NOTE ON 2D SCHRODINGER OPERATORS WITH ?-INTERACTIONS ON ANGLES AND CROSSING LINES

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NOTE ON 2D SCHRODINGER OPERATORS WITH 5-INTERACTIONS ON ANGLES AND CROSSING LINES
V. Lotoreichik
Technische Universitat Graz, Institut fur Numerische Mathematik Steyrergasse 30, 8010 Graz, Austria
lotoreichik@math. tugraz. at
PACS 03. 65. -w
In this note we sharpen the lower bound previously obtained by Lobanov et al [LLP10] for the spectrum of the 2D Schrodinger operator with a ?-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D Schrodinger operator with a ?-interaction supported on crossing straight lines. The latter operators arise in the three-body quantum problem with ?-interactions between particles.
Keywords: Schrodinger operators, ?-interactions, spectral estimates.
1. Introduction
Self-adjoint Schrodinger operators with-interactions supported on sufficiently regular hypersurfaces can be defined via closed, densely defined, symmetric and lower-semibounded quadratic forms using the first representation theorem, see [BEKS94], [BLL13]
-interactions on angles. In our first model the support of the-interaction is the set Ev C R2, which consists of two rays meeting at the common origin and constituting the angle f e as in Figure 1.
Fig. 1. The angle Ev of degree f e (0,^j
The quadratic form in L2(R2)
M/] := \Vf hL2(R2-c2) — a\fdoma^ := H 1(R2), (1)
is closed, densely defined, symmetric and lower-semibounded, where /is the trace of / on E^, and the constant a & gt- 0 is called the strength of interaction. The corresponding self-adjoint operator in L2(R2) we denote by Av. Known spectral properties of this operator include explicit representation of the essential spectrum aess (A^) = [-a2/4, and some
information on the discrete spectrum:ad (Av) ^ 1 if and only if & lt- = n. These two statements can be deduced from more general results by Exner and Ichinose [EI01]. They are complemented by Exner and Nemcova [EN03] with the limiting property § od (Av) ^ as & lt- ^ 0+.
In [LLP10] the author obtained jointly with Igor Lobanov and Igor Yu. Popov a general result, which implies the lower bound on the spectrum of Av
a2
inf a (Av) ^ - 4. • (2)
4sin2(& lt-/2)
This bound is close to optimal for & lt- close to n, whereas in the limit & lt-p ^ 0+ the bound tends to -to. In the present note we sharpen this bound. Namely, we obtain
a2
inf a (Av) ^ -n.f /02 • (3)
(1 + sin (& lt-/2))2
The new bound yields that the operators Av are uniformly lower-semibounded with respect to & lt- and
inf a (Av) ^ -a2
holds for all & lt- e (0,n]. This observation agrees well with physical expectations. Note that separation of variables yields that inf o (An) = -a2/4 and in this case the lower bound in (3) coincides with the exact spectral bottom.
For sufficiently sharp angles upper bounds on inf a (A^) were obtained by Brown, Eastham and Wood [BEW08]. See also Open Problem 7.3 [E08] related to the discrete spectrum of Av for & lt- close to n.
-interactions on crossing straight lines. We also consider an analogous model with the-interaction supported on the set = r U r2, where r and r2 are two straight lines, which cross at the angle & lt- e (0,n) as in Figure 2.
Fig. 2. The straight lines r and r2 crossing at the angle of degree p g (0,^)
The corresponding self-adjoint operator Bv in L2(R2) can be defined via the closed, densely defined, symmetric and lower-semibounded quadratic form in L2(R2)
M/] := \Vf \h (R2-C2) — a\f Ir, |||2(r^ dom b^ := H ^R2), (4)
where a & gt- 0 is the strength of interaction. According to [EN03] it is known that
ess (B^) = [-a2/4, +?) and that $ad (Bv) ^ 1.
In this note we obtain the lower bound
a2
infa (Bv) & gt- ¦, (5)
1 + sin f
using the same method as for the operator A^. Separation of variables yields inf a (Bn/2) = -a2/2, and in this case the lower bound in the estimate (5) coincides with the exact spectral bottom.
Upper bounds on inf a (B^) were obtained in [BEW08, BEW09]. The operators of the type Bv arise in the one-dimensional quantum three-body problem after excluding the center of mass, see Cornean, Duclos and Ricaud [CDR06,CDR08] and the references therein.
We want to stress that our proofs are of elementary nature and we do not use any reduction to integral operators acting on interaction supports Ev and r^.
2. Sobolev spaces on wedges
In this section Q c R2 is a wedge with an angle of f e (0,2n). The Sobolev space H 1(Q) is defined as usual, see [McL, Chapter 3]. For any / e H 1(Q) the trace /|an e L2(dQ) is well-defined as in [McL, Chapter 3] and [M87].
Proposition 2.1. [LP08, Lemma 2. 6] Let Q be a wedge with angle of degree f e (0,n]. Then for any / e H 1(Q) the estimate
Y 2
\W\l2(Q-C2) — Y\/^nWL^Q) ^ - sin2(f/2) \/L2(Q)
holds for all y & gt- 0.
Proposition 2.2. [LP08, Lemma 2. 8] Let Q be a wedge with angle of degree f e (n, 2n). Then for any / e H 1(Q) the estimate
\V/\L2(Q-C2) — Y\/|дпWl2(дп) ^ -^Y2/\l2(n)
holds for all y & gt- 0.
Propositions 2.1 and 2.2 are variational equivalents of spectral results from [LP08].
3. A lower bound on the spectrum of Av
In the next theorem we sharpen the bound (2) using only properties of the Sobolev space H1 on wedges and some optimization.
Theorem 3.1. Let the self-adjoint operator Av be associated with the quadratic form given in (1). Then the estimate
a2
inf a (Ap) ^ - --
& quot- (l + sin (f/2))2
holds.
Proof. The angle separates the Euclidean space R2 into two wedges Q1 and Q2 with
angles of degrees f and 2n — f, see Figure 3.
The underlying Hilbert space can be decomposed as
L2(R2) = L2(Q1) © L2(Q2).
Fig. 3. The angle separates the Euclidean space R2 into two wedges Q and Q2
Any f G dom can be written as the orthogonal sum f1 © f2 with respect to that decomposition of L2(R2). Note that f1 g H 1(^1) and that f2 g H 1(^2). Clearly,
i2(R2) = \fl\l2(ni) + \/2l2(Q2), (6)
Il^f \ L2(R2-C2) = \V/l\ L2(Qi-C2) + \V/2 L2 (H2-C2) —
The coupling constant can be decomposed as a = 3 + (a — 3) with some optimization parameter / g [0,a] and the relation
a\f ^\L2(S,) = 3\f1 |dQi WL2 (dQi) + (a — 3)/2|dQ2 lL2(dn2) — (7)
holds. According to Proposition 2. 1
2
ii)
and according to Proposition 2. 2
\^f2 L2(I2-C2) — (a — 3)\f2|fln2L2(dQ2) ^ -(a — 3)2W/2WL2(П2)• (9)
The observations (6), (7) and the estimates (8), (9) imply
Mf ] ^ - max{-J& amp-/2), (a — 3)2
l|Vf1|| L2(Qi-€ 2) _? llf1| 9Qi ! L2(dQi) ^ & quot- sin2(^/2) 11 A 11 L2(Qi) '- (8)
2 2)
Making optimization with respect to 3, we observe that the maximum between the two values in the estimate above is minimal, when these two values coincide. That is
smfe = (a — 3)2,
which is equivalent to
o = a sin (^/2) (10)
3 = (1+sin (^/2)) '- (10)
resulting in the final estimate
aV [/] ^ - (1+sina (^/2))2 W/ llL2(R2) —
This final estimate implies the desired spectral bound. ?
Remark 3.2. Note that the previously known lower bound (2) was derived from the proof of the last theorem if we choose /3 = a/2, which is the optimal choice in our proof only for & lt- = n as we see from (10).
4. A lower bound on the spectrum of B^
In the next theorem we obtain a lower bound on the spectrum of the self-adjoint operator Bv using the same idea as in Theorem 3.1.
Theorem 4.1. Let the self-adjoint operator Bv be associated with the quadratic form given in (4). Then the estimate
inf a (Bv) ^ -
a
1 + sin p
holds.
Proof. The crossing straight lines r and r2 separate the Euclidean space R2 into four wedges {Qk}k=1. Namely, the wedges Q and Q2 with angles of degree p and the wedges Q3 and Q4 with angles of degree n — p, see Figure 4.
Fig. 4. The crossing straight lines r1 and r2 separate the Euclidean space R2 into four wedges {Qk}?=1
The underlying Hilbert space can be decomposed as
4
L2(R2) = 0 Lnk).
k=i
Any f G dom can be written as the orthogonal sum ®=1fk with respect to that decomposition of L2(R2). Note that fk G Hfor k = 1, 2,3,4. Clearly,
4
If IL2(R2) =? fk\h (ak), Wf ||L2(r2-C2) =? \Vfk\h (ak -C2). (11)
k=1 k=1
The coupling constant can be decomposed as a = / + (a — /) with some optimization parameter / e [0, a] and the relation
a\f ^\2L2(rv) = 3 ll/l|oQi\2L2(dQ1) + 3\hdii2\2L2{dQ2)
+ (a — 3)\/s| dQs L2(dns) + (a — 3)\/4U4\ L2(dQ4)
(12)
holds. According to Proposition 2. 1
\V/l\L2(Qi-C2) — 3/l|dniL2(dQi) ^ - sivL2((p/2)/lL2(Qi)'- (13)
L2 (H2-C2)
3 /2| 9^2L2(dQ2) ^ sin2(^/2) WhWL2^) —
Also according to Proposition 2. 1
||Vf3|L2(n3-C2) & quot- (a — ?)|f3|dQ3 ||L2(dQ3) L2(Q4-C2) — (a — ?)|f4|dQ4 HL2(dfi4)
^ -& gt- -
(a-?)2 cos2(^/2)
(a-?)2
cos2(^/2)
^4|L2(Q4) —
(14)
The observations (11), (12) and the estimates (13), (14) imply
Mf ] ^ maX j sin2/(^/2) '- cO2^) }
2
L2(R2) •
Making optimization with respect to 3, we observe that the maximum between the two values in the estimate above is minimal, when these two values coincide. That is
?2
(a -?)2
which is equivalent to resulting in the final estimate
sin2(^/2) cos2(f/2) '-
n _ atan (^/2)
? = (1+tan (^/2)) '-
Mf] ^ - 1+sin (^) ^ ^(R2) •
(15)
This final estimate implies the desired spectral bound.
?
Remark 4.2. The result of Theorem 4.1 complements [CDR08, Theorem 4. 6(iv)], where the bound
inf j (Bv) ^ -a2
for all p g (0,n) was obtained.
Acknowledgements
The author is grateful to Jussi Behrndt, Sylwia Kondej, Igor Lobanov, Igor Yu. Popov, and Jonathan Rohleder for discussions. The work was supported by Austrian Science Fund (FWF): project P 25 162-N26 and partially supported by the Ministry of Education and Science of Russian Federation: project 14. B37. 21. 0457.
References
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