1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials. (English) Zbl 1270.34198

Golovaty, Yuriy

Integral Equations Oper. Theory 75, No. 3, 341-362 (2013).

For real \({L_\infty(\mathbb{R})}\)-functions \({\Phi}\) and \({\Psi}\) of compact support, the author prove the norm resolvent convergence, as \({\varepsilon}\) and \({\nu}\) tend to 0, of a family \({S_{\varepsilon \nu}}\) of one-dimensional Schrödinger operators on the line of the form \[ S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left( \frac{x}{\nu} \right), \] provided the ratio \({\nu/\varepsilon}\) has a finite or infinite limit. The limit operator \(S _{0}\) depends on the shape of \({\Phi}\) and \({\Psi}\) as well as on the limit of the ratio \({\nu/\varepsilon}\). If the potential \({\alpha\Phi}\) possesses a zero-energy resonance, then \(S _{0}\) describes a nontrivial point interaction at the origin. Otherwise \(S _{0}\) is the direct sum of Dirichlet half-line Schrödinger operators.

Reviewer: Bilender P. Allahverdiev (Isparta)

MSC:

34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B09	Boundary eigenvalue problems for ordinary differential equations
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

1D Schrödinger operator; resonance; short range interaction; point interaction; \(\delta \)-potential; \(\delta ^{\prime}\)-potential; distributional potential; solvable model; norm resolvent convergence

References:

[1]	Aktosun T., Klaus M.: Small-energy asymptotics for the Schrödinger equation on the line. Inverse Probl. 17(4), 619–632 (2001) · Zbl 0989.81129
[2]	Albeverio S, Cacciapuoti C, Finco D: Coupling in the singular limit of thin quantum waveguides. J. Math. Phys. 48(3), 032103 (2007) · Zbl 1137.81330
[3]	Albeverio S., Gesztesy F., Høegh-Krohn R.: The low energy expansion in nonrelativistic scattering theory. Annales de l’I. H. P. sect. A 37(1), 1–28 (1982) · Zbl 0528.35076
[4]	Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H.: Solvable Models in Quantum Mechanics, 2nd revised ed. AMS Chelsea Publ., Providence (2005) · Zbl 1078.81003
[5]	Albeverio S., Gesztesy F., Høegh-Krohn R., Streit L.: Charged particles with short range interactions. Annales de l’I. H. P. sect. A. 38(3), 263–293 (1983) · Zbl 0555.35111
[6]	Albeverio S., Høegh-Krohn R.: Point interactions as limits of short range interactions. J. Oper. Theory 6, 313–339 (1981) · Zbl 0501.35023
[7]	Albeverio S., Kurasov P.: Singular Perturbations of Differential Operators and Schrö-dinger Type Operators. Cambridge University Press, Cambridge (2000) · Zbl 0945.47015
[8]	Bollé D., Gesztesy F., Danneels C.: Threshold scattering in two dimensions. Annales de l I. H. P. section A 48, 175–204 (1988) · Zbl 0696.35040
[9]	Bollé D., Gesztesy F., Klaus M.: Scattering theory for one-dimensional systems with $${\(\backslash\)int dx V(x) = 0}$$ . J. Math. Anal. Appl. 122, 496–518 (1987) · Zbl 0617.35105
[10]	Bollé D., Gesztesy F., Wilk S.F.J.: A complete treatment of low-energy scattering in one dimension. J. Oper. Theory 13, 3–31 (1985) · Zbl 0567.47008
[11]	Brasche J.F., Figari R., Teta A.: Singular Schrödinger operators as limits of point interaction Hamiltonians. Potential Anal. 8(2), 163–178 (1998) · Zbl 0907.35037
[12]	Brasche, J.F., Nizhnik, L.: One-dimensional Schrödinger operators with {\(\delta\)}’-interactions on a set of Lebesgue measure zero. arXiv:1112.2545 [math.FA] · Zbl 1301.47034
[13]	Cacciapuoti C., Exner P.: Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide. J. Phys. A Math. Theor. 40(26), F511–F523 (2007) · Zbl 1136.81442
[14]	Cacciapuoti C., Finco D.: Graph-like models for thin waveguides with Robin boundary conditions. Asymptot. Anal. 70(3–4), 199–230 (2010) · Zbl 1227.35238
[15]	Cacciapuoti, C.: Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains. arXiv:1102.3767v2 [math-ph]
[16]	Christiansen P.L., Arnbak H.C., Zolotaryuk A.V., Ermakov V.N., Gaididei Y.B.: On the existence of resonances in the transmission probability for interactions arising from derivatives of Diracs delta function. J. Phys. A Math. Gen. 36, 7589–7600 (2003) · Zbl 1047.81567
[17]	Deift P., Trubowitz E.: Inverse scattering on the line. Commun. Pure Appl. Math. 32, 121–251 (1979) · Zbl 0395.34019
[18]	Exner P., Neidhardt H., Zagrebnov V.A.: Potential approximations to {\(\delta\)}’: an inverse Klauder phenomenon with norm-resolvent convergence. Commun. Math. Phys. 224(3), 593–612 (2001) · Zbl 1161.81375
[19]	Golovaty Yu., Man’ko S.: Solvable models for the Schrödinger operators with {\(\delta\)}’-like potentials. Ukr. Math. Bull. 6(2), 169–203 (2009)
[20]	Golovaty, Yu., Hryniv, R.: On norm resolvent convergence of Schrödinger operators with {\(\delta\)}’-like potentials. J. Phys. A Math. Theor. 43(15), 155204 (2010) [Corrigendum J. Phys. A: Math. Theor. 44, 049802 (2011)] · Zbl 1187.81104
[21]	Golovaty, Yu., Hryniv, R.: Norm resolvent convergence of singularly scaled Schrödinger operators and {\(\delta\)}’-potentials. Proc. R. Soc. Edinb. A. arXiv:1108.5345 [math.SP] (accepted for publication) · Zbl 1297.34096
[22]	Golovaty Yu.: Schrödinger operators with ({\(\alpha\)}{\(\delta\)}’ + {\(\beta\)}{\(\delta\)})-like potentials: norm resolvent convergence and solvable models. Methods Funct. Anal. Topol. 18(3), 243–255 (2012)
[23]	Goriunov A., Mikhailets V.: Regularization of singular Sturm–Liouville equations. Methods Funct. Anal. Topol. 16(2), 120–130 (2010) · Zbl 1221.47083
[24]	Goriunov A., Mikhailets V.: Resolvent convergence of SturmLiouville operators with singular potentials. Math. Notes 87(2), 287–292 (2010) · Zbl 1205.34120
[25]	Ismagilov R.S., Kostyuchenko A.G.: Spectral asymptotics for the Sturm–Liouville operator with point interaction. Funct. Anal. Appl. 44(4), 253–258 (2010) · Zbl 1271.47039
[26]	Klaus M.: Some applications of the Birman–Schwinger principle. Helvetica Phys. Acta 55, 49–68 (1982)
[27]	Klaus M.: Low-energy behaviour of the scattering matrix for the Schrödinger equation on the line. Inverse Probl. 4(2), 505–512 (1988) · Zbl 0669.34030
[28]	Klaus M., Simon B.: Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two body case. Ann. Phys. 130, 251–281 (1980) · Zbl 0455.35112
[29]	Klaus M., Simon B.: Coupling constant thresholds in nonrelativistic quantum mechanics II. Two cluster thresholds in N-body systems. Commun. Math. Phys. 78, 153–168 (1980) · Zbl 0462.35082
[30]	Kostenko A., Malamud M.: 1-D Schrödinger operators with local point interactions on a discrete set. J. Differ. Equ. 249, 253–304 (2010) · Zbl 1195.47031
[31]	Man’ko S.S.: On {\(\delta\)}’-like potential scattering on star graphs. J. Phys. A Math. Theor 43(44), 445304 (2010) · Zbl 1202.81201
[32]	Mikhailets V., Molyboga V.: One-dimensional Schrödinger operators with singular periodic potentials. Methods Funct. Anal. Topol. 14(2), 184–200 (2008) · Zbl 1164.47047
[33]	Mikhailets V., Molyboga V.: Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials. Methods Funct. Anal. Topol. 15(1), 31–40 (2009) · Zbl 1199.34453
[34]	Nizhnik L.P.: A one-dimensional Schrödinger operator with point interactions on Sobolev spaces. Funct. Anal. Appl. 40(2), 143–147 (2006) · Zbl 1104.34341
[35]	Šeba P.: Schrödinger particle on a half line. Lett. Math. Phys. 10(1), 21–27 (1985) · Zbl 0591.47039
[36]	Šeba P.: Some remarks on the {\(\delta\)}’-interaction in one dimension. Rep. Math. Phys. 24(1), 111–120 (1986) · Zbl 0638.70016
[37]	Savchuk A.M., Shkalikov A.A.: Sturm–Liouville operators with singular potentials. Math. Notes 66(6), 741–753 (1999) · Zbl 0968.34072
[38]	Savchuk A.M., Shkalikov A.A.: Sturm–Liouville operators with distribution potentials. Tr. Mosk. Mat. O. va 64,–159212 (2003) · Zbl 1066.34085
[39]	Zolotaryuk A.V.: Two-parametric resonant tunneling across the {\(\delta\)}’(x) potential. Adv. Sci. Lett. 1, 187–191 (2008)
[40]	Zolotaryuk A.V.: Point interactions of the dipole type defined through a three-parametric power regularization. J. Phys. A Math. Theor. 43, 105302 (2010) · Zbl 1205.81093
[41]	Zolotaryuk A.V.: Boundary conditions for the states with resonant tunnelling across the {\(\delta\)}’-potential. Phys. Lett. A 374(15–16), 1636–1641 (2010) · Zbl 1236.81094