CC BY
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NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2018, 9 (2), P. 212-214
N wells at a circle. Splitting of lower eigenvalues
T. F. Pankratova
ITMO University, 49 Kronverkskiy, St. Petersburg, 197101, Russia [email protected]
PACS 32.30-r; 03.65-w; 73.21.Fg; 78.67.De; 31.15-xr; 05.45.xt DOI 10.17586/2220-8054-2018-9-2-212-214
A stationary Schrodinger operator on R2 with a potential V having N nondegenerate minima which divide a circle of radius ro into N equal parts is considered. Some sufficient asymptotic formulae for lower energy levels are obtained in a simple example. The ideology of our research is based on an abstract theorem connecting modes and quasi-modes of some self-adjoint operator A and some more detailed investigation of low energy levels in one well (in Rd).
Keywords: Shrodinger operator, potential, splitting, eigenvalues and eigenfunctions.
Received: 19 December 2017 Revised: 22 December 2017
1. Introduction. Modes and quasi-modes
We use terms modes and quasi-modes following V.I. Arnold [1]. An eigenvalue and eigenvector of some operator A, i.e. a pair (A, u) which satisfies equation Au = Au exactly, is called a mode. Some value and vector which satisfy this equation approximately with some error of order e is called a quasi-mode. More precisely, the result is as follows:
Let A be a self-adjoint operator in a Hilbert space H, A0 - a real value, orthonormal vectors ui,u2, ...,uN g D(A), Q is a positive constant, e = max ||(A - A0) uj||, 0 < 4%/3Ne < Q, A1,..., AN are the eigenvalues of the
1<i<N
matrix M with the inputs {Mik} = {(Aui,uk}} ((-, •} means a scalar product in H), every eigenvalue is counted according to its multiplicity.
Theorem 1. Suppose the interval I = [A0 — Q, A0 + Q] contains at most N eigenvalues of A. Then, the interval I1 = A0 — Q + 4%/3Ne, A0 + Q — 4%/3Ne contains exactly N eigenvalues of A. There exist constants p and q such that if 0 < e < p then, any interval Sj = [Aj — qe2, Aj + qe2] is included in I1 and contains an eigenvalue
N
of A. Any connected component of the set Sj contains exactly as many eigenvalues of A as there are intervals
j=1
Sj forming it.
Theorem 1 allows us to describe eigenvectors and eigenvalues of A based on the knowledge only of its quasi-modes. If Sj does not intersect with Sj+1, the distance between their middle points gives us a good approximation of the distance between the two nearest eigenvalues. The first proposition of Theorem 1 guaranties the absence of additional eigenvalue of A in our interval.
2. A self-adjoint Schrodinger operator on Rd
Let us consider the Schrodinger equation:
h2
—— Au + Vu = Eu, (1)
d d 2
where A = ^ is the Laplace operator, V is a real valued function defined on Rd having nondegenerate
i=1 i
minima (wells) with some kind of symmetry, h (small parameter) is the Planck constant (in special system of units). Let A be the corresponding Schrodinger operator defined by the left hand side of equation (1) in L2(Rd).
If V in (1) has a finite number of identical wells which differ only by space translations and V (x) > C beyond the region of the wells where C exceeds the value of V at minimum, lower part of the spectrum of operator A is organized in the following way. There is a set of finite groups of eigenvalues (each of them is related
N wells at a circle. Splitting of lower eigenvalues
213
to some quantum vector n G the distance between the groups being of the order h, and the distance between eigenvalues in each group, the splitting, being exponentially small with respect to h.
It is possible to find explicit formulae for the widths of these splittings using semi-classical asymptotics for each well. The problem was considered in different ways by different authors and almost completely solved in one dimensional case [1-8]. The case d > 1 is much more complicated. There are many results obtained in this area (see [9-17] and the list is far from exhaustive). The semiclassical asymptotics of the discrete spectrum and strict estimates of the splittings are described in [9] and other works of these authors (using the theory of pseudo differential operators). The semiclassical expansion for the eigenfunctions and the rigorous asymptotics for the splitting widths in the lowest levels were obtained in [10] (with the use of Maslov's canonical operator). The possibility to solve this problem in that case was discussed during the Diffraction Day Conference 2014 in the talk of A. Anikin and M. Rouleux [12].
In the present work, in order to write down strict asymptotic formulae for splittings in two-dimensional case, one has to use Theorem 1. It is necessary to find a sufficiently accurate semiclassical approximation to eigenstates for a single well in some vicinity of a minimum, independent of h. Such an approximation was constructed in [11,13]. The formal series on powers of h were obtained. Coefficients in all terms were found in some domain independent of h. Terms for eigenfunctions are analytic for analytic potential. If we truncate the series at the m-th term the remaining sums satisfy the equation (1) with an error of the order of hm+1 exp(-S/h), where S is a nonnegative function defined in [11]. The possibility to take m as large as we like and exponential decreasing of all terms beyond some vicinity of a minimum allows one to construct sufficient quasi-modes. Each quasi-mode has to be constructed from semiclassical approximations of lower eigenfunctions in the region of the bottom of each well vanishing beyond it.
In this work, a simple example is considered. Here, the circle containing N minima of V is f line of minimum of the corresponding functional b and it is easy to find b in a plain form.
3. An example. N wells at a circle
Let d = 2. Let V in equation (1) in polar coordinates be of the following form:
V = "2"(r - ro) +ysm ~y ,
(2)
w1, w2 are some positive Diophantine numbers. (This means that there exist positive numbers a and g such that
for any k G Z2, k = 0, |(k, w)| > jf-).
|k|a
( 2nj \
It is easy to see that the points Mj ( r0; j, j = 0,1,..., N — 1, are nondegenerate minima of V, Mj G r, r is a circle r = r0 and
V (r, = V r; ^ +
2j N J.
(3)
We put a Cartesian system of coordinates (xj; yj) in the vicinity of the bottom of each well in such a way that Mj = Mj (0; 0) in this coordinates, axis xj is tangential to a circle r at the point Mj and yj is normal to it. One can find the following Taylor series for V:
v (xj) = ^ 0%2 + ¿2y2) + £ vfcx|,
fc|> 3
Xj = (xj; yj), k =(ki; ki), Xj1 = xk1 yk2, |k| = ki + ki, 7i > 0, i = 1,2,
in a vicinity of Mj. The form of this series does not depend on j because of equality (3).
In order to use Theorem 1, let us find semiclassical approximations for some first quantum vectors
n = (n1,n2), n1 = 0,1,...; n2 = 0,1,...; in each domain Dj = {|xj| < 7, |yj| < 7}. They are the same for all Dj, j = 0,1, ...N — 1. Let us take numbers 7 and 7 such that two neighboring domains Dj and Dj+1 intersect.
n (2j + 1)'
j,j+1,
Let domain Gj,j+1 = Dj P| Dj+1 be such an intersection. Let the point Mj = Mj ^r0; n (2j+ 1) ^ j = 0,1,..., N — 1. Then, we multiply Un by cutting functions x[j] = X[j] (xj, yj) = x1j] (xj) x2j] (yj), where
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T. F. Pankratova
(xj) and X2] (Vj) are smooth cutting functions, i.e.
(x 0 — J 1 |xj1 - Y' xj (v 0 — J 1 |yj1 - 7' 7 x[j] = U[j]
X1 (xj) — 1 n U.l^ „, ^ x2 (yj) - 1 n l„.l ^ A _L „„ — un .
0' |xj |> Y + £1, A2Vj 1 0' |vj |> 7 + £2' nA n Function ujj'] is equal to zero beyond rectangular {|xj| > y + £1' |VjI > 7 + £2}- We construct N quasi-modes
N
Un,k, k — 1'...' N, as a linear combination of cut-off functions un, i.e. U„ik — ^ ajikujj], k — 1'...' N. We find
j=i
numbers ojik in order to orthonormalize the system {U„ik}fc=1. Now, we can use Theorem 1 in a way similar to one presented in [8].
We find that for our example with N wells (eq. (2)) for each quantum vector N eigenvalues Ek, k — 1'...' N, of operator A has the following form:
En — En + mL"]+ O (£2) '
where: _
11
En — ^ En,jhj; En,i — ^ni + 0 wi + ^«2 + ^ ' ()
— a • exp (-h-1b) • cos^^-' k — 1'...'N' — 0 (£) ' b — V2VdS'
Mfc_iMfc
Mk_iMk is a line of minimum of functional b. In our case it is a part of the circle r. At this circle,
/— N^ 4
V2V = w2 sin ———, dS = r0d^. Hence, b = — r0w2.
Now, we can write down the splitting formula for lower eigenvalues of operator A:
AEn = Ek+1 - En = dk exp (-b) (1 + O (h)), k = 1,..., N. One can regard this example as a simple model for some possibly more complicated situation. References
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