Periodic Schrödinger operators with δ -potentials

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1 Networ meeting April 23-28, TU Graz Periodic Schrödinger operators with δ -potentials Andrii Khrabustovsyi Institute for Analysis, Karlsruhe Institute of Technology, Germany CRC 1173 Wave phenomena: analysis and numerics, Karlsruhe Institute of Technology, Germany joint wor with Pavel Exner 1/22

2 Preliminaries Periodic operators and their spectrum It is nown that the spectrum of self-adjoint periodic differential operators has a band structure, i.e. the spectrum is a locally finite union of compact intervals called bands. The open interval (α, β) is called a gap if (α, β) σ(h) = and α, β σ(h). In general the presence of gaps in the spectrum is not guaranteed! Example: σ( R n) = [0, ). Problem 1 For a given class L of periodic differential operators to construct the operator H L with at least one gap in the spectrum 2/22

3 Preliminaries References Scalar elliptic operators of the form b (a ) with periodic a and b - A. Figotin, P. Kuchment, SIAM J. Appl. Math. 56 (1996). - R. Hempel, K. Lienau, Commun. PDEs 25 (2000). - V. Zhiov, St. Petersb. Math. J. 16 (2005). - V. Hoang, M. Plum, C. Wieners, ZAMP 60 (2009). Laplace-Beltrami operators on periodic Riemannian manifolds - E. B. Davies, E. M. Harrell, J. Differ. Equ. 66 (1987). - O. Post, J. Differ. Equ. 187 (2003). - P. Exner, O. Post, J. Geom. Phys. 54 (2005). Laplacians posed in noncompact periodic domains - S. Nazarov, K. Ruotsalainen, J. Tasinen, J. Math. Sci. 181(2012). Periodic operators posed in domains with waveguide geometry - K. Yoshitomi, J. Differ. Equ. 142 (1998). - G. Cardone, S. Nazarov, C. Perugia, Math. Nachr. 283 (2010). - D. Borisov, K. Panrashin, J. Phys. A 46 (2013). Maxwell operators - A. Figotin, P. Kuchment, SIAM J. Appl. Math. 56 (1996) - N. Filonov, Commun. Math. Phys. 240 (2003). 3/22

4 Preliminaries Control of spectral gaps Problem 2 To construct the operator H L having gaps which are close (in some natural sense) to preassigned intervals Laplace-Beltrami operators on periodic Riemannian manifolds [A. K., J. Differ. Equations 252(3) (2012)] Scalar elliptic operators in divergence form [A. K., Asympt. Analysis 82(1-2) (2013)] Laplacians posed in noncompact periodic domains [A. K., E. Khruslov, Math. Meth. Appl. Sci. 38(1) (2015)], [A. K., J. Math. Phys. 55(12) (2014)] Periodic quantum graphs [D. Barseghyan, A. K., J. Phys. A 48(25) (2015)] 4/22

5 Preliminaries Control of spectral gaps: example D ε S ε ε m N is a given number the sets S ε (j {1,..., m} is fixed, i Z n ) are distributed ε-periodically in R n (ε > 0) each set S ε has the form ε(s j + i) \ D ε, where S j are fixed surfaces without a boundary, D ε are small holes the radius of D ε is equal to d j ε n n 2 (n 3) or e 1/d j ε 2 (n = 2) ( m Ω ε = R n \ i Z n j=1 ) S ε We denote by H ε = Ω ε the Neumann Laplacian in Ω ε. One has [A.K., 2014]: the operator H ε has at least m gaps as ε is small enough, the first m gaps converge as ε 0 to certain intervals (a j, b j ), whose closures are pairwise disjoint; the next gaps (if any) go to infinity, one can completely control the location of the intervals (a j, b j ) via a suitable choice of the numbers d j and the surfaces S j. 5/22

6 Preliminaries Our goal The goal To study this problem for periodic Schrödinger operators with singular potentials 6/22

$Preliminaries Example: gaps in the spectrum of Schrödinger operator G B (0, 1) n an open domain G := i Z n (B + i) Ω := R n \ G H ε := R n + ε 1 1 Ω, ε > 0.$

7 Preliminaries Example: gaps in the spectrum of Schrödinger operator G B (0, 1) n an open domain G := i Z n (B + i) Ω := R n \ G H ε := R n + ε 1 1 Ω, ε > 0. H is the Dirichlet Laplacian in G One can prove 1 that H ε norm resolvent converges to H. Since σ(h) = {λ }, 0 < λ 1 λ 2 λ, the spectrum =1 of H ε has at least m gaps provided ε is small enough. 1 R. Hempel, I. Herbst, Commun. Math. Phys. 169 (1995), /22

8 The operator H ε Notations: m N ε > 0 - a small parameter Y := (0, 1) n B j, j = 1,..., m be Lipschitz domains satisfying B 0 := Y \ m B j j=1 B j1 B j2 =, m B j Y j=1 S j := B j, j = 1,..., m B ε := ε(b j + i), i Z n, j = 1,..., m S ε := B ε the surfaces supporting our potential Ω ε := R n \ B ε 8/22

9 The operator H ε Let us define accurately the Schrödinger operator H ε = + V ε with a singular potential defined by the following formal expression: m V ε = q j ε 1 δ S ε, δ S ε, q j are positive constants. i Z n j=1 In what follows by (f) + (respectively, (f) + ) we denote the trace of the function f on S ε, when we approach this surface from outside (respectively, inside). In the space L 2 (R n ) we define the sesquilinear form h ε by m ( ) ( h ε [u, v] = u vdx+ q 1 j ε (u) + (u) (v) + R n i Z n j=1 S ε (u) ) ds with dom(h ε ) = H1 (R n ) := H 1 (Ω ε ) H 1 (B ε i,j ). 9/22

10 The operator H ε The form h ε is symmetric, densely defined, closed and positive. By H ε we denote the operator associated with the form h ε, i.e. (H ε u, v) L 2 (R n ) = h ε [u, v], u dom(h ε ), v dom(h ε ). The functions u from dom(h ε ) satisfy (see, e.g., 2 ): ( ) ± u u H1 (R n ), u L 2 (R n ), L 2 (S ε n ), H ε u = u, ( ) + ( u u = n n ) =: ( ) ( ) u u, q j ε 1 + ( ) (u) (u) + n n = 0. 2 J. Behrndt, P. Exner, V. Lotoreichi, Rev. Math. Phys. 26 (2014), /22

11 Notations For j = 1,..., m we set: a j := q 1 j S j B j 1. It is assumed that the numbers a j are pairwise non-equivalent. We renumber them in the ascending order: a j < a j+1, j = 1,..., m + 1. We consider the following equation (with unnown λ C): F (λ) = 0, where F (λ) := B 0 m i=1 q 1 j S j λ q 1 j S j B j 1. It has exactly m roots b j satisfying (after appropriate renumbering) a j < b j < a j+1, j = 1,..., m 1, a m < b m <. 11/22

12 Convergence theorem Theorem 1 Let L > 0 be an arbitrary number. Then the spectrum of the operator H ε in [0, L] has the following structure for ε small enough: σ(h ε ) [0, L] = [0, L] \ m ( aj (ε), b j (ε) ), where the endpoints of the intervals ( a j (ε), b j (ε) ) satisfy the relations j=1 lim a j(ε) = a j, ε 0 lim b j (ε) = b j, j = 1,..., m. ε 0 12/22

13 Control of gaps edges Theorem 2 Let L > 0 be an arbitrarily large number and let (α j, β j ), j = 1,..., m be any arbitrary intervals satisfying 0 < α 1, α j < β j < α j+1, j = 1, m 1, α m < β m < L. Suppose that the sets B j, j = 1,..., m, satisfy Then one has m B j = 1 B j j=1 β j α j α j i=1,m i j ( βi α j α i α j ). a j = α j, b j = β j, j = 1,..., m provided q j = B j, j = 1,..., m. α j S j 13/22

14 Setch of the proof Preliminaries We rescale the problem to Y-periodic. Namely, we consider the operator H ε = ε 2 + i Z n j=1 m q j δ S, δ S, where S := S j + i. It is clear that σ( H ε ) = σ(h ε ). We introduce the following forms in L 2 (Y): { h ε,n : dom(h ε,n ) = u L 2 (Y) : u H 1 (B j ), j = 1, m, u H 1 (Y \ m h ε N [u, v] = 1 ε 2 Y u vdx + m 1 q j=1 j S j ( (u) + j (u) j ) ( (v) + j h ε,d : dom(h ε,d ) = { u h ε,n : u Y = } 0, h ε [u, v] = D hε [u, v] N h ε,+ : dom(h ε,+ ) = { u h ε,n : } u is periodic, h ε + [u, v] = hε [u, v] N h ε, : dom(h ε, ) = { u h ε,n : } u is antiperiodic, h ε [u, v] = h ε [u, v] N j=1 (v) j } B j ), ) ds 14/22

15 Setch of the proof Preliminaries We denote by H ε,n, H ε,d, H ε,+, H ε, the operators associated with these forms. The spectra of these operators are purely discrete. { } We denote by λ { } ε,n N, λ { } ε,d N, λ { } ε,+ N, λ ε, N the corresponding sequences of eigenvalues, renumbered in the ascending order and with account of multiplicity. 15/22

16 Step 1: Braceting Using Floquet-Bloch theory and minimax principle we get: σ(h ε ) = N L ε, L ε are compact intervals satisfying [λ ε,+, λ ε, ] L ε [λε,n, λ ε,d ] Our goal it to prove that 0, = 1 lim ε 0 λε,n = lim λ ε,+ = ε 0 b 1, 2 m + 1, m + 2 lim ε 0 λε,d = lim λ ε, a, 1 m = ε 0, m + 1 (1) (2) (1) + (2) = Theorem 1 16/22

17 Step 2: Resolvent convergence of the operators H ε, The forms h ε, increases monotonically as ε decreases. We introduce the limit forms h by { dom(h ) = h [u, v] := lim ε 0 h ε, [u, v] u dom(h ε, ) : sup h ε, [u, u] < ε The forms h are positive and closed (see 3 ). }, We denote by H the operators acting in dom(h ) L 2 (Y) being associated with these forms. 3 B. Simon, J. Funct. Anal. 28 (1978), no. 3, /22

18 Step 2 (continuation) Finally, we define the resolvents of these operators: R (H + I) 1 on dom(h := ) L 2 (Y) 0 on L 2 (Y) dom(h ) L 2 (Y) Then (again see 2 ) f L 2 (Y) : (H ε, + I) 1 f R f L 0 as ε 0. (3) 2 (Y) Moreover, since (H ε 1, + I) 1 (H ε 2, + I) 1 as ε 1 ε 2, and (H ε, + I) 1 and R are compact operators, one can upgrade (3) to norm convergence (see Theorem VIII-3.5 from 4 ): (H ε, + I) 1 R L(L 0 as ε 0. (4) 2 (Y)) 4 T. Kato, Perturbation theory for linear operators, Springer, New-Yor, /22

19 Step 2 (continuation) One has: m dom(h N ) = dom(h + ) = u(x) = u j 1 Bj (x), u j are constants j=0 m H N u = H + S j m u = q j B 0 (u 0 u ) 1 S j B 0 (x) + q j B j (u j u 0 )1 Bj (x) =1 m H D u = H u = dom(h ) = u(x) = u j 1 Bj (x), u j are constants j=1 m H D u = H S j u = q j B j u j1 Bj (x) j=1 We denote the eigenvalues of these operators by j=1 λ N, λ+, = 1, m + 1, λd, λ, = 1, m. 19/22

20 Step 2 (continuation) It follows from (3) that lim ε 0 (λε,n/+ + 1) 1 = or, equivalently, lim ε 0 (λε,d/ + 1) 1 = lim ε 0 λε,n/+ = lim ε 0 λε,d/ = (λ N/+ + 1) 1, 1 m + 1 0, m + 2 (λ D/ + 1) 1, 1 m 0, m + 1 λ N/+, 1 m + 1, m + 2 λ D/, 1 m, m /22

21 Step 3: Analysis of matrices It is easy to see that λ ε,d The eigenvalues λ ε,n = λ ε,+ = λ ε, = q 1 S B 1 = a. are the roots of the equation det(h N λi) = 0, where the matrix H is as follows: m q 1 S j j B 0 1 q 1 1 S 1 B 0 1 q 1 2 S 2 B qm 1 S m B 0 1 j=1 q 1 1 H := S 1 B 1 1 q 1 1 S 1 B q 1 2 S 2 B q 1 2 S 2 B qm 1 S m B m qm 1 S m B m 1 After some algebra we obtain: m ( det(h N λi) = λ q 1 j S j B j 1 λ ) m B 0 j=1 whence λ N/+ 1 = 0, λ N/+ = b 1 as = 2,..., m + 1. i=1 λ q 1 j q 1 j S j S j B j, 1 21/22

22 Than you for your attention! 22/22

ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS

ON THE SPECTRUM OF NARROW NEUMANN WAVEGUIDE WITH PERIODICALLY DISTRIBUTED δ TRAPS PAVEL EXNER 1 AND ANDRII KHRABUSTOVSKYI 2 Abstract. We analyze a family of singular Schrödinger operators describing a