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proceedings of the american mathematical society Volume 102, Number 2, February 1988 MULTIPLICITIES OF THE EIGENVALUES O
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proceedings of the american mathematical Volume
102, Number
society
2, February
1988
MULTIPLICITIES OF THE EIGENVALUES OF THE DISCRETE SCHRÖDINGER EQUATION IN ANY DIMENSION DAN BURGHELEA AND THOMAS KAPPELER (Communicated
by Walter Littman)
ABSTRACT. The following von Neumann-Wigner type result is proved: The set of potentials a: T —► R (r Ç ZN), with the property that the corresponding discrete Schrödinger equation A¿ + a has multiple eigenvalues when considered with certain boundary conditions, is an algebraic set of codimension> 2 within
Rr.
1. Introduction. The classical theorem of von Neumann and Wigner [10] shows that, within the space S of real symmetric nxn matrices (n > 2), the ones with multiple eigenvalue(s) form a real algebraic set of codimension 2. This implies, in particular, that the set of all real symmetric matrices with simple spectrum is pathwise connected, locally pathwise connected, and dense in S. Recently, Lax [7] shows that in a three-dimensional vector space of (n x n) symmetric matrices there exists at least a one-dimensional subspace of matrices with multiple eigenvalues ( "crossing" of eigenvalues). These results were refined and generalized by Friedland,
Robbin, and Sylvester [2]. In this paper the subset of all real symmetric matrices is considered which come from the discrete Schrödinger equation A 2. To be more precise, let N and nx,...,n^ be arbitrary positive numbers which define a subset T in ZN in the following way:
r :={z = (zx,...,zN):l
2, where Ker S (a) denotes the kernel of S (a). Now dim Ker S (a) > 2 iff all the (n2 -1 ) x (n2 -1 ) submatrices of S (a) are 2 singular. It follows that T is an algebraic set in R" and can thus be decomposed in its irreducible components X¿, T = U¿=i lilt suffices to show that codimaT¿ > 3 for 1 < i < M and any regular point a in Ti (cf. e.g. [8, p. 41]). To simplify notation let Tj be Tx. Choose an arbitrary regular point a0 in Tx. The idea of the proof is to express, in a neighborhood of o° in Tx, two coefficients out of a¿¿ as functions of the others, and then to show that there is a third equation among the remaining coefficients which holds on this neighborhood and which does not hold identically on Rn ~2. Let us denote by [a, i] the number (a - l)n -Yi (1 < a, i < n). For an arbitrary n2 x n2 matrix M we denote by M((a,i), (ß,j)) the (n2 — 1) x (n2 - 1) submatrix of M by eliminating the [a, i]th row and the [/?, j]th column. Moreover, we define b[a,i} ~a(a,i). Step 1. Let us define F0(a) := det S(a) and Fk(a) := dFk_x(a)/dbk (1 < k < n2) as well as Gk(a) := Fk-X(a) —bkFk(a) (1 < k < n2). Clearly Fk(a) and Gk(a) are independent of bx,..., bk. In particular, we have
Fk_x(a) = bkFk(a) + Gk(a)
and det S(a) = bxFx(a) -YGx(a).
Fx(a) vanishes identically on T. Now let us assume that F2(a),... ,Fk(a) are all vanishing identically in a certain neighborhood of a0 in Tx. Due to the fact that a0 is regular there are two possibilities: (1) there is a neighborhood of o° in Tx such that F2(a),... ,Fk(a) and Fk+X(a) do vanish identically; or (2) there exists a neighborhood of o° in Tx such that Fk+X vanishes at most on a real analytic variety of codim > codim Tx -Y 1 contained in this neighborhood. If the second possibility holds then one can solve the equation 0 = bk+xFk+x(a)-Y Gk+X(a) for bk+x in a neighborhood of A0 in Tx except on a set of points of lower dimension. Now Fn2 (a) — det
(1 0
Vo.
0
■•■ 0\ 1 0
o i)
and thus we conclude that there exists a smallest k, 1 < k < n2, such that bk can be expressed as a real analytic function of bk+x,..., bni in a neighborhood V of a0 in Tx except on a real analytic variety of lower dimension, contained in V. Step 2. Let us assume that in Step 1 we could express a(a,i) as a real analytic function of the remaining coefficients. Then consider 0 = detS((a, i), (a,i)).
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EIGENVALUES OF THE DISCRETE SCHRÖDINGER
EQUATION
259
Applying the same procedure for S((a, i), (a,i)) as was applied for S(a) in Step 1, we conclude that there exists a neighborhood V of a0 in Tx and (ß, j) such that a(ß,j) can be expressed as a function of a(i,k) ^ (a,i) and (7, A;)^ (ß,j) except on a set of lower dimension in V. Step 3. Now let us assume that there exists a neighborhood V of a0 in Tx and (a,i),(ß,j) ((a,i) t¿ (ß,j)) such that a(a,i) can be expressed as a real analytic function of the remaining coefficients of a, and a(ß,j) can be expressed as a real analytic function of the other coefficients different from a(a,i), except on a set of points of lower dimension in V. Then let us consider the equation
0 = detS((a,k),(ß,j)) which holds on T. This is a polynomial in 0(7, k) with (7, k) ^ (a,i) and ^ (ß,j). It thus suffices to show that det S((a, i), (ß,j))
This will be done with the following Lemma. LEMMA. detS((a,i),(ß,j))
is not identically zero on R™ ~2.
D
is not identically zero on R" ~2.
PROOF. Clearly det S((a, i), (ß, j)) is a polynomial in 0(7, k). We have to show that degdetS((cM),(/?,y))>l. Without loss of any generality we may and do assume that a < ß and i < j. It then follows that detS((a,i), (ß,j)) contains the following monomial:
I
\ I
n
\l