
Although the theory is well developed the case where an infinite
number or even a few zero range potentials are present is still not
considered in full details. In the present article we are going to discuss
properties of these models in relation to inverse problems and Wignervon Neumann theorem.
The celebrated von Neumann-Wigner theorem [10] describes the
probability that a finite dimensional matrix has a degenerate eigenvalue. This probability is lower than one may expect, namely the
co-dimension of the set of matrices having double eigenvalue is always
higher than 1. It follows that for a time-dependent Hamiltonian the
probability that two energy curves cross each other is extremely low,
and this phenomenon is called level’s repulsion. Usually two levels
cross each other only if the corresponding eigenfunctions have different symmetries. Hence one expects to observe degenerate eigenvalues
for Hamiltonians having certain symmetries only. In this paper we
are going to study this phenomena using Hamiltonians with point interactions in R1 and R3 . These operators are widely used in quantum
mechanics and atomic physics to model different physical processes (see
[5, 2, 3] and numerous references there).
Every Hamiltonian with N point interactions may have at most
N eigenvalues. Straightforward analysis shows that no eigenvalue of
multiplicity N can appear. The main goal of this paper is to study the
possibility for operators with point interactions to have eigenvalues of
the highest possible multiplicity N − 1.
The paper is organized as follows. Schrödinger operators with point
interactions are defined rigorously in Section 2 following mainly [2,
5]. Since two local point interactions cannot produce any degenerate
eigenvalue (even of multiplicity 2), we study level’s repulsion for two
centers in Section 3. For the cases of three, four and five centers the
degeneracies of the eigenvalues are studied in Section 4. It is shown
that the maximal degeneracy can be observed in the case where the
configuration has a certain symmetry.
2. Hamiltonians with delta interactions
Schrödinger operators with N local delta interactions at the points{y n }N
n=1
are formally defined by
(1)
Lα = −∆ +
N
X
n=1
αn δ(· − y n ),
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
3
where ∆ is the Laplace operator and δ(·−y n ) is the delta function with
the support at the point y n . Without loss of generality we suppose that
all points y n are different.
In what follows we give precise definition for the operator Lα . Note
that the operator corresponding to the formal expression (1) is uniquely
defined in R1 , but to define this operator in R3 one has to take into
account extra assumptions.
Consider the Laplace operators
(2)
d2
∂2
∂2
∂2
−∆1 = − 2 in L2 (R1 ); and −∆3 = − 2 − 2 − 2 in L2 (R3 ),
dx
∂x1 ∂x2 ∂x3
which are self-adjoint when defined on the Sobolev spaces W22 (Rj ). Here
x and ~x = (x1 , x2 , x3 ) denote the coordinates in R1 and R3 respectively.
Then the operator corresponding to the formal expression (1) is one
of the self-adjoint extensions of the symmetric restrictions −∆j0 of
−∆j , j = 1, 3 to the set of functions vanishing at the interaction
points
(3)
−∆j0 = −∆j |{ψ∈W22 (Rj ):ψ(yn )=0,n=1,2,...,N } .
The deficiency elements for λ = −χ2 are just solutions to the equations
−∆j0∗ g + χ2 g = δ(x − y n )
(4)
n
n
e−χ|x−y |
e−χ|x−y |
1
n
3
n
g (x, y ) =
and g (x, y ) =
, n = 1, 2, ..., N.
2χ
4π|x − y n |
Hence the deficiency indices of the restricted operators are equal to
(N, N ). The domains of the adjoint operators are given by
1
Dom (−∆10∗ ) = W22 (R1 \ {y n }N
n=1 ) ∩ C(R );
(5)
Dom (−∆30∗ ) = W22 (R3 \ {y n }N
n=1 ).
To describe self-adjoint extensions of −∆j0 we are going to use the
boundary values for the functions from the domain of the adjoint operator
1
ψ(x) ∼x→yn − |x − y n | ψ−n + ψ0n + o(1), for R1 ;
2
1
ψ−n + ψ0n + o(1), for R3 .
4π|x − y n |
Then the boundary forms of the adjoint operators are given by
(6)
(7)
ψ(x) ∼x→yn
h(−∆j0∗ )u, vi − hu, (−∆j0∗ )vi =
N
X
n=1
(u0n v̄−n − u−n v̄0n ) .
4
YU.N.DEMKOV AND P.KURASOV
Definition 1. The operator Ljα is the restriction of the adjoint
operator −∆j0∗ to the set of functions from ψ ∈ Dom (−∆j0∗ ) satisfying
the boundary conditions
~0 = −α−1 ψ
~− ,
(8)
ψ
~− = (ψ−1 , ψ−2 , ..., ψ−N )T and ψ
~0 = (ψ01 , ψ02 , ..., ψ0N )T are vecwhere ψ
tors of boundary values of the function ψ. In other words the operator
Ljα is the Laplace operator defined on the domain of functions satisfying
(8).
In dimension one it is possible to prove that the operator corresponding to the formal expression (1) is given by Definition 1. In
dimension three it is necessary to use certain additional assumptions
like the homogeneity properties of the Laplace operator and the delta
distribution in order to show that the self-adjoint operator corresponding to (1) is given by Definition 1 (see [3], Section 1.5.1 for details).
We do not want to dwell on this point, since our further studies are
based on Definition 1 and are independent of these assumptions.
In what follows only local delta interactions will be considered, i.e.
interactions corresponding to the diagonal matrix α. For exhausting description of local point interactions see [8]. (It is possible to show that
non-diagonal matrices α in (8) lead to nonlocal interactions.) Without loss of generality we suppose that all coefficients αn are different from zero. If it is not the case then the set of singular points
y n , n = 1, 2, ..., N can simply be reduced.
The resolvent of the perturbed operator Lα can be calculated using
Kreins formula [6, 7, 9], since each of the operators Ljα is a finite dimensional perturbation of the Laplace operator −∆j in the resolvent
sense. Hence the essential spectrum of Lα is purely absolutely continuous and coincides with the interval [0, ∞). (It has multiplicity 2 in
R1 and infinite multiplicity in R3 .) The number of negative eigenvalues
cannot exceed N (the rank of the perturbation). In addition straightforward analysis shows that no positive eigenvalues occur. The discrete
spectrum of the operator is given by the zeroes of the perturbation determinant appearing in Kreins formula.
Another way to obtain the equation for the discrete spectrum is to
consider the following Ansatz for the eigenfunction
(9)
ψ=
N
X
an g j (x, yn ),
n=1
j
where g (x, y) are the Green functions for the Laplace operator given
by (4). The function ψ presented by (9) satisfies the eigenfunction
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
5
equation for λ = −χ2
(10)
Lj0∗ ψ = −χ2 ψ
for any value of the complex parameters an . It is an eigenfunction if
and only if it in addition satisfies the boundary condition (8). Consider
the boundary values of the Greens functions
(11)

1/2χ,
m=n


½

1, m = n
1
1
n
m
g 1 = g 1 (x − y n ) ⇒ g−m
=
,
g0m
=
0, m 6= n
e−χ|y −y |


m 6= n

2χ

−χ/4π,
m=n


½

1, m = n
3
3
n
m
=
=
g 1 = g 1 (x − y n ) ⇒ g−m
, g0m
e−χ|y −y |
0, m 6= n


m 6= n

4π|y n − y m |
The dispersion equations defining the discrete spectrum in R1 and R3
are
¡
¢
(12)
det 1 + G1 + 2χα−1 = 0;
(13)
¡
¢
det −χ + G3 + 4πα−1 = 0;
respectively, where Gj are the following Hermitian N × N matrices
½ −χ|yn −ym |
e
, n 6= m
1
,
Gnm =
0,
n=m
(
n
m
(14)
e−χ|y −y |
n 6= m
3
n −y m | ,
|y
Gnm =
.
0,
n=m
In what follows we are going to study solutions to these dispersion
equations concentrating our attention to the degeneracy of the eigenvalues.
In addition to exponentially decreasing at infinity eigenfunctions
(corresponding to negative eigenvalues) there exist solutions decreasing
power-like (corresponding to the eigenvalue zero). Decreasing of these
functions at infinity is related to their angular dependance. Spherically
symmetric functions are decreasing like c/r and therefore are not normalizable. All other values of the angular momentum are admissible
(for E = 0).
6
YU.N.DEMKOV AND P.KURASOV
3. Level’s repulsion for two centers in R1 and R3 .
Two local delta interactions cannot produce a degenerate eigenvalue. Therefore we are studying here the level repulsion. The operator with two delta potentials can be parameterized by three real
parameters: d > 0 - the distance between the centers, αj , j = 1, 2 - the
strengths of the delta interactions.
Consider first the case of two point centers in dimension one. The
dispersion equation is given by the following formula in this case
(15)
(1 + 2χα1−1 )(1 + 2χα2−1 ) − e−2χd = 0.
It is convenient to use the following two parameters
1
γi = − αi , i = 1, 2.
2
Then the dispersion equation is
(16)
(17)
(χ − γ1 )(χ − γ2 )
− e−2χd = 0.
γ1 γ2
Let us denote the corresponding operator by L1 (γ1 , γ2 ).
The dispersion equation for two centers in R3 is given by
(18)
(−χ + 4πα1−1 )(−χ + 4πα2−1 ) −
e−2χd
= 0.
d2
Let us introduce two new parameters
(19)
γj = 4π/αj , j = 1, 2.
We get the dispersion equation
(20)
(χ − γ1 )(χ − γ2 ) −
e−2χd
= 0.
d2
The corresponding operator will be denoted by L3 (γ1 , γ2 ).
The parameters γj just introduced for one- and three-dimensional
problems can be interpreted as the energies of the bound states associated with each of the two point centers.
Consider the Schrödinger operator with one delta interaction
−∆ + αδ(x).
Then the corresponding operator has exactly one bound state with the
energy
(21)
E = −γ 2 = −
α2
, provided α < 0 in dimension one
4
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
7
and
(4π)2
, provided α > 0 in dimension three.
α2
The energies corresponding to single interactions can be obtained
from the dispersion equations (17) and (20) considering the limit d →
∞. The exponential function tends to zero and the two dispersion equations transform into the following equation
(22)
E = −γ 2 = −
(χ − γ1 )(χ − γ2 ) = 0,
having two solutions χ = γ1,2 and defining the energies of the two
bound states E1 = −γ12 and E2 = −γ22 provided that γ1,2 > 0.
Another way to obtain these bound states is to consider the limit
where the interaction at one of the centers vanishes. Note that vanishing interaction αj = 0 corresponds formally to γj = 0 for R1 and to
γj = ∞ for R3 . The limits of the equations (17) and (20) when γ2 → 0,
respectively γ2 → ∞ are given by
χ − γ1 = 0.
The last equation determines the unique bound state with the energy
E1 = −γ12 (provided γ1 < 0).
We find it more convenient to use parameters γj instead of αj to
parameterize the operators with delta interactions.
Let us study the number of eigenvalues depending on the values of
the two parameters. Without loss of generality we can assume that
γ1 ≥ γ2 . We introduce the following notation
(χ − γ1 )(χ − γ2 )
P 1 (χ) =
;
γ1 γ2
P 3 (χ) = (χ − γ1 )(χ − γ2 ).
The eigenvalues of the operator L(γ1 , γ2 ) correspond to positive (real)
solutions of the dispersion equation. Note that equation (17) has one
”parasite” solution χ = 0 which is not physical, since no eigenfunction
corresponds to E = 0 in this case. The corresponding function does
not belong to the Hilbert space.
Let us study the three cases covering all possibilities (the cases when
γ1 or γ2 are equal to zero can be excluded from the consideration),
separately for one and three dimensional point interactions
• γ2 ≤ γ1 < 0
For positive χ the functions P 1 (χ) and e−2χd satisfy the
following inequalities
P 1 (χ) ≥ 1 ≥ e−2χd .
8
YU.N.DEMKOV AND P.KURASOV
The latter inequality is strong for χ 6= 0. Therefore equation
(17) has no solution on the interval (0, ∞) in this case.
The function P 3 is growing to infinity for positive χ and
the function e−2χd /d is decreasing. Comparing the values of
the functions at the origin one can deduce that eq. (20) has
one solution iff
1
(23)
γ1 γ2 < 2 .
d
• γ2 < 0 < γ 1
The functions P 1 (χ) and e−2χd are equal to 1 at χ = 0.
Their second derivatives are negative and positive, respectively. Therefore equation (17) has at most one positive solution and this solution exists if and only if
d −2χd
d 1
(e
)|χ=0 <
P (χ)|χ=0
dχ
dχ
1
1
⇒
+
< 2d.
γ1 γ2
The solution belongs to the interval (0, γ1 ). Let us denote this
solution by χ1 .
Equation (20) always has one solution in this case, since
the function P 3 is growing to infinity and has positive zero and
the function e−2χd /d2 is positive and decreases to zero. The
solution belongs to the interval (γ1 , ∞).
• 0 < γ 2 ≤ γ1
The function P 1 is equal to zero at the points χ = γ1 and
χ = γ2 . It increases to +∞ on the interval (γ1 , ∞). Therefore
there exists a solution to the dispersion equation (17) on the
interval (γ1 , ∞). This solution will be denoted by χ1 in what
follows. The second solution is situated on the interval (0, γ2 )
if and only if the following condition is satisfied
d −2χd
d 1
(e
)|χ=0 <
P (χ)|χ=0
dχ
dχ
1
1
⇒
+
< 2d.
γ1 γ2
The second solution will be denoted by χ2 .
Similarly equation (20) has two solutions χ1 and χ2 situated on the intervals (0, γ2 ) and (γ1 , ∞) if and only if the
following condition is satisfied
1
(24)
γ1 γ2 > 2 .
d
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
9
Otherwise the equation has unique solution situated on the
interval (γ1 , ∞).
Thus we proved once more that the discrete spectra of the two
problems consist of at most two distinct eigenvalues situated on the
negative half-axis. Let us study the inverse spectral problem for these
operator:
Reconstruct the coupling constants γ1 and γ2 so that the operators
L1 (γ1 , γ2 ) and L3 (γ2 , γ3 ) have given −χ21 = E1 < E2 = −χ22 < 0 as
eigenvalues provided that the distance d between the interaction points
is fixed.
Without loss of generality we can suppose that d = 1. This problem
can be solved in the following way.
Consider first the dispersion equation corresponding to the one dimensional problem. Suppose that χ1 and χ2 are solutions of (17). Then
the parameters γ1,2 satisfy the following system of equations

 (χ1 − γ1 )(χ1 − γ2 ) = γ1 γ2 e−2χ1
(25)
,

−2χ2
(χ2 − γ1 )(χ2 − γ2 ) = γ1 γ2 e
which can be written as follows

χ1 χ2 (χ1 −χ2 )

 γ1 γ2 = χ1 (1−e−2χ2 )−χ2 (1−e−2χ1 ) ≡ A(χ1 , χ2 )
(26)
.

 γ + γ = χ21 (1−e−2χ2 )−χ22 (1−e−2χ1 ) ≡ B(χ , χ )
1
2
1
2
χ1 (1−e−2χ2 )−χ2 (1−e−2χ1 )
Each of the two equations is linear with respect to γi . Therefore this
system of equations can be solved for example by resolving the first of
the two equations with respect to γ1 and substituting it into the second
equation. One obtains in this way the quadratic equation, which can
always be solved:
(27)
γ 2 − Bγ + A = 0.
But one cannot guarantee that the solutions, i.e. parameters γ1,2 are
real. Only real parameters γj define a self-adjoint operator. Hence the
energies of the bound states are not arbitrary, but satisfy the inequality
(28)
D(χ1 , χ2 ) ≡ B 2 (χ1 , χ2 ) − 4A(χ1 , χ2 ) ≥ 0.
Similarly for the three dimensional problem we have the system of
equations
½
(χ1 − γ1 )(χ1 − γ2 ) = e−2χ1 ,
(29)
(χ2 − γ1 )(χ2 − γ2 ) = e−2χ2 ;
10
YU.N.DEMKOV AND P.KURASOV
(
(30)
⇒
−2χ1
−2χ2
2e
γ1 γ2 = χ1 χ2 + χ1 e χ1 −χ
≡ A(χ1 , χ2 ),
−χ2
e−2χ1 −e−2χ2
γ1 + γ2 = χ1 + χ2 − χ1 −χ2
≡ B(χ1 , χ2 ).
Again there exists a self-adjoint operator if and only if the discriminant
given by (28) is non-negative.
Theorem 1. Let −χ21 < −χ22 be two eigenvalues of the operator
L (γ1 , γ2 ), j = 1, 3. Assume that the energy E1 = −χ21 is fixed. Then all
possible values of χ2 fills in the interval [0, χmax
] where χmax
= χmax
(χ1 )
2
2
2
is the value of χ2 corresponding to the symmetric interaction γ1 = γ2 .
This χmax
is the unique solution to the following equations
2
j
max
(31)
1 − e−χ2
χmax
2
=
1 + e−χ1
max
and χmax
+ e−χ2 = χ1 − e−χ1 ,
2
χ1
for R1 and R3 respectively.
Proof. We are going to prove the theorem for the one and three
dimensional cases separately.
Dimension one. The energies χ1 and χmax
corresponding to the
2
symmetric case γ1 = γ2 ≡ γ can be calculated from the following
equation
(χ − γ)2 = γ 2 e−2χ ⇒ χ − γ = ±γe−χ
max
1 − e−χ2
1 + e−χ1
=
.
χ1
χmax
2
To prove the theorem it is enough to show that the discriminant of
the system (27) is positive for all χ2 < χmax
and negative for χmax
< χ2 .
2
2
Taking into account that the discriminant is equal to zero for χ2 =
χmax
( γ1 = γ2 in this case) it is sufficient to show that the derivative
2
∂
D(χ1 , χ2 ) is negative. Consider the function f (x) = (1 − e−2x )/x.
∂χ2
Direct calculations show that
⇒
0 ≤ f (x) ≤ 2, −2 ≤ f 0 (x) ≤ 0, 0 ≤ f 00 (x),
provided x > 0. It is easy to see that
χ1 − χ2
A=
.
f (χ2 ) − f (χ1 )
Then the main value theorem implies 0 ≤ A(χ1 , χ2 ) ≤ 1/2. Using the
fact that B = χ1 + A(χ1 , χ2 )f (χ1 ) the derivative of the discriminant
can be evaluated
∂A(χ1 , χ2 )
∂D(χ1 , χ2 )
= 2 ((χ1 + A(χ1 , χ2 )f (χ1 ))f (χ1 ) − 2)
.
∂χ2
∂χ2
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
11
To prove that the derivative
theorem again
∂A(χ1 ,χ2 )
∂χ2
is positive we use the main value
∂A(χ1 , χ2 )
f (χ1 ) − (f (χ2 ) + (χ1 − χ2 )f 0 (χ2 ))
=
∂χ2
(f (χ1 ) − f (χ2 ))2
and take into account that the second derivative of f is positive. The
expression in the brackets is negative
χ1 f (χ1 ) + A(χ1 , χ2 )f 2 (χ1 ) − 2 ≤ χ1 f (χ1 ) + f 2 (χ1 )/2 − 2 ≤ 0.
The last inequality follows directly from the properties of the function
f (x). It follows that D(χ1 , χ2 ) is positive for χ2 < χmax
and negative
2
max
for χ2 < χ2 .
Dimension three We introduce new parameters
½
½
ξ = 12 (γ1 + γ2 )
γ1 = ξ + η
⇒
(32)
1
γ2 = ξ − η
η = 2 (γ1 − γ2 )
Dispersion equation (20) is then given by
(χ − ξ)2 − η 2 − e−2χ = 0.
(33)
Since χ1 is a solution of the last equation we get
(34)
Q3 (ξ, χ) ≡ (χ − ξ)2 − (χ1 − ξ)2 − e−2χ + e−2χ1 = 0.
Then the partial derivative
∂χ2
∂η
can be estimated
∂χ
η(χ1 − χ2 )
|χ=χ2 =
< 0,
∂η
(χ1 − ξ)(χ2 − ξ + e−2χ2 )
since
χ2 − ξ + e−χ = −
p
e−2χ2 + η 2 + e−2χ2 ≤ e−χ2 (e−χ2 − 1) < 0.
It follows that χ2 attains its maximal value when η = 0 i.e. in the
symmetric case γ1 = γ2 ≡ γ. This case corresponds to χ1 and χ2 = χmax
2
satisfying the equation
max
(χ − γ)2 = e−2χ ⇒ χ1 − χmax
= e−χ1 + e−χ2 .
2
the following estimate holds
Hence for all χ2 : 0 ≤ χ2 ≤ χmax
2
(35)
χ1 − χ2 ≥ e−χ1 + e−χ2 .
. The
Let us show that the discriminant is positive for all χ2 ≤ χmax
2
discriminant of the quadratic equation (30) on γ1 and γ2 is
¡
¢ (e−2χ1 − e−2χ2 )2
D(χ1 , χ2 ) = (χ1 − χ2 )2 − 2 e−2χ1 + e−2χ2 +
.
(χ1 − χ2 )2
12
YU.N.DEMKOV AND P.KURASOV
2
(e−2χ1 −e−2χ2 )
The sum of the first and the third terms (χ1 − χ2 )2 +
(χ1 −χ2 )2
2
−2χ
−2χ
(e 1 −e 2 )
can be estimated by (e−χ1 + e−χ2 )2 + (e−χ1 +e−χ2 )2 taking into account
that (χ1 − χ2 )2 ≥ e−2χ1 − e−2χ2 in accordance with (35). Then the
discriminant can be estimated from below
¡ −2χ1
¢ (e−2χ1 − e−2χ2 )2
−χ1
−χ2 2
−2χ2
D(χ1 , χ2 ) ≥ (e
+e ) −2 e
= 0.
+e
+
(e−χ1 + e−χ2 )2
It follows that the system (30) has real solutions for any 0 ≤ χ2 ≤
χmax
(χ1 ).
¤
2
The theorem states that the system of two local point interactions
never has a multiple eigenvalue. The distance between the eigenenergies is minimal in the symmetric case γ1 = γ2 . This can be illustrated
by the following figure.
5
5
4
4
3
3
x2
x2
2
2
1
1
0
0
0
1
2
3
4
5
0
1
2
3
x1
x1
R1
R3
4
5
Figure 1. Level’s repulsion for two centers.
The area lying between the two curved lines is forbidden, i.e. it is
impossible to find two point interactions at the distance d = 1 determining two eigenvalues in this region. In the limit χ1,2 → ∞ the curves
approach the line χ1 = χ2 . It follows that two deep eigenvalues can be
situated rather close to each other. The curves crosses the corresponding axes at the same point which is the unique solution of the equation
x = 1 + e−x .
We have proven that the Schrödinger operator with two local point
interactions cannot have a degenerate eigenvalue. Moreover the two
eigenvalues cannot be situated arbitrarily close to each other. This is
a certain generalization of von Neumann-Wigner theorem [10]. Note
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
13
that the last statement holds due to the special form of the boundary
conditions described by diagonal matrices. Considering nonlocal point
interactions one can obtain operators with two negative eigenvalues
situated arbitrarily.
4. Degenerate eigenvalues for several centers
The one-dimensional Schrödinger operator cannot have degenerate
eigenvalues (except the case where the operator can be represented by
an orthogonal sum of operators on two intervals). Therefore we restrict
our consideration to the case of the three-dimensional Schrödinger operator with point interaction described in Section 2. Moreover we are
going to study the maximal possible degeneracy for the sake of simplicity. Low-order degeneracies can be described by studying clusters
consisting of a smaller number of centers.
Let us consider the possibility for the maximal degeneracy N − 1
for the system of N point potentials. The equation determining the
eigenvalues in this system reads as follows


e−χd12
e−χd13
e−χd1N
−χ + γ1
...
d12
d13
d1N
 e−χd21

e−χd23
e−χd2N
−χ + γ2
...


d21
d23
d2N


31
e−χd32
e−χd3N
(36)
det  e−χd
 = 0,
−χ + γ3 ...
d31
d32
d3N




...
...
...
...
...
−χd
−χd
−χd
N
2
N
3
N
1
e
e
e
... −χ + γN
dN 1
dN 2
dN 3
where γj = 4παj−1 . This equation determines at most N negative eigenvalues. We are interested in the case where one of these eigenvalues
has maximal possible multiplicity N − 1. This happens if and only if
all rows in the matrix are parallel, i.e. the determinants of all 2 × 2
minors are zero. Let us reckon for which number of centers this is
possible. For large enough N (N ≥ 3) configuration of the centers
is determined by Ngeom = 3(N − 2) distances. In addition there are
Next = N parameters determining the extensions. Hence the matrix
contains Npar = Ngeom + Next + 1 = 4N − 5 parameters including in
addition the spectral parameter χ. The total number of 2 × 2 minors is
)2 , but the number of independent equations reduces to
equal to ( n(n−1)
2
n(n−1)
due to the symmetry and special form of the matrix. Hence in
2
general there are Ncon = n(n−1)
constraints on the parameters, which
2
guarantee the maximal degeneracy of the eigenvalue. Since the number
of constraints is growing quadratically, but the number of parameters
- just linearly, it is impossible to find a configuration leading to the
maximal degeneracy for sufficiently large N . In the table below we
14
YU.N.DEMKOV AND P.KURASOV
calculated the maximal expected dimension D of the set of parameters
which guarantee the maximal degeneracy.
N Ngeom Next Npar Ncon D
1
0
1
2
1
1
2
1
2
4
1
3
3
3
3
7
3
4
4
6
4
11
6
5
5
9
5
15
10
5
6
12
6
19
15
4
7
15
7
23
21
2
8
18
8
27
28
∅
We see that for low N our ”naiv” calculations give the correct result. Thus in the case of one point interaction there is a one-parameter
family of extensions having an eigenvalue. For two point interactions
the family of operators having an eigenvalue is described by three parameters: two extension parameters and one distance.
For large N (N ≥ 8) this table indicates that no eigenvalue of
maximal multiplicity is possible. In the current section we are going to
study the case of intermediate values of N.
Three centers. There are 9 minors. Since the matrix is symmetric, only 6 minors are independent. Due to the special structure of the
matrix, the number of equations reduces to 3
d23 χ(d23 −d13 −d12 )
−χ + γ1 =
e
d12 d13
d13 χ(d13 −d12 −d23 )
(37)
−χ + γ2 =
e
d12 d23
d12 χ(d12 −d13 −d23 )
−χ + γ3 =
e
d13 d23
Theorem 2. For any configuration {y 1 , y 2 , y 3 } of three points in
R3 and any negative number E = −χ2 there exists a unique set of parameters α1 , α2 , α3 such that the Schrödinger operator with three delta interactions of strengthes α1 , α2 , α3 concentrated at y 1 , y 2 , y 3 respectively
has a degenerate eigenvalue with the energy E = −χ2 .
Proof. Consider the system of equations (37) for arbitrary set of
positive parameters d12 , d13 , d23 , χ. These equations allow one to calculate directly 3 positive real numbers γj = 4π/αj and hence reconstruct
the parameters determining the delta interactions at the points y j . ¤
The theorem shows that for any fixed configuration of points supporting delta interactions the set of parameters determining such interactions leading to double eigenvalues can be parameterized by one
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
15
real parameter - the energy of the degenerate eigenvalue. Then the set
of parameters leading to a degenerate eigenvalue can be described by
4 parameters as it was predicted by the table.
Four centers. There are 36 minors, but due to the symmetry of
the matrix only 21 minors are independent. Special form of the matrix
reduces the number of independent equations to 6. It is natural to
divide these equations into two systems of four and two equations in
each system respectively

d23 χ(d23 −d13 −d12 )

 −χ + γ1 =
e



d13 d12


d34 χ(d34 −d24 −d23 )


 −χ + γ2 =
e
d24 d23
(38)
d14 χ(d14 −d34 −d13 )


e
−χ
+
γ
=

3

d34 d13



d12 χ(d12 −d14 −d24 )


e
 −χ + γ4 =
d14 d24
and
(39)
eχ(d12 +d34 ) d12 d34 = eχ(d13 +d24 ) d13 d24 = eχ(d14 +d23 ) d14 d23 .
The second system in the general case describes a certain relation between the distances and the energy parameter. It follows that not all
configurations of four centers lead to a triple eigenvalue. In general if
this configuration is admissible then it determines uniquely the possible energy of the triple bound state (except special cases described by
Theorem 3). Then the distances and the energy of the triple eigenvalue
can be used to determine the strengthes of the point interactions from
the first four equations (38).
For the system of four centers let us introduce some perimetric
coordinates - the sums of lengthes of opposite edges in the tetrahedron
determined by y 1 , y 2 , y 3 , y 4
(40)
D12 = d12 + d34 ,
D13 = d13 + d24 ,
D14 = d14 + d23 .
These coordinates can be used to calculate the perimeters of all three
possible four-angles in an easy way: the perimeters are equal to the
sums of the corresponding two perimetric coordinates.
Theorem 3. Consider the Schrödinger operator in L2 (R3 ) with
four point interactions of strengthes α1 , α2 , α3 , α4 situated at the points
y 1 , y 2 , y 3 , y 4 . This operator has a triple eigenvalue if and only if one of
the following conditions hold:
16
YU.N.DEMKOV AND P.KURASOV
a) If all three perimetric coordinates are different, then the distances
between the centers should satisfy one of the following three equivalent
conditions
(41)
ln d12 + ln d34 − ln d13 − ln d24
ln d12 + ln d34 − ln d14 − ln d23
=
< 0,
d12 + d34 − d13 − d24
d12 + d34 − d14 − d23
(42)
ln d13 + ln d24 − ln d14 − ln d23
ln d12 + ln d34 − ln d13 − ln d24
=
< 0,
d12 + d34 − d13 − d24
d13 + d24 − d14 − d23
(43)
ln d12 + ln d34 − ln d14 − ln d23
ln d13 + ln d24 − ln d14 − ln d23
=
< 0.
d12 + d34 − d14 − d23
d13 + d24 − d14 − d23
The energy of the triple eigenvalue is determined uniquely by the geometry of the centers
µ
(44)
E=−
ln d12 + ln d34 − ln d14 − ln d23
d12 + d34 − d14 − d23
¶2
.
The unique values of the constants αj are determined by (38) (taking
into account (19)).
b) If two of the perimetric coordinates coincide, say D12 − D13 = d12 +
d34 − d13 − d24 = 0, then the lengthes in these pairs should be equal, i.e.
½
(45)
d12 = d13
d34 = d24
½
or
d12 = d24
d34 = d13
Then the triple eigenvalue exists only if condition (41) holds, its value is
uniquely determined by the geometry and is given by (44). The unique
values of the constants αj are determined by (38) (using (19)).
c) If all three parametric coordinates are equal then the triple eigenvalue
exists if and only if the four centers are situated at the vertices of a
tetrahedron with at least one side given by a regular triangle and the
three other sides equal. The energy of the triple eigenvalue is arbitrary
negative and the values of the parameters αj (all equal) are uniquely
determined by this energy.
Proof. Let us consider all three cases separately.
a) If all three parametric coordinates are different then the parameter χ which determines the energy of the triple bound state can be
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
17
calculated from (39) using three different equalities as follows
ln d12 + ln d34 − ln d13 − ln d24
χ = −
,
d12 + d34 − d13 − d24
ln d12 + ln d34 − ln d14 − ln d23
,
χ = −
d12 + d34 − d14 − d23
ln d13 + ln d24 − ln d14 − ln d23
χ = −
.
d13 + d24 − d14 − d23
Excluding χ from the last equations in three different ways we get
(41,42,43) taking into account that χ must be positive. Then the parameters αj (or γj ) can be calculated from (38).
b) If any two of three perimetric coordinates coinside, say D12 =
D13 , then the system of equations (39) is equivalent to the following
two equations
ln d12 + ln d34 − ln d13 − ln d24
ln d12 + ln d34 − ln d14 − ln d23
−
=−
> 0;
d12 + d34 − d13 − d24
d12 + d34 − d14 − d23
d12 d34 = d13 d24 .
Taking into account that the two perimetric coordinates coincide we
conclude that the later equation implies (45) and that the energy of
the triple eigenvalue is given by (44). The extension parameters are
again determined by (38).
c) If all three perimetric coordinates coincide then (39) is equivalent
to
d12 d34 = d13 d24 = d14 d23 ,
and it follows that the four centers form a tetrahedron with at least
one side given by a regular triangle. The three remaining sides are
equal. In this case equation (39) is satisfied for arbitrary value of
the energy parameter E = −χ2 . After choosing this parameter equal
to arbitrary negative number one can calculate unique values of the
strength parameters from (38).
¤
The first family of point interactions is described by 5 independent
parameters as it was expected (see the table). The other two families
are described by 4 and 3 parameters respectively. The last family is the
most interesting, since it includes the regular tetrahedron - the most
symmetric configuration of four centers.
Five centers. We study the possibility that this system has an
eigenvalue of multiplicity 4. It is not obvious that the system of equations is solvable. The table predicts that the family of solutions is
described by 5 parameters. We are able to show that there exists a
two-parameter family. Consider the most symmetric configuration of 5
points:
18
YU.N.DEMKOV AND P.KURASOV
points y 2 , y 3 , y 4 , y 5 are situated at the corners of a regular tetrahedron,
point y 1 is situated in the center of the tetrahedron.
We denote by d the length of the tetrahedron’s edges and by r
the radius of the described sphere containing the four vertices of the
tetrahedron. Then the system (36) reads as follows


e−χr
e−χr
e−χr
e−χr
−χ + γ1
r
r
r
r
 e−χr

e−χd
e−χd
e−χd
−χ + γ2


r
d
d
d
−χd
−χd
−χd
 e−χr

e
e
e
(46) det 
−χ
+
γ
 = 0.
3
r
d
d
d
 e−χr

−χd
−χd
−χd
e
e
e


−χ + γ4
r
d
d
d
e−χr
e−χd
e−χd
e−χd
−χ + γ5
r
d
d
d
All rows of the matrix are linearly dependent (the eigenvalue has multiplicity 4) if and only if
−χ + γ2 = −χ + γ3 = −χ + γ4 = −χ + γ5 =
e−χd
d
⇒ γ2 = γ3 = γ4 = γ5 = χ + e−χd /d
and
µ
−χ + γ1 =
e−χr
r
¶2
/
e−χd
d
⇒ γ1 = χ + 2 eχ(d−2r) .
d
r
Hence for any χ one can find parameters αj , j = 1, ..., 5 such that
the Schrödinger operator with 5 point interactions has an eigenvalue of
multiplicity 4. The last family appears to be the most interesting, since
it conyains the regular tetrahedron - the most symmetric configuration
of four centers.
Six and more centers. Consider the system of six centers. The
table predicts that the set of parameters leading to the maximal degeneracy in this case is described by 4 parameters. Let us examine the
most symmetric configuration of six centers - the points situated at the
summits of the octahedron. The matrix takes the following form


√
2d
e−χd
e−χ
e−χd
e−χd
e−χd
√
−χ + γ1
d
d
d
d
d
2d
√
 e−χd

2d
e−χd
e−χ
e−χd
e−χd


√
−χ
+
γ
d
2


d√
d
d
d
2d
 e−χ 2d

−χd
−χd
−χd
−χd
e
e
e
e
√


−χ + γ3
d
d
d
d
2d

,
√
2d
−χ
−χd
−χd
−χd
 e−χd

e√
e
e
e
−χ
+
γ


4
d
d
d
d√
2d


−χd
−χd
−χd
−χd
−χ
2d
e
e
e
e√
 e

−χ
+
γ
5
d
d
d
d


2d
e−χd
d
e−χd
d
e−χd
d
e−χd
d
√
2d
e−χ
√
2d
−χ + γ6
VON NEUMANN-WIGNER THEOREM:LEVEL’S REPULSION AND DEGENERATE EIGENVALUES.
19
where d is the length of the octahedrons edge. The rows are parallel
only if
√
e−χd
e−χ 2d
ln 2
= √
,⇒ χ = − √
< 0.
d
2d
2d( 2 − 1)
But the parameter χ has to be positive. It follows that this system
cannot have an eigenvalue of multiplicity 5. The last equation determines a resonance instead of the eigenvalue. We conjecture that the
system of 6 point interactions cannot have an eigenvalue of multiplicity 5. Similarly we do not expect eigenvalues of the multiplicity N − 1
for any system N point interaction, provided N > 6. We are going to
return back to this problem in one of our forthcoming publications.
Conclusions The authors are gratefull to The Swedish Royal Academy of Sciences and Foundation for Basic Research (Vetenskapsrådet)
for financial support.
References
[1] M.N. Adamov, Yu.N. Demkov, V.D. Ob”edkov, and T.K. Rebane, Model of a
small-radius potential for molecular systems, Teor. Eksp. Khimiya, 4 (1968),
147–153.
[2] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable models in
quantum mechanics. Second edition. With an appendix by Pavel Exner. AMS
Chelsea Publishing, Providence, RI, 2005.
[3] S. Albeverio and P. Kurasov, Singular perturbations of differential operators.
Solvable Schrödinger type operators, London Mathematical Society Lecture
Note Series, 271. Cambridge University Press, Cambridge, 2000.
[4] Yu.N. Demkov, G.F. Drukarev, and V.V. Kuchinskii, Dissociation of negative
ions in short range potential approximation, J. Exp. Theor. Phys., 58 (1970),
944–951.
[5] Yu. Demkov and V. Ostrovsky, Zero-range potentials and their applications
in atomic physics, Plenum, New York, 1988 (Translation from the 1975
Russian original: Demkov, .N., Ostrovski, V.N., Metod potencialov
nulevogo radiusa v atomno fizike, Izd–vo Leningradskogo Universiteta, Leningrad, 1975 )
[6] M. Krein, On Hermitian operators whose deficiency indices are 1, Comptes
Rendue (Doklady) Acad. Sci. URSS (N.S.), 43, 131-134, 1944.
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II, Comptes Rendue (Doklady) Acad. Sci. URSS (N.S.), 44, 131-134, 1944.
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conditions for wave equations with point interactions. Proc. Amer. Math. Soc.
133 (2005), 3071–3078.
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[10] J. von Neumann and E. Wigner, Über das Verhalten von Eigenwerten bei
adiabatischen Prozessen, Phys. Zeit., 30 (1929), 467-470.
20
YU.N.DEMKOV AND P.KURASOV
Particulars about the authors
Pavel Kurasov
(corresponding author)
Physics Research Institute, St.Petersburg University
Russia, 198904 St.Petersburg, St. Peterhof Ul’yanovskaya 1, St.Petersburg
University, Physics Research Institute, Laboratory of Quantum Networks
and
Dept. of Mathematics, Lund University, 221 00 Lund, Sweden
e-mail: [email protected]
Contact phone number: (812) 2340791
Yurii N. Demkov
Physics Research Institute, St.Petersburg University
Russia, 198904 St.Petersburg, St. Peterhof Ul’yanovskaya 1, St.Petersburg
University, Physics Research Institute, Dept. of Quantum Mechanics
Contact phone number: (812) 2320267
Title: von Neumann-Wigner theorem: level’s repulsion and degenerate eigenvalues.
Authors: Yu.N.Demkov and P.Kurasov
Abstract Spectral properties of Schrödinger operators with point
interactions are investigated. Attention is focused on the interplay between the level’s repulsion (von Neumann-Wigner theorem) and the
symmetry of the points configuration. Explicit solvability of the problem allows to observe level’s repulsion for two centers. For larger number of centers the families of point interactions leading to the highest
possible degeneracy is investigated.
UDC 517.984, 530.2.
Keywords: Wigner-von Neumann theorem, zero-range potentials,
extension theory, inverse spectral problem.