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Nanosystems: Physics, Chemistry, Mathematics Ministry of Education and Science of the Russian Federation Saint Petersburg National Research University of Information Technologies, Mechanics, and Optics NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS 2019, volume 10(6) Наносистемы: физика, химия, математика 2019, том 10, № 6 NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS ADVISORY BOARD MEMBERS Chairman: V.N. Vasiliev (St. Petersburg, Russia), V.M. Buznik (Moscow, Russia); V.M. Ievlev (Voronezh, Russia), P.S. Kop’ev(St. Petersburg, Russia), N.F. Morozov (St. Petersburg, Russia), V.N. Parmon (Novosibirsk, Russia), A.I. Rusanov (St. Petersburg, Russia), EDITORIAL BOARD Editor-in-Chief: I.Yu. Popov (St. Petersburg, Russia) Section Co-Editors: Physics – V.M. Uzdin (St. Petersburg, Russia), Chemistry, material science –V.V. Gusarov (St. Petersburg, Russia), Mathematics – I.Yu. Popov (St. Petersburg, Russia). Editorial Board Members: V.M. Adamyan (Odessa, Ukraine); O.V. Al’myasheva (St. Petersburg, Russia); A.P. Alodjants (Vladimir, Russia); S. Bechta (Stockholm, Sweden); J. Behrndt (Graz, Austria); M.B. Belonenko (Volgograd, Russia); A. Chatterjee (Hyderabad, India); S.A. Chivilikhin (St. Petersburg, Russia); A.V. Chizhov (Dubna, Russia); A.N. Enyashin (Ekaterinburg, Russia), P.P. Fedorov (Moscow, Russia); E.A. Gudilin (Moscow, Russia); V.K. Ivanov (Moscow, Russia), H. Jónsson (Reykjavik, Iceland); A.A. Kiselev (Durham, USA); Yu.S. Kivshar (Canberra, Australia); S.A. Kozlov (St. Petersburg, Russia); P.A. Kurasov (Stockholm, Sweden); A.V. Lukashin (Moscow, Russia); I.V. Melikhov (Moscow, Russia); G.P. Miroshnichenko (St. Petersburg, Russia); I.Ya. Mittova (Voronezh, Russia); V.V. Pankov (Minsk, Belagus); K. Pankrashkin (Orsay, France); A.V. Ragulya (Kiev, Ukraine); V. Rajendran (Tamil Nadu, India); A.A. Rempel (Ekaterinburg, Russia); V.Ya. Rudyak (Novosibirsk, Russia); D Shoikhet (Karmiel, Israel); P Stovicek (Prague, Czech Republic); V.M. Talanov (Novocherkassk, Russia); A.Ya. Vul’ (St. Petersburg, Russia); A.V. Yakimansky (St. Petersburg, Russia), V.A. Zagrebnov (Marseille, France). Editors: I.V. Blinova; A.I. Popov; A.I. Trifanov; E.S. Trifanova (St. Petersburg, Russia), R. Simoneaux (Philadelphia, Pennsylvania, USA). Address: University ITMO, Kronverkskiy pr., 49, St. Petersburg 197101, Russia. Phone: +7(812)312-61-31, Journal site: http://nanojournal.ifmo.ru/, E-mail: [email protected] AIM AND SCOPE The scope of the journal includes all areas of nano-sciences. Papers devoted to basic problems of physics, chemistry, material science and mathematics inspired by nanosystems investigations are welcomed. Both theoretical and experimental works concerning the properties and behavior of nanosystems, problems of its creation and application, mathematical methods of nanosystem studies are considered. The journal publishes scientific reviews (up to 30 journal pages), research papers (up to 15 pages) and letters (up to 5 pages). All manuscripts are peer-reviewed. Authors are informed about the referee opinion and the Editorial decision. CONTENT From Editorial Board 607 MATHEMATICS S. Fassari, F. Rinaldi Exact calculation of the trace of the Birman-Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential 608 T.H. Rasulov, E.B. Dilmurodov Threshold analysis for a family of 2 × 2 operator matrices 616 E.G. Fedorov, A.I. Popov, I.Y. Popov Metric graph version of the FitzHugh-Nagumo model 623 PHYSICS F.D. Kiselev, E.Y. Samsonov, A.V. Gleim Modeling of linear optical controlled-Z quantum gate with dimensional errors of passive components 627 V.G. Shemanin, E.V. Kolpakova, A.B. Atkarskaya, O.V. Mkrtychev SiO2 barrier layer influence on the glass composites with oxide nano films laser ablation destruction 632 CHEMISTRY AND MATERIAL SCIENCE O.V. Alexeeva, S.S. Kozlov, L.L. Larina, O.I. Shevaleevskiy Pt nanoparticle-functionalized RGO counter electrode for efficient dye-sensitized solar cells 637 A.V. Ankudinov On the accuracy of the probe–sample contact stiffness measured by an atomic force microscope 642 A.N. Bugrov, R.Yu. Smyslov, T.V. Khamova, D.A. Kirilenko, I.A. Rodionov Phosphors with different morphology, formed under hydrothermal 3+ conditions on the basis of ZrO2:Eu nanocrystallites 654 E.N. Bykova, I. V. Gofman, E.M. Ivankova, A. L. Nikolaeva, A. V. Yakimansky, O.S. Ivanova, A.E. Baranchikov, V.K. Ivanov Influence of nanoparticles of various types as fillers on resistance to hydrolysis of films of heat-resistant polyimide 666 А.V. Egorov, D.А. Kozlov, Yu.B. Mamaeva, A.K. Petrov, А.V. Garshev, P.V. Evdokimov, Ya.Yu. Filippov, N.K. Оrlov,rlov, V.I. Putlayev, A.V. Chetvertukhin, I.Yu. Mikhailov, S.V. Polyakov, A.A. Fedyanin The influence of wet milling of aluminum and aluminum alloys powder screenings on the characteristics of the aluminum-based pastes 674 I.A. Kodintsev, K.D. Martinson, A.A. Lobinsky, V.I. Popkov SILD synthesis of the efficient and stable electrocatalyst based on CoO-NiO solid solution toward hydrogen production 681 M. Kumari, Y.C. Sharma Effect of doping with ‘se’ on structural, optical, electrical and thermoelectric properties of multilayers of Bi2Te2.7Se0.3 / Sb2Te3 to enhance thermoelectric performance 686 K.D. Martinson, V.A. Ivanov, M.I. Chebanenko, V.V. Panchuk, V.G. Semenov, V.I. Popkov Facile combustion synthesis of TbFeO3 nanocrystals with hexagonal and orthorhombic structure 694 Najmeh Mahdavipour, Mohammad Reza Bozorgmehr, Mohammad Momen-Heravi The effect of carbon nanotube on the structure of H-NS protein DNA complex: molecular dynamics approach 701 Sudha Maurya, Sandhya Pillai A comparative study of the effect of solvents on the optical, structural and morphological properties of ZnO-GO nanocomposites synthesized by sol-gel method 711 A.I. Mikhailov, V.F. Kabanov, M.V. Gavrikov Methodology of analyzing the InSb semiconductor quantum dots parameters 720 N.K. Orlov, D.A. Kozlov, S.A. Tikhonova, A.A. Tikhonov, A.V. Garshev, P.V. Evdokimov, V.I. Putlayev, A.A. Vasyakov, A.V. Chetvertukhin, A.K. Petrov, I.Yu. Mikhailov, S.V. Polyakov, Fedyanin A.A. Experimental evaluation of chemical resistance of steatite products towards aluminum-based melts 725 V.V. Zlobin, A.A. Krasilin, O.V. Almjasheva Effect of heterogeneous inclusions on the formation of TiO2 nanocrystals in hydrothermal conditions 733 Information for authors 740 The Editorial Board of the Journal congratulates Andrei Andreyevich Rempel, a member of our editorial board, on his election to the prestigious role of Academician of the Russian Academy of Sciences. We wish him continued success in his academic career. NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2019, 10 (6), P. 608–615 Exact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential S. Fassari1;2;3, F. Rinaldi2;3;4 1Department of Higher Mathematics, ITMO University, St. Petersburg, Russia 2CERFIM, PO Box 1132, CH-6601 Locarno, Switzerland 3Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi,Via Plinio 44, I-00193 Rome, Italy 4Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Naples, Italy [email protected], [email protected] DOI 10.17586/2220-8054-2019-10-6-608-615 By taking advantage of Wang’s results on the scalar product of four eigenfunctions of the 1D harmonic oscillator, we explicitly calculate the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by a Gaussian potential, showing that it can be written as a ratio of Gamma functions. Keywords: Gaussian potential, Birman–Schwinger operator, trace class operator, harmonic oscillator. Received: 4 November 2019 1. Introduction As is well known, the harmonic oscillator is one of the very few solvable quantum models, that is to say its eigenfunctions and eigenvalues can be expressed analytically. As a consequence, one of the most fundamental chapters in any quantum mechanics textbook, such as [1], is devoted to its detailed description. This remarkable property has led theoretical physicists to study various types of models involving perturbations of the harmonic oscillator over many decades. In our inevitably limited summary of the related literature, we have chosen to restrict ourselves to time independent perturbations of the one-dimensional harmonic oscillator. The numerous contributions spanning many decades can be essentially divided into the following groups: i) polynomial perturbations; ii) nonpolynomial perturbations; iii) point perturbations. The x4-perturbation was the first polynomial perturbation to be extensively investigated also because of its con- d2 nections with Quantum Field Theory. As the Hamiltonian − + x2 + βx4 is one of the key topics in [2, 3], we dx2 refer the interested reader to those textbooks and the related references cited therein (see [4] in particular). We remind the reader that the non-analyticity of its eigenvalues as functions of β was a well-known thorny issue in the early days of quantum mechanics. However, it is possible to exploit the Borel summability method to get around this obstacle (see Theorems XII.19, XII.21 and Example 1 for both theorems in [3]). A modified Borel transform allows the exten- d2 sion of the method to Hamiltonians of the type − + x2 + βx2m; m ≥ 3 (see Example 3 for Theorems XII.20, dx2 XII.21 in [3]). Another noteworthy polynomial perturbation is the one leading to the double well oscillator whose d2 d2 Hamiltonian is given by − + x2 + 2βx2 + β2x4 = − + x2(1 + βx)2. In particular, the analyticity of its dx2 dx2 eigenvalues is dealt with in Example 6 following Theorems XII.16.5 in [3] and the related references cited therein. Another important branch of polynomial perturbations is the one of the type βx2m+1; m ≥ 0, which had its origin d2 in the early investigations on the Hamiltonian of the real cubic oscillator − + x2 + βx3 (see, e.g., [5, 6]) and dx2 was initially studied in [7, 8]. Further interest has been drawn into the cubic anharmonic oscillator by the fact that the 2 d 2 1 3 Hamiltonian − + x + iβ 2 x ; β > 0 is PT-symmetric (see [9, 10]).
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